a practical guide to forecasting financial market...

JWBK021-FM JWBK021-Poon March 22, 2005 4:30 Char Count= 0

A Practical Guide to ForecastingFinancial Market Volatility

Ser-Huang Poon

iii


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Library of Congress Cataloging-in-Publication Data

Poon, Ser-Huang.A practical guide for forecasting financial market volatility / Ser Huang

Poon.p. cm. — (The Wiley finance series)

Includes bibliographical references and index.ISBN-13 978-0-470-85613-0 (cloth : alk. paper)ISBN-10 0-470-85613-0 (cloth : alk. paper)1. Options (Finance)—Mathematical models. 2. Securities—Prices—Mathematical models. 3. Stock price forecasting—Mathematical models. I. Title.

II. Series.HG6024.A3P66 2005332.64′01′5195—dc22 2005005768

British Library Cataloguing in Publication Data

A catalogue record for this book is available from the British Library

ISBN-13 978-0-470-85613-0 (HB)ISBN-10 0-470-85613-0 (HB)

Typeset in 11/13pt Times by TechBooks, New Delhi, IndiaPrinted and bound in Great Britain by TJ International Ltd, Padstow, CornwallThis book is printed on acid-free paper responsibly manufactured from sustainable forestryin which at least two trees are planted for each one used for paper production.

iv


Contents

Foreword by Clive Granger xiii

Preface xv

1 Volatility Definition and Estimation 11.1 What is volatility? 11.2 Financial market stylized facts 31.3 Volatility estimation 10

1.3.1 Using squared return as a proxy fordaily volatility 11

1.3.2 Using the high–low measure to proxy volatility 121.3.3 Realized volatility, quadratic variation

and jumps 141.3.4 Scaling and actual volatility 16

1.4 The treatment of large numbers 17

2 Volatility Forecast Evaluation 212.1 The form of Xt 212.2 Error statistics and the form of εt 232.3 Comparing forecast errors of different models 24

2.3.1 Diebold and Mariano’s asymptotic test 262.3.2 Diebold and Mariano’s sign test 272.3.3 Diebold and Mariano’s Wilcoxon sign-rank test 272.3.4 Serially correlated loss differentials 28

2.4 Regression-based forecast efficiency andorthogonality test 28

2.5 Other issues in forecast evaluation 30

vii


viii Contents

3 Historical Volatility Models 313.1 Modelling issues 313.2 Types of historical volatility models 32

3.2.1 Single-state historical volatility models 323.2.2 Regime switching and transition exponential

smoothing 343.3 Forecasting performance 35

4 Arch 374.1 Engle (1982) 374.2 Generalized ARCH 384.3 Integrated GARCH 394.4 Exponential GARCH 414.5 Other forms of nonlinearity 414.6 Forecasting performance 43

5 Linear and Nonlinear Long Memory Models 455.1 What is long memory in volatility? 455.2 Evidence and impact of volatility long memory 465.3 Fractionally integrated model 50

5.3.1 FIGARCH 515.3.2 FIEGARCH 525.3.3 The positive drift in fractional integrated series 525.3.4 Forecasting performance 53

5.4 Competing models for volatility long memory 545.4.1 Breaks 545.4.2 Components model 555.4.3 Regime-switching model 575.4.4 Forecasting performance 58

6 Stochastic Volatility 596.1 The volatility innovation 596.2 The MCMC approach 60

6.2.1 The volatility vector H 616.2.2 The parameter w 62

6.3 Forecasting performance 63

7 Multivariate Volatility Models 657.1 Asymmetric dynamic covariance model 65


Contents ix

7.2 A bivariate example 677.3 Applications 68

8 Black–Scholes 718.1 The Black–Scholes formula 71

8.1.1 The Black–Scholes assumptions 728.1.2 Black–Scholes implied volatility 738.1.3 Black–Scholes implied volatility smile 748.1.4 Explanations for the ‘smile’ 75

8.2 Black–Scholes and no-arbitrage pricing 778.2.1 The stock price dynamics 778.2.2 The Black–Scholes partial differential equation 778.2.3 Solving the partial differential equation 79

8.3 Binomial method 808.3.1 Matching volatility with u and d 838.3.2 A two-step binomial tree and American-style

options 858.4 Testing option pricing model in practice 868.5 Dividend and early exercise premium 88

8.5.1 Known and finite dividends 888.5.2 Dividend yield method 888.5.3 Barone-Adesi and Whaley quadratic

approximation 898.6 Measurement errors and bias 90

8.6.1 Investor risk preference 918.7 Appendix: Implementing Barone-Adesi and Whaley’s

efficient algorithm 92

9 Option Pricing with Stochastic Volatility 979.1 The Heston stochastic volatility option pricing model 989.2 Heston price and Black–Scholes implied 999.3 Model assessment 102

9.3.1 Zero correlation 1039.3.2 Nonzero correlation 103

9.4 Volatility forecast using the Heston model 1059.5 Appendix: The market price of volatility risk 107

9.5.1 Ito’s lemma for two stochastic variables 1079.5.2 The case of stochastic volatility 1079.5.3 Constructing the risk-free strategy 108


x Contents

9.5.4 Correlated processes 1109.5.5 The market price of risk 111

10 Option Forecasting Power 11510.1 Using option implied standard deviation to forecast

volatility 11510.2 At-the-money or weighted implied? 11610.3 Implied biasedness 11710.4 Volatility risk premium 119

11 Volatility Forecasting Records 12111.1 Which volatility forecasting model? 12111.2 Getting the right conditional variance and forecast

with the ‘wrong’ models 12311.3 Predictability across different assets 124

11.3.1 Individual stocks 12411.3.2 Stock market index 12511.3.3 Exchange rate 12611.3.4 Other assets 127

12 Volatility Models in Risk Management 12912.1 Basel Committee and Basel Accords I & II 12912.2 VaR and backtest 131

12.2.1 VaR 13112.2.2 Backtest 13212.2.3 The three-zone approach to backtest

evaluation 13312.3 Extreme value theory and VaR estimation 135

12.3.1 The model 13612.3.2 10-day VaR 13712.3.3 Multivariate analysis 138

12.4 Evaluation of VaR models 139

13 VIX and Recent Changes in VIX 14313.1 New definition for VIX 14313.2 What is the VXO? 14413.3 Reason for the change 146

14 Where Next? 147


Contents xi

Appendix 149

References 201

Index 215


Foreword

If one invests in a financial asset today the return received at some pre-specified point in the future should be considered as a random variable.Such a variable can only be fully characterized by a distribution func-tion or, more easily, by a density function. The main, single and mostimportant feature of the density is the expected or mean value, repre-senting the location of the density. Around the mean is the uncertainty orthe volatility. If the realized returns are plotted against time, the jaggedoscillating appearance illustrates the volatility. This movement containsboth welcome elements, when surprisingly large returns occur, and alsocertainly unwelcome ones, the returns far below the mean. The well-known fact that a poor return can arise from an investment illustratesthe fact that investing can be risky and is why volatility is sometimesequated with risk.

Volatility is itself a stock variable, having to be measured over a periodof time, rather than a flow variable, measurable at any instant of time.Similarly, a stock price is a flow variable but a return is a stock variable.Observed volatility has to be observed over stated periods of time, suchas hourly, daily, or weekly, say.

Having observed a time series of volatilities it is obviously interestingto ask about the properties of the series: is it forecastable from its ownpast, do other series improve these forecasts, can the series be mod-eled conveniently and are there useful multivariate generalizations ofthe results? Financial econometricians have been very inventive and in-dustrious considering such questions and there is now a substantial andoften sophisticated literature in this area.

The present book by Professor Ser-Huang Poon surveys this literaturecarefully and provides a very useful summary of the results available.

xiii


xiv Foreword

By so doing, she allows any interested worker to quickly catch up withthe field and also to discover the areas that are still available for furtherexploration.

Clive W.J. GrangerDecember 2004


Preface

Volatility forecasting is crucial for option pricing, risk management andportfolio management. Nowadays, volatility has become the subject oftrading. There are now exchange-traded contracts written on volatility.Financial market volatility also has a wider impact on financial regula-tion, monetary policy and macroeconomy. This book is about financialmarket volatility forecasting. The aim is to put in one place models, toolsand findings from a large volume of published and working papers frommany experts. The material presented in this book is extended from tworeview papers (‘Forecasting Financial Market Volatility: A Review’ inthe Journal of Economic Literature, 2003, 41, 2, pp. 478–539, and ‘Prac-tical Issues in Forecasting Volatility’ in the Financial Analysts Journal,2005, 61, 1, pp. 45–56) jointly published with Clive Granger.

Since the main focus of this book is on volatility forecasting perfor-mance, only volatility models that have been tested for their forecastingperformance are selected for further analysis and discussion. Hence, thisbook is oriented towards practical implementations. Volatility modelsare not pure theoretical constructs. The practical importance of volatil-ity modelling and forecasting in many finance applications means thatthe success or failure of volatility models will depend on the charac-teristics of empirical data that they try to capture and predict. Giventhe prominent role of option price as a source of volatility forecast, Ihave also devoted much effort and the space of two chapters to coverBlack–Scholes and stochastic volatility option pricing models.

This book is intended for first- and second-year finance PhD studentsand practitioners who want to implement volatility forecasting modelsbut struggle to comprehend the huge volume of volatility research. Read-ers who are interested in more technical aspects of volatility modelling

xv


xvi Preface

could refer to, for example, Gourieroux (1997) on ARCH models,Shephard (2003) on stochastic volatility and Fouque, Papanicolaou andSircar (2000) on stochastic volatility option pricing. Books that coverspecific aspects or variants of volatility models include Franses and vanDijk (2000) on nonlinear models, and Beran (1994) and Robinson (2003)on long memory models. Specialist books that cover financial time se-ries modelling in a more general context include Alexander (2001),Tsay (2002) and Taylor (2005). There are also a number of edited seriesthat contain articles on volatility modelling and forecasting, e.g. Rossi(1996), Knight and Satchell (2002) and Jarrow (1998).

I am very grateful to Clive for his teaching and guidance in the lastfew years. Without his encouragement and support, our volatility surveyworks and this book would not have got started. I would like to thank allmy co-authors on volatility research, in particular Bevan Blair, NamwonHyung, Eric Jondeau, Martin Martens, Michael Rockinger, Jon Tawn,Stephen Taylor and Konstantinos Vonatsos. Much of the writing herereflects experience gained from joint work with them.

JWBK021-01 JWBK021-Poon March 15, 2005 13:28 Char Count= 0

1Volatility Definition and

Estimation

1.1 WHAT IS VOLATILITY?

It is useful to start with an explanation of what volatility is, at leastfor the purpose of clarifying the scope of this book. Volatility refersto the spread of all likely outcomes of an uncertain variable. Typically,in financial markets, we are often concerned with the spread of assetreturns. Statistically, volatility is often measured as the sample standarddeviation

σ =√√√√ 1

T − 1

T∑t=1

(rt − µ)2, (1.1)

where rt is the return on day t , and µ is the average return over the T -dayperiod.

Sometimes, variance, σ 2, is used also as a volatility measure. Sincevariance is simply the square of standard deviation, it makes no differ-ence whichever measure we use when we compare the volatility of twoassets. However, variance is much less stable and less desirable thanstandard deviation as an object for computer estimation and volatilityforecast evaluation. Moreover standard deviation has the same unit ofmeasure as the mean, i.e. if the mean is in dollar, then standard devi-ation is also expressed in dollar whereas variance will be expressed indollar square. For this reason, standard deviation is more convenient andintuitive when we think about volatility.

Volatility is related to, but not exactly the same as, risk. Risk is associ-ated with undesirable outcome, whereas volatility as a measure strictlyfor uncertainty could be due to a positive outcome. This important dif-ference is often overlooked. Take the Sharpe ratio for example. TheSharpe ratio is used for measuring the performance of an investment bycomparing the mean return in relation to its ‘risk’ proxy by its volatility.

1


2 Forecasting Financial Market Volatility

The Sharpe ratio is defined as

Sharpe ratio =

(Averagereturn, µ

)−

(Risk-free interestrate, e.g. T-bill rate

)Standard deviation of returns, σ

.

The notion is that a larger Sharpe ratio is preferred to a smaller one. Anunusually large positive return, which is a desirable outcome, could leadto a reduction in the Sharpe ratio because it will have a greater impacton the standard deviation, σ , in the denominator than the average return,µ, in the numerator.

More importantly, the reason that volatility is not a good or perfectmeasure for risk is because volatility (or standard deviation) is onlya measure for the spread of a distribution and has no information onits shape. The only exception is the case of a normal distribution or alognormal distribution where the mean, µ, and the standard deviation,σ , are sufficient statistics for the entire distribution, i.e. with µ and σ

alone, one is able to reproduce the empirical distribution.This book is about volatility only. Although volatility is not the sole

determinant of asset return distribution, it is a key input to many im-portant finance applications such as investment, portfolio construction,option pricing, hedging, and risk management. When Clive Granger andI completed our survey paper on volatility forecasting research, therewere 93 studies on our list plus several hundred non-forecasting paperswritten on volatility modelling. At the time of writing this book, thenumber of volatility studies is still rising and there are now about 120volatility forecasting papers on the list. Financial market volatility is a‘live’ subject and has many facets driven by political events, macroecon-omy and investors’ behaviour. This book will elaborate some of thesecomplexities that kept the whole industry of volatility modelling andforecasting going in the last three decades. A new trend now emergingis on the trading and hedging of volatility. The Chicago Board of Ex-change (CBOE) for example has started futures trading on a volatilityindex. Options on such futures contracts are likely to follow. Volatilityswap contracts have been traded on the over-the-counter market wellbefore the CBOE’s developments. Previously volatility was an input toa model for pricing an asset or option written on the asset. It is now theprincipal subject of the model and valuation. One can only predict thatvolatility research will intensify for at least the next decade.


Volatility Definition and Estimation 3

1.2 FINANCIAL MARKET STYLIZED FACTS

To give a brief appreciation of the amount of variation across differentfinancial assets, Figure 1.1 plots the returns distributions of a normally

(a) Normal N(0,1)

−4 −3 −2 −1 0 1 2 3 4

−4 −3 −2 −1 0 1 2 3 4

5

(b) Daily returns on S&P100Jan 1965 – Jul 2003

−5 −4 −3 −2 −1 0 1 2 3 4 5

(c) £ vs. yen daily exchange rate returnsSep 1971 – Jul 2003

(d) Daily returns on Legal & General shareJan 1969 – Jul 2003

−10 −5 0 5 10

(e) Daily returns on UK Small Cap IndexJan 1986 – Jul 2003

−4 −3 −2 −1 0 1 2 3 4

(f) Daily returns on silverAug 1971 – Jul 2003

−10 −5 0 5 10

Figure 1.1 Distribution of daily financial market returns. (Note: the dotted line isthe distribution of a normal random variable simulated using the mean and standarddeviation of the financial asset returns)



distributed random variable, and the respective daily returns on the USStandard and Poor market index (S&P100),1 the yen–sterling exchangerate, the share of Legal & General (a major insurance company in theUK), the UK Index for Small Capitalisation Stocks (i.e. small compa-nies), and silver traded at the commodity exchange. The normal distri-bution simulated using the mean and standard deviation of the financialasset returns is drawn on the same graph to facilitate comparison.

From the small selection of financial asset returns presented in Fig-ure 1.1, we notice several well-known features. Although the asset re-turns have different degrees of variation, most of them have long ‘tails’ ascompared with the normally distributed random variable. Typically, theasset distribution and the normal distribution cross at least three times,leaving the financial asset returns with a longer left tail and a higher peakin the middle. The implications are that, for a large part of the time, finan-cial asset returns fluctuate in a range smaller than a normal distribution.But there are some occasions where financial asset returns swing in amuch wider scale than that permitted by a normal distribution. This phe-nomenon is most acute in the case of UK Small Cap and silver. Table 1.1provides some summary statistics for these financial time series.

The normally distributed variable has a skewness equal to zero anda kurtosis of 3. The annualized standard deviation is simply

√252σ ,

assuming that there are 252 trading days in a year. The financial assetreturns are not adjusted for dividend. This omission is not likely to haveany impact on the summary statistics because the amount of dividendsdistributed over the year is very small compared to the daily fluctuationsof asset prices. From Table 1.1, the Small Cap Index is the most nega-tively skewed, meaning that it has a longer left tail (extreme losses) thanright tail (extreme gains). Kurtosis is a measure for tail thickness andit is astronomical for S&P100, Small Cap Index and silver. However,these skewness and kurtosis statistics are very sensitive to outliers. Theskewness statistic is much closer to zero, and the amount of kurtosisdropped by 60% to 80%, when the October 1987 crash and a smallnumber of outliers are excluded.

Another characteristic of financial market volatility is the time-varying nature of returns fluctuations, the discovery of which led toRob Engle’s Nobel Prize for his achievement in modelling it. Figure 1.2plots the time series history of returns of the same set of assets presented

1 The data for S&P100 prior to 1986 comes from S&P500. Adjustments were made when the two series weregrafted together.


Tabl

e1.

1Su

mm

ary

stat

istic

sfo

ra

sele

ctio

nof

finan

cial

seri

es

N(0

,1)

S&P1

00Y

en/£

rate

Leg

al&

Gen

eral

UK

Smal

lCap

Silv

er

Star

tdat

eJa

n65

Sep

71Ja

n69

Jan

86A

ug71

Num

ber

ofob

serv

atio

ns80

0096

7573

3876

8444

3277

71D

aily

aver

agea

00.

024

−0.0

210.

043

0.02

20.

014

Dai

lySt

anda

rdD

evia

tion

10.

985

0.71

52.

061

0.64

82.

347

Ann

ualiz

edav

erag

e0

6.06

7−5

.188

10.7

275.

461

3.54

3A

nnua

lized

Stan

dard

Dev

iatio

n15

.875

15.6

3211

.356

32.7

1510

.286

37.2

55Sk

ewne

ss0

−1.3

37−0

.523

0.02

6−3

.099

0.38

7K

urto

sis

337

.140

7.66

46.

386

42.5

6145

.503

Num

ber

ofou

tlier

sre

mov

ed1

59

Skew

ness

b−0

.055

−0.9

17−0

.088

Kur

tosi

sb7.

989

13.9

7215

.369

aR

etur

nsno

tadj

uste

dfo

rdi

vide

nds.

bT

hese

two

stat

istic

alm

easu

res

are

com

pute

daf

ter

the

rem

oval

ofou

tlier

s.A

llse

ries

have

anen

dda

teof

22Ju

ly,2

003.

5


(a)

Nor

mal

ly d

istr

ibut

ed r

ando

m v

aria

ble

N(0

,1)

−10−8−6−4−202468

(b)

Dai

ly r

etur

ns o

n S

&P

100

−10−8−6−4−20246810

1965

0104

1969

0207

1973

0208

1977

0126

1981

0122

1985

0103

1988

1221

1992

1207

1996

1119

2000

1107

(c)

Yen

to £

exc

hang

e ra

te r

etur

ns

−8−6−4−20246

1971

0831

1974

0903

1977

0221

1979

0615

1981

0715

1983

0921

1985

1107

1987

1222

1990

0316

1992

0505

1994

0614

1996

0531

1998

0511

2000

0418

2002

0401

(d)

Dai

ly r

etur

ns o

n L

egal

& G

ener

al's

sha

re

−15

−10−505101520

1969

0106

1971

0614

1973

1017

1976

0406

1978

1011

1981

0302

1983

0615

1985

1015

1988

0202

1990

0424

1992

0610

1994

0718

1996

0905

1998

0916

2000

1004

2002

1015

(e)

Dai

ly r

etur

ns U

K S

mal

l Cap

Ind

ex

−12−8−4048

1986

0102

1987

0310

1988

0516

1989

0721

1990

0927

1991

1204

1993

0211

1994

0420

1995

0628

1996

0903

1997

1110

1999

0120

2000

0328

2001

0607

2002

0814

(f)

Dai

ly r

etur

ns o

n si

lver

−40

−30

−20

−10010203040

1971

0818

1973

0914

1975

0916

1977

0930

1979

1030

1981

1110

1983

1213

1985

1223

1988

0105

1990

0105

1992

0114

1994

0117

1996

0117

1998

0210

2000

0317

3740

0

Fig

ure

1.2

Tim

ese

ries

ofda

ilyre

turn

son

asi

mul

ated

rand

omva

riab

lean

da

colle

ctio

nof

finan

cial

asse

ts

6



in Figure 1.1. The amplitude of the returns fluctuations represents theamount of variation with respect to a short instance in time. It is clearfrom Figures 1.2(b) to (f) that fluctuations of financial asset returns are‘lumpier’ in contrast to the even variations of the normally distributedvariable in Figure 1.2(a). In the finance literature, this ‘lumpiness’ iscalled volatility clustering. With volatility clustering, a turbulent trad-ing day tends to be followed by another turbulent day, while a tranquilperiod tends to be followed by another tranquil period. Rob Engle (1982)is the first to use the ARCH (autoregressive conditional heteroscedastic-ity) model to capture this type of volatility persistence; ‘autoregressive’because high/low volatility tends to persist, ‘conditional’ means time-varying or with respect to a point in time, and ‘heteroscedasticity’ is atechnical jargon for non-constant volatility.2

There are several salient features about financial market returns andvolatility that are now well documented. These include fat tails andvolatility clustering that we mentioned above. Other characteristics doc-umented in the literature include:

(i) Asset returns, rt , are not autocorrelated except possibly at lag onedue to nonsynchronous or thin trading. The lack of autocorrelationpattern in returns corresponds to the notion of weak form marketefficiency in the sense that returns are not predictable.

(ii) The autocorrelation function of |rt | and r2t decays slowly and

corr (|rt | , |rt−1|) > corr(r2

t , r2t−1

). The decay rate of the auto-

correlation function is much slower than the exponential rate ofa stationary AR or ARMA model. The autocorrelations remainpositive for very long lags. This is known as the long memoryeffect of volatility which will be discussed in greater detail inChapter 5. In the table below, we give a brief taste of the finding:

∑ρ(|r |) ∑

ρ(r 2)∑

ρ(ln|r |) ∑ρ(|T r |)

S&P100 35.687 3.912 27.466 41.930Yen/£ 4.111 1.108 0.966 5.718L&G 25.898 14.767 29.907 28.711Small Cap 25.381 3.712 35.152 38.631Silver 45.504 8.275 88.706 60.545

2 It is worth noting that the ARCH effect appears in many time series other than financial time series. In factEngle’s (1982) seminal work is illustrated with the UK inflation rate.



(iii) The numbers reported above are the sum of autocorrelations for thefirst 1000 lags. The last column, ρ(|T r |), is the autocorrelation ofabsolute returns after the most extreme 1% tail observations weretruncated. Let r0.01 and r0.99 be the 98% confidence interval of theempirical distribution,

T r = Min [r, r0.99] , or Max [r, r0.01] . (1.2)

The effect of such an outlier truncation is discussed in Huber (1981).The results reported in the table show that suppressing the largenumbers markedly increases the long memory effect.

(iv) Autocorrelation of powers of an absolute return are highest at powerone: corr (|rt | , |rt−1|) > corr

(rd

t , rdt−1

), d �= 1. Granger and Ding

(1995) call this property the Taylor effect, following Taylor (1986).We showed above that other means of suppressing large numberscould make the memory last longer. The absolute returns |rt | andsquared returns r2

t are proxies of daily volatility. By analysing themore accurate volatility estimator, we note that the strongest auto-correlation pattern is observed among realized volatility. Figure 1.3demonstrates this convincingly.

(v) Volatility asymmetry: it has been observed that volatility increases ifthe previous day returns are negative. This is known as the leverageeffect (Black, 1976; Christie, 1982) because the fall in stock pricecauses leverage and financial risk of the firm to increase. The phe-nomenon of volatility asymmetry is most marked during large falls.The leverage effect has not been tested between contemporaneousreturns and volatility possibly due to the fact that it is the previ-ous day residuals returns (and its sign dummy) that are includedin the conditional volatility specification in many models. With theavailability of realized volatility, we find a similar, albeit slightlyweaker, relationship in volatility and the sign of contemporaneousreturns.

(vi) The returns and volatility of different assets (e.g. different companyshares) and different markets (e.g. stock vs. bond markets in oneor more regions) tend to move together. More recent research findscorrelation among volatility is stronger than that among returns andboth tend to increase during bear markets and financial crises.

The art of volatility modelling is to exploit the time series proper-ties and stylized facts of financial market volatility. Some financial timeseries have their unique characteristics. The Korean stock market, for



(a) Autocorrelation of daily returns on S&P100

−0.3

−0.1

0.1

0.3

0.5

0.7

0.9

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36

(b) Autocorrelation of daily squared returns on S&P100

−0.3

−0.1

0.1

0.3

0.5

0.7

0.9

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36

(c) Autocorrelation of daily absolute returns on S&P100

−0.3

−0.1

0.1

0.3

0.5

0.7

0.9

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36

(d) Autocorrelation of daily realized volatility of S&P100

−0.3

−0.1

0.1

0.3

0.5

0.7

0.9

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36

Figure 1.3 Aurocorrelation of daily returns and proxies of daily volatility of S&P100.(Note: dotted lines represent two standard errors)

example, clearly went through a regime shift with a much higher volatil-ity level after 1998. Many of the Asian markets have behaved differentlysince the Asian crisis in 1997. The difficulty and sophistication of volatil-ity modelling lie in the controlling of these special and unique featuresof each individual financial time series.



1.3 VOLATILITY ESTIMATION

Consider a time series of returns rt , t = 1, · · · , T , the standard de-viation, σ , in (1.1) is the unconditional volatility over the T period.Since volatility does not remain constant through time, the conditionalvolatility, σt,τ is a more relevant information for asset pricing and riskmanagement at time t . Volatility estimation procedure varies a great dealdepending on how much information we have at each sub-interval t , andthe length of τ , the volatility reference period. Many financial time seriesare available at the daily interval, while τ could vary from 1 to 10 days(for risk management), months (for option pricing) and years (for in-vestment analysis). Recently, intraday transaction data has become morewidely available providing a channel for more accurate volatility esti-mation and forecast. This is the area where much research effort hasbeen concentrated in the last two years.

When monthly volatility is required and daily data is available,volatility can simply be calculated using Equation (1.1). Many macro-economic series are available only at the monthly interval, so the currentpractice is to use absolute monthly value to proxy for macro volatility.The same applies to financial time series when a daily volatility estimateis required and only daily data is available. The use of absolute valueto proxy for volatility is the equivalent of forcing T = 1 and µ = 0 inEquation (1.1). Figlewski (1997) noted that the statistical properties ofthe sample mean make it a very inaccurate estimate of the true mean es-pecially for small samples. Taking deviations around zero instead of thesample mean as in Equation (1.1) typically increases volatility forecastaccuracy.

The use of daily return to proxy daily volatility will produce a verynoisy volatility estimator. Section 1.3.1 explains this in a greater detail.Engle (1982) was the first to propose the use of an ARCH (autoregres-sive conditional heteroscedasticity) model below to produce conditionalvolatility for inflation rate rt ;

rt = µ + εt , εt ∼ N(

0,√

ht

).

εt = zt

√ht ,

ht = ω + α1ε2t−1 + α2ε

2t−2 + · · · . (1.3)

The ARCH model is estimated by maximizing the likelihood of {εt}.This approach of estimating conditional volatility is less noisy than theabsolute return approach but it relies on the assumption that (1.3) is the



true return-generating process, εt is Gaussian and the time series is longenough for such an estimation.

While Equation (1.1) is an unbiased estimator for σ 2, the square rootof σ 2 is a biased estimator for σ due to Jensen inequality.3 Ding, Grangerand Engle (1993) suggest measuring volatility directly from absolute re-turns. Davidian and Carroll (1987) show absolute returns volatility spec-ification is more robust against asymmetry and nonnormality. There issome empirical evidence that deviations or absolute returns based mod-els produce better volatility forecasts than models that are based onsquared returns (Taylor, 1986; Ederington and Guan, 2000a; McKenzie,1999). However, the majority of time series volatility models, especiallythe ARCH class models, are squared returns models. There are methodsfor estimating volatility that are designed to exploit or reduce the influ-ence of extremes.4 Again these methods would require the assumptionof a Gaussian variable or a particular distribution function for returns.

1.3.1 Using squared return as a proxy for daily volatility

Volatility is a latent variable. Before high-frequency data became widelyavailable, many researchers have resorted to using daily squared returns,calculated from market daily closing prices, to proxy daily volatility.Lopez (2001) shows that ε2

t is an unbiased but extremely impreciseestimator of σ 2

t due to its asymmetric distribution. Let

Yt = µ + εt , εt = σt zt , (1.4)

and zt ∼ N (0, 1). Then

E[ε2

t

∣∣�t−1] = σ 2

t E[

z2t

∣∣�t−1] = σ 2

t

since z2t ∼ χ2

(1). However, since the median of a χ2(1) distribution is 0.455,

ε2t is less than 1

2σ2t more than 50% of the time. In fact

Pr

(ε2

t ∈[

1

2σ 2

t ,3

2σ 2

t

])= Pr

(z2

t ∈[

1

2,

3

2

])= 0.2588,

which means that ε2t is 50% greater or smaller than σ 2

t nearly 75% ofthe time!

3 If rt ∼ N(0, σ 2

t

), then E (|rt |) = σt

√2/π . Hence, σ t = |rt |/

√2/π if rt has a conditional normal distri-

bution.4 For example, the maximum likelihood method proposed by Ball and Torous (1984), the high–low method

proposed by Parkinson (1980) and Garman and Klass (1980).



Under the null hypothesis that returns in (1.4) are generated by aGARCH(1,1) process, Andersen and Bollerslev (1998) show that thepopulation R2 for the regression

ε2t = α + βσ 2

t + υt

is equal to κ−1 where κ is the kurtosis of the standardized residuals and κ

is finite. For conditional Gaussian error, the R2 from a correctly specifiedGARCH(1,1) model cannot be greater than 1/3. For thick tail distribu-tion, the upper bound for R2 is lower than 1/3. Christodoulakis andSatchell (1998) extend the results to include compound normals and theGram–Charlier class of distributions confirming that the mis-estimationof forecast performance is likely to be worsened by nonnormality knownto be widespread in financial data.

Hence, the use of ε2t as a volatility proxy will lead to low R2 and under-

mine the inference on forecast accuracy. Blair, Poon and Taylor (2001)report an increase of R2 by three to four folds for the 1-day-ahead fore-cast when intraday 5-minutes squared returns instead of daily squaredreturns are used to proxy the actual volatility. The R2 of the regressionof |εt | on σ intra

t is 28.5%. Extra caution is needed when interpreting em-pirical findings in studies that adopt such a noisy volatility estimator.Figure 1.4 shows the time series of these two volatility estimates overthe 7-year period from January 1993 to December 1999. Although theoverall trends look similar, the two volatility estimates differ in manydetails.

1.3.2 Using the high–low measure to proxy volatility

The high–low, also known as the range-based or extreme-value, methodof estimating volatility is very convenient because daily high, low, open-ing and closing prices are reported by major newspapers, and the cal-culation is easy to program using a hand-held calculator. The high–lowvolatility estimator was studied by Parkinson (1980), Garman and Klass(1980), Beckers (1993), Rogers and Satchell (1991), Wiggins (1992),Rogers, Satchell and Yoon (1994) and Alizadeh, Brandt and Diebold(2002). It is based on the assumption that return is normally distributedwith conditional volatility σt . Let Ht and Lt denote, respectively, thehighest and the lowest prices on day t . Applying the Parkinson (1980)H -L measure to a price process that follows a geometric Brownian



(a) Conditional variance proxied by daily squared returns

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

04/01/1993 04/01/1994 04/01/1995 04/01/1996 04/01/1997 04/01/1998 04/01/1999

(b) Conditional variance derived as the sum of intraday squared returns

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

04/01/1993 04/01/1994 04/01/1995 04/01/1996 04/01/1997 04/01/1998 04/01/1999

Figure 1.4 S&P100 daily volatility for the period from January 1993 to December1999

motion results in the following volatility estimator (Bollen and Inder,2002):

σ 2t = (ln Ht − ln Lt )2

4 ln 2The Garman and Klass (1980) estimator is an extension of Parkinson(1980) where information about opening, pt−1, and closing, pt , pricesare incorporated as follows:

σ 2t = 0.5

(ln

Ht

Lt

)2

− 0.39

(ln

pt

pt−1

)2

.

We have already shown that financial market returns are not likely tobe normally distributed and have a long tail distribution. As the H -Lvolatility estimator is very sensitive to outliers, it will be useful to ap-ply the trimming procedures in Section 1.4. Provided that there are nodestabilizing large values, the H -L volatility estimator is very efficient



and, unlike the realized volatility estimator introduced in the next sec-tion, it is least affected by market microstructure effect.

1.3.3 Realized volatility, quadratic variation and jumps

More recently and with the increased availability of tick data, the termrealized volatility is now used to refer to volatility estimates calculatedusing intraday squared returns at short intervals such as 5 or 15 minutes.5

For a series that has zero mean and no jumps, the realized volatility con-verges to the continuous time volatility. To understand this, we assumefor the ease of exposition that the instantaneous returns are generated bythe continuous time martingale,

dpt = σt dWt , (1.5)

where dWt denotes a standard Wiener process. From (1.5) the con-ditional variance for the one-period returns, rt+1 ≡ pt+1 − pt , is∫ t+1

t σ 2s ds which is known as the integrated volatility over the period t

to t + 1. Note that while asset price pt can be observed at time t , thevolatility σt is an unobservable latent variable that scales the stochasticprocess dWt continuously through time.

Let m be the sampling frequency such that there are m continuouslycompounded returns in one unit of time and

rm,t ≡ pt − pt− 1/m (1.6)

and realized volatility

RVt+1 =∑

j=1,···,mr2

m,t+ j/m .

If the discretely sampled returns are serially uncorrelated and the samplepath for σt is continuous, it follows from the theory of quadratic variation(Karatzas and Shreve, 1988) that

p limm→∞

(∫ t+1

tσ 2

s ds −∑

j=1,···,mr2

m,t+ j/m

)= 0.

Hence time t volatility is theoretically observable from the sample pathof the return process so long as the sampling process is frequent enough.

5 See Fung and Hsieh (1991) and Andersen and Bollerslev (1998). In the foreign exchange markets, quotesfor major exchange rates are available round the clock. In the case of stock markets, close-to-open squared returnis used in the volatility aggregation process during market close.



When there are jumps in price process, (1.5) becomes

dpt = σt dWt + κt dqt ,

where dqt is a Poisson process with dqt = 1 corresponding to a jump attime t , and zero otherwise, and κt is the jump size at time t when thereis a jump. In this case, the quadratic variation for the cumulative returnprocess is then given by∫ t+1

tσ 2

s ds +∑

t<s≤t+1

κ2 (s) , (1.7)

which is the sum of the integrated volatility and jumps.In the absence of jumps, the second term on the right-hand side of (1.7)

disappears, and the quadratic variation is simply equal to the integratedvolatility. In the presence of jumps, the realized volatility continues toconverge to the quadratic variation in (1.7)

p limm→∞

(∫ t+1

tσ 2

s ds +∑

t<s≤t+1

κ2 (s) −m∑

j=1

r2m,t+ j/m

)= 0. (1.8)

Barndorff-Nielsen and Shephard (2003) studied the property of the stan-dardized realized bipower variation measure

BV [a,b]m,t+1 = m[(a+b)/2−1]

m−1∑j=1

∣∣rm,t+ j/m∣∣a ∣∣rm,t+ ( j+1)/m

∣∣b , a, b ≥ 0.

They showed that when jumps are large but rare, the simplest case wherea = b = 1,

µ−21 BV [1,1]

m,t+1 = µ−21

m−1∑j=1

∣∣rm,t+ j/m∣∣ ∣∣rm,t+ ( j+1)/m

∣∣ →∫ t+1

tσ 2

s ds

where µ1 = √2/π . Hence, the realized volatility and the realized

bipower variation can be substituted into (1.8) to estimate the jumpcomponent, κt . Barndorff-Nielsen and Shephard (2003) suggested im-posing a nonnegative constraint on κt . This is perhaps too restrictive.For nonnegative volatility, κt + µ−2

1 BVt > 0 will be sufficient.Characteristics of financial market data suggest that returns measured

at an interval shorter than 5 minutes are plagued by spurious serialcorrelation caused by various market microstructure effects includingnonsynchronous trading, discrete price observations, intraday periodic



volatility patterns and bid–ask bounce.6 Bollen and Inder (2002), Ait-Sahalia, Mykland and Zhang (2003) and Bandi and Russell (2004) havegiven suggestions on how to isolate microstructure noise from realizedvolatility estimator.

1.3.4 Scaling and actual volatility

The forecast of multi-period volatilityσT,T + j (i.e. for j period) is taken tobe the sum of individual multi-step point forecasts s

j=1hT + j |T . Thesemulti-step point forecasts are produced by recursive substitution andusing the fact that ε2

T +i |T = hT +i |T for i > 0 and ε2T +i |T = ε2

T +i forT + i ≤ 0. Since volatility of financial time series has complex struc-ture, Diebold, Hickman, Inoue and Schuermann (1998) warn that fore-cast estimates will differ depending on the current level of volatility,volatility structure (e.g. the degree of persistence and mean reversionetc.) and the forecast horizon.

If returns are i id (independent and identically distributed, or strictwhite noise), then variance of returns over a long horizon can be derivedas a simple multiple of single-period variance. But, this is clearly not thecase for many financial time series because of the stylized facts listed inSection 1.2. While a point forecast of σ T −1,T | t−1 becomes very noisyas T → ∞, a cumulative forecast, σ t,T | t−1, becomes more accuratebecause of errors cancellation and volatility mean reversion except whenthere is a fundamental change in the volatility level or structure.7

Complication in relation to the choice of forecast horizon is partlydue to volatility mean reversion. In general, volatility forecast accu-racy improves as data sampling frequency increases relative to forecasthorizon (Andersen, Bollerslev and Lange, 1999). However, for forecast-ing volatility over a long horizon, Figlewski (1997) finds forecast errordoubled in size when daily data, instead of monthly data, is used to fore-cast volatility over 24 months. In some cases, where application is ofvery long horizon e.g. over 10 years, volatility estimate calculated using

6 The bid–ask bounce for example induces negative autocorrelation in tick data and causes the realizedvolatility estimator to be upwardly biased. Theoretical modelling of this issue so far assumes the price processand the microstructure effect are not correlated, which is open to debate since market microstructure theorysuggests that trading has an impact on the efficient price. I am grateful to Frank de Jong for explaining this to meat a conference.

7 σ t,T | t−1 denotes a volatility forecast formulated at time t − 1 for volatility over the period from t to T . Inpricing options, the required volatility parameter is the expected volatility over the life of the option. The pricingmodel relies on a riskless hedge to be followed through until the option reaches maturity. Therefore the requiredvolatility input, or the implied volatility derived, is a cumulative volatility forecast over the option maturity andnot a point forecast of volatility at option maturity. The interest in forecasting σ t,T | t−1 goes beyond the risklesshedge argument, however.



weekly or monthly data is better because volatility mean reversion isdifficult to adjust using high frequency data. In general, model-basedforecasts lose supremacy when the forecast horizon increases with re-spect to the data frequency. For forecast horizons that are longer than6 months, a simple historical method using low-frequency data over aperiod at least as long as the forecast horizon works best (Alford andBoatsman, 1995; and Figlewski, 1997).

As far as sampling frequency is concerned, Drost and Nijman (1993)prove, theoretically and for a special case (i.e. the GARCH(1,1) process,which will be introduced in Chapter 4), that volatility structure should bepreserved through intertemporal aggregation. This means that whetherone models volatility at hourly, daily or monthly intervals, the volatilitystructure should be the same. But, it is well known that this is not thecase in practice; volatility persistence, which is highly significant indaily data, weakens as the frequency of data decreases. 8 This furthercomplicates any attempt to generalize volatility patterns and forecastingresults.

1.4 THE TREATMENT OF LARGE NUMBERS

In this section, I use large numbers to refer generally to extreme values,outliers and rare jumps, a group of data that have similar characteristicsbut do not necessarily belong to the same set. To a statistician, there arealways two ‘extremes’ in each sample, namely the minimum and themaximum. The H -L method for estimating volatility described in theprevious section, for example, is also called the extreme value method.We have also noted that these H -L estimators assume conditional dis-tribution is normal. In extreme value statistics, normal distribution is butone of the distributions for the tail. There are many other extreme valuedistributions that have tails thinner or thicker than the normal distribu-tion’s. We have known for a long time now that financial asset returns arenot normally distributed. We also know the standardized residuals fromARCH models still display large kurtosis (see McCurdy and Morgan,1987; Milhoj, 1987; Hsieh, 1989; Baillie and Bollerslev, 1989). Con-ditional heteroscedasticity alone could not account for all the tail thick-ness. This is true even when the Student-t distribution is used to construct

8 See Diebold (1988), Baillie and Bollerslev (1989) and Poon and Taylor (1992) for examples. Note thatNelson (1992) points out separately that as the sampling frequency becomes shorter, volatility modelled usingdiscrete time model approaches its diffusion limit and persistence is to be expected provided that the underlyingreturns is a diffusion or a near-diffusion process with no jumps.



the likelihood function (see Bollerslev, 1987; Hsieh, 1989). Hence, in theliterature, the extreme values and the tail observations often refer to thosedata that lie outside the (conditional) Gaussian region. Given that jumpsare large and are modelled as a separate component to the Brownianmotion, jumps could potentially be seen as a set similar to those tailobservations provided that they are truly rare.

Outliers are by definition unusually large in scale. They are so largethat some have argued that they are generated from a completely dif-ferent process or distribution. The frequency of occurrence should bemuch smaller for outliers than for jumps or extreme values. Outliersare so huge and rare that it is very unlikely that any modelling effortwill be able to capture and predict them. They have, however, undueinfluence on modelling and estimation (Huber, 1981). Unless extremevalue techniques are used where scale and marginal distribution are of-ten removed, it is advisable that outliers are removed or trimmed beforemodelling volatility. One such outlier in stock market returns is the Oc-tober 1987 crash that produced a 1-day loss of over 20% in stock marketsworldwide.

The ways that outliers have been tackled in the literature largely de-pend on their sizes, the frequency of their occurrence and whether theseoutliers have an additive or a multiplicative impact. For the rare andadditive outliers, the most common treatment is simply to remove themfrom the sample or omit them in the likelihood calculation (Kearns andPagan, 1993). Franses and Ghijsels (1999) find forecasting performanceof the GARCH model is substantially improved in four out of five stockmarkets studied when the additive outliers are removed. For the raremultiplicative outliers that produced a residual impact on volatility, adummy variable could be included in the conditional volatility equationafter the outlier returns has been dummied out in the mean equation(Blair, Poon and Taylor, 2001).

rt = µ + ψ1 Dt + εt , εt =√

ht zt

ht = ω + βht−1 + αε2t−1 + ψ2 Dt−1

where Dt is 1 when t refers to 19 October 1987 and 0 otherwise. Per-sonally, I find a simple method such as the trimming rule in (1.2) veryquick to implement and effective.

The removal of outliers does not remove volatility persistence. In fact,the evidence in the previous section shows that trimming the data using(1.2) actually increases the ‘long memory’ in volatility making it appear



to be extremely persistent. Since autocorrelation is defined as

ρ (rt , rt−τ ) = Cov (rt , rt−τ )

V ar (rt ),

the removal of outliers has a great impact on the denominator, reducesV ar (rt ) and increases the individual and the cumulative autocorrelationcoefficients.

Once the impact of outliers is removed, there are different views abouthow the extremes and jumps should be handled vis-a-vis the rest of thedata. There are two schools of thought, each proposing a seeminglydifferent model, and both can explain the long memory in volatility. Thefirst believes structural breaks in volatility cause mean level of volatilityto shift up and down. There is no restriction on the frequency or the sizeof the breaks. The second advocates the regime-switching model wherevolatility switches between high and low volatility states. The means ofthe two states are fixed, but there is no restriction on the timing of theswitch, the duration of each regime and the probability of switching.Sometimes a three-regime switching is adopted but, as the number ofregimes increases, the estimation and modelling become more complex.Technically speaking, if there are infinite numbers of regimes then thereis no difference between the two models. The regime-switching modeland the structural break model will be described in Chapter 5.


2

Volatility Forecast Evaluation

Comparing forecasting performance of competing models is one of themost important aspects of any forecasting exercise. In contrast to theefforts made in the construction of volatility models and forecasts, littleattention has been paid to forecast evaluation in the volatility forecastingliterature. Let Xt be the predicted variable, Xt be the actual outcome andεt = Xt − Xt be the forecast error. In the context of volatility forecast,Xt and Xt are the predicted and actual conditional volatility. There aremany issues to consider:

(i) The form of Xt : should it be σ 2t or σt ?

(ii) Given that volatility is a latent variable, the impact of the noiseintroduced in the estimation of Xt , the actual volatility.

(iii) Which form of εt is more relevant for volatility model selection; ε2

t , |εt | or |εt |/Xt ? Do we penalize underforecast, Xt < Xt ,more than overforecast, Xt > Xt ?

(iv) Given that all error statistics are subject to noise, how do we knowif one model is truly better than another?

(v) How do we take into account when Xt and Xt+1 (and similarly forεt and Xt ) cover a large amount of overlapping data and are seriallycorrelated?

All these issues will be considered in the following sections.

2.1 THE FORM OF Xt

Here we argue that Xt should be σt , and that if σt cannot be estimatedwith some accuracy it is best not to perform comparison across predictivemodels at all. The practice of using daily squared returns to proxy dailyconditional variance has been shown time and again to produce wrongsignals in model selection.

Given that all time series volatility models formulate forecasts basedon past information, they are not designed to predict shocks that are new

21



to the system. Financial market volatility has many stylized facts. Oncea shock has entered the system, the merit of the volatility model dependson how well it captures these stylized facts in predicting the volatility ofthe following days. Hence we argue that Xt should be σt . Conditionalvariance σ 2

t formulation gives too much weight to the errors caused by‘new’ shocks and especially the large ones, distorting the less extremeforecasts where the models are to be assessed.

Note also that the square of a variance error is the fourth power ofthe same error measured from standard deviation. This can complicatethe task of forecast evaluation given the difficulty in estimating fourthmoments with common distributions let alone the thick-tailed ones infinance. The confidence interval of the mean error statistic can be verywide when forecast errors are measured from variances and worse if theyare squared. This leads to difficulty in finding significant differencesbetween forecasting models.

Davidian and Carroll (1987) make similar observations in their studyof variance function estimation for heteroscedastic regression. Usinghigh-order theory, they show that the use of square returns for modellingvariance is appropriate only for approximately normally distributed data,and becomes nonrobust when there is a small departure from normal-ity. Estimation of the variance function that is based on logarithmictransformation or absolute returns is more robust against asymmetryand nonnormality.

Some have argued that perhaps Xt should be lnσt to rescale the size ofthe forecast errors (Pagan and Schwert, 1990). This is perhaps one steptoo far. After all, the magnitude of the error directly impacts on optionpricing, risk management and investment decision. Taking the logarithmof the volatility error is likely to distort the loss function which is directlyproportional to the magnitude of forecast error. A decision maker mightbe more risk-averse towards the larger errors.

We have explained in Section 1.3.1 the impact of using squared returnsto proxy daily volatility. Hansen and Lunde (2004b) used a series ofsimulations to show that ‘. . . the substitution of a squared return forthe conditional variance in the evaluation of ARCH-type models canresult in an inferior model being chosen as [the] best with a probabilityconverges to one as the sample size increases . . . ’. Hansen and Lunde(2004a) advocate the use of realized volatility in forecast evaluation butcaution the noise introduced by market macrostructure when the intradayreturns are too short.


Volatility Forecast Evaluation 23

2.2 ERROR STATISTICS AND THE FORM OF εt

Ideally an evaluation exercise should reflect the relative or absolute use-fulness of a volatility forecast to investors. However, to do that oneneeds to know the decision process that require these forecasts and thecosts and benefits that result from using better forecasts. Utility-basedcriteria, such as that used in West, Edison and Cho (1993), require someassumptions about the shape and property of the utility function. In prac-tice these costs, benefits and utility function are not known and one oftenresorts to simply use measures suggested by statisticians.

Popular evaluation measures used in the literature includeMean Error (ME)

1

N

N∑t=1

εt = 1

N

N∑t=1

(σ t − σt ) ,

Mean Square Error (MSE)

1

N

N∑t=1

ε2t = 1

N

N∑t=1

(σ t − σt )2 ,

Root Mean Square Error (RMSE)√√√√ 1

N

N∑t=1

ε2t =

√√√√ 1

N

N∑t=1

(σ t − σt )2,

Mean Absolute Error (MAE)

1

N

N∑t=1

|εt | = 1

N

N∑t=1

|σ t − σt | ,

Mean Absolute Percent Error (MAPE)

1

N

N∑t=1

|εt |σt

= 1

N

N∑t=1

|σ t − σt |σt

.

Bollerslev and Ghysels (1996) suggested a heteroscedasticity-adjusted version of MSE called HMSE where

HMSE = 1

N

N∑t=1

[σt

σ t− 1

]2



This is similar to squared percentage error but with the forecast errorscaled by predicted volatility. This type of performance measure is notappropriate if the absolute magnitude of the forecast error is a majorconcern. It is not clear why it is the predicted and not the actual volatilitythat is used in the denominator. The squaring of the error again will givegreater weight to large errors.

Other less commonly used measures include mean logarithm of ab-solute errors (MLAE) (as in Pagan and Schwert, 1990), the Theil-Ustatistic and one based on asymmetric loss function, namely LINEX:Mean Logarithm of Absolute Errors (MLAE)

1

N

N∑t=1

ln |εt | = 1

N

N∑t=1

ln |σ t − σt |

Theil-U measure

Theil-U =

N∑t=1

(σ t − σt )2

N∑t=1

(σ B M

t − σt)2

, (2.1)

where σ B Mt is the benchmark forecast, used here to remove the effect of

any scalar transformation applied to σt .LINEX has asymmetric loss function whereby the positive errors are

weighted differently from the negative errors:

LINEX = 1

N

N∑t=1

[exp {−a (σ t − σt )} + a (σ t − σt ) − 1]. (2.2)

The choice of the parameter a is subjective. If a > 0, the function isapproximately linear for overprediction and exponential for underpre-diction. Granger (1999) describes a variety of other asymmetric lossfunctions of which the LINEX is an example. Given that most investorswould treat gains and losses differently, the use of asymmetric loss func-tions may be advisable, but their use is not common in the literature.

2.3 COMPARING FORECAST ERRORSOF DIFFERENT MODELS

In the special case where the error distribution of one forecastingmodel dominates that of another forecasting model, the comparison is



straightforward (Granger, 1999). In practice, this is rarely the case, andmost comparisons of forecasting results are made based on the errorstatistics described in Section 2.2. It is important to note that these er-ror statistics are themselves subject to error and noise. So if an errorstatistic of model A is higher than that of model B, one cannot concludethat model B is better than A without performing tests of significance.For statistical inference, West (1996), West and Cho (1995) and Westand McCracken (1998) show how standard errors for ME, MSE, MAEand RMSE may be derived taking into account serial correlation in theforecast errors and uncertainty inherent in volatility model parameterestimates.

If there are T number of observations in the sample and T is large,there are two ways in which out-of-sample forecasts may be made.Assume that we use n number of observations for estimation and makeT − n number of forecasts. The recursive scheme starts with the sample{1, · · · , n} and makes first forecast at n + 1. The second forecast forn + 2 will include the last observation and form the information set{1, · · · , n + 1}. It follows that the last forecast for T will include allbut the last observation, i.e. the information set is {1, · · · , T − 1}. Inpractice, the rolling scheme is more popular, where a fixed number ofobservations is used in the estimation. So the forecast for n + 2 will bebased on information set {2, · · · , n + 1}, and the last forecast at T will bebased on {T − n, · · · , T − 1}. The rolling scheme omits information inthe distant past. It is also more manageable in terms of computation whenT is very large. The standard errors developed by West and co-authorsare based on asymptotic theory and work for recursive scheme only. Forsmaller sample and rolling scheme forecasts, Diebold and Mariano’s(1995) small sample methods are more appropriate.

Diebold and Mariano (1995) propose three tests for ‘equal accuracy’between two forecasting models. The tests relate prediction error tosome very general loss function and analyse loss differential derivedfrom errors produced by two competing models. The three tests includean asymptotic test that corrects for series correlation and two exact fi-nite sample tests based on sign test and the Wilcoxon sign-rank test.Simulation results show that the three tests are robust against non-Gaussian, nonzero mean, serially and contemporaneously correlatedforecast errors. The two sign-based tests in particular continue to workwell among small samples. The Diebold and Mariano tests have beenused in a number of volatility forecasting contests. We provide the testdetails here.



Let {Xi t}Tt=1 and {X j t}T

t=1 be two sets of forecasts for {Xt}Tt=1 from

models i and j respectively. Let the associated forecast errors be {eit}Tt=1

and {e jt}Tt=1. Let g (·) be the loss function (e.g. the error statistics in

Section 2.2) such that

g(Xt , Xi t

) = g (eit ) .

Next define loss differential

dt ≡ g (eit ) − g(e jt

).

The null hypothesis is equal forecast accuracy and zero loss differentialE(dt ) = 0.

2.3.1 Diebold and Mariano’s asymptotic test

The first test targets on the mean

d = 1

T

T∑t=1

|g(eit ) − g(e jt )|

with test statistic

S1 = d√1

T2π fd (0)

S1 ∼ N (0, 1)

2π fd (0) =T −1∑

τ=−(T −1)

1

(τ

S (T )

)γ d (τ )

γ d (τ ) = 1

T

T∑t=|τ |+1

(dt − d

) (dt−|τ | − d

).

The operator 1 (τ/S (T )) is the lag window, and S (T ) is the truncationlag with

1

(τ

S (T )

)=

1 for

∣∣∣∣ τ

S (T )

∣∣∣∣ ≤ 1

0 otherwise.

Assuming that k-step ahead forecast errors are at most (k − 1)-dependent, it is therefore recommended that S (T ) = (k − 1). It is notlikely that fd (0) will be negative, but in the rare event that fd (0) < 0,



it should be treated as zero and the null hypothesis of equal forecastaccuracy be rejected automatically.

2.3.2 Diebold and Mariano’s sign test

The sign test targets on the median with the null hypothesis that

Med (d) = Med(g(eit ) − g

(e jt

))= 0.

Assuming that dt ∼ iid, then the test statistic is

S2 =T∑

t=1

I+ (dt )

where

I+ (dt ) ={

1 if dt > 00 otherwise

.

For small sample, S2 should be assessed using a table for cumulativebinomial distribution. In large sample, the Studentized verson of S2 isasymptotically normal

S2a = S2 − 0.5T√0.25T

a∼ N (0, 1).

2.3.3 Diebold and Mariano’s Wilcoxon sign-rank test

As the name indicates, this test is based on both the sign and the rank ofloss differential with test statistic

S3 =T∑

t=1

I+ (dt ) rank (|dt |)

represents the sum of the ranks of the absolute values of the positiveobservations. The critical values for S3 have been tabulated for smallsample. For large sample, the Studentized verson of S3 is again asymp-totically normal

S2a =S3 − T (T + 1)

4√T (T + 1) (2T + 1)

24

a∼ N (0, 1) .



2.3.4 Serially correlated loss differentials

Serial correlation is explicitly taken care of in S1. For S2 and S3 (andtheir asymptotic counter parts S2a and S3a), the following k-set of lossdifferentials have to be tested jointly{

di j,1, di j,1+k, di j,1+2k, · · ·},{

di j,2, di j,2+k, di j,2+2k, · · ·},

...{di j,k, di j,2k, di j,3k, · · ·

}.

A test with size bounded by α is then tested k times, each of size α/k,on each of the above k loss-differentials sequences. The null hypothesisof equal forecast accuracy is rejected if the null is rejected for any of thek samples.

2.4 REGRESSION-BASED FORECAST EFFICIENCYAND ORTHOGONALITY TEST

The regression-based method for examining the informational contentof forecasts is by far the most popular method in volatility forecasting.It involves regressing the actual volatility, Xt , on the forecasts literature,Xt , as shown below:

Xt = α + β Xt + υt . (2.3)

Conditioning upon the forecast, the prediction is unbiased only if α = 0and β = 1.

Since the error term, υt , is heteroscedastic and serially correlated whenoverlapping forecasts are evaluated, the standard errors of the parameterestimates are often computed on the basis of Hansen and Hodrick (1980).Let Y be the row matrix of regressors including the constant term. In(2.3), Yt = (

1 Xt)

is a 1 × 2 matrix. Then

� = T −1T∑

t=1

υ2t Y

′t Yt

+T −1T∑

k=1

T∑t=k+1

Q (k, t) υkυt

(Y

′t Yk + Y

′kYt

),



where υk and υt are the residuals for observation k and t from the regres-sion. The operator Q (k, t) is an indicator function taking the value 1 ifthere is information overlap between Yk and Yt . The adjusted covariancematrix for the regression coefficients is then calculated as

� =(

Y′Y)−1

�(

Y′Y)−1

. (2.4)

Canina and Figlewski (1993) conducted some simulatation studies andfound the corrected standard errors in (2.4) are close to the true values,and the use of overlapping data reduced the standard error between one-quarter and one-eighth of what would be obtained with only nonover-lapping data.

In cases where there are more than one forecasting model, additionalforecasts are added to the right-hand side of (2.3) to check for incremen-tal explanatory power. Such a forecast encompassing test dates backto Theil (1966). Chong and Hendry (1986) and Fair and Shiller (1989,1990) provide further theoretical exposition of such methods for testingforecast efficiency. The first forecast is said to subsume information con-tained in other forecasts if these additional forecasts do not significantlyincrease the adjusted regression R2. Alternatively, an orthogonality testmay be conducted by regressing the residuals from (2.3) on other fore-casts. If these forecasts are orthogonal, i.e. do not contain additionalinformation, then the regression coefficients will not be different fromzero.

While it is useful to have an unbiased forecast, it is important todistinguish between bias and predictive power. A biased forecast canhave predictive power if the bias can be corrected. An unbiased forecastis useless if all forecast errors are big. For Xi to be considered as a goodforecast, Var(υt ) should be small and R2 for the regression should tendto 100%. Blair, Poon and Taylor (2001) use the proportion of explainedvariability, P , to measure explanatory power

P = 1 −∑(

Xi − Xi)2∑

(Xi − µX )2. (2.5)

The ratio in the right-hand side of (2.5) compares the sum of squaredprediction errors (assuming α = 0 and β = 1 in (2.3)) with the sumof squared variation of Xi . P compares the amount of variation in theforecast errors with that in actual volatility. If prediction errors are small,



P is closer to 1. Given that a regression model that produces (2.5) is morerestrictive than (2.3), P is likely to be smaller than conventional R2. Pcan even be negative since the ratio on the right-hand side of (2.5) canbe greater than 1. A negative P means that the forecast errors havea greater amount of variation than the actual volatility, which is not adesirable characteristic for a well-behaved forecasting model.

2.5 OTHER ISSUES IN FORECAST EVALUATION

In all forecast evaluations, it is important to distinguish in-sample andout-of-sample forecasts. In-sample forecast, which is based on param-eters estimated using all data in the sample, implicitly assumes parameterestimates are stable across time. In practice, time variation of parameterestimates is a critical issue in forecasting. A good forecasting modelshould be one that can withstand the robustness of out-of-sample test –a test design that is closer to reality.

Instead of striving to make some statistical inference, model perfor-mance could be judged on some measures of economic significance. Ex-amples of such an approach include portfolio improvement derived frombetter volatility forecasts (Fleming, Kirby and Ostdiek, 2000, 2002).Some papers test forecast accuracy by measuring the impact on optionpricing errors (Karolyi, 1993). In the latter case, pricing error in theoption model will be cancelled out when the option implied volatility isreintroduced into the pricing formula. So it is not surprising that evalu-ation which involves comparing option pricing errors often prefers theimplied volatility method to all other time series methods.

Research in financial market volatility has been concentrating onmodelling and less on forecasting. Work on combined forecast is rare,probably because the groups of researchers in time series models andoption pricing do not seem to mix. What has not yet been done in theliterature is to separate the forecasting period into ‘normal’ and ‘excep-tional’ periods. It is conceivable that different forecasting methods arebetter suited to different trading environment and economic conditions.


3

Historical Volatility Models

Compared with the other types of volatility models, the historical volatil-ity models (HIS) are the easiest to manipulate and construct. The well-known Riskmetrics EWMA (equally weighted moving average) modelfrom JP Morgan is a form of historical volatility model; so are modelsthat build directly on realized volatility that have became very popu-lar in the last few years. Historical volatility models have been shownto have good forecasting performance compared with other time seriesvolatility models. Unlike the other two time series models (viz. ARCHand stochastic volatility (SV)) conditional volatility is modelled sepa-rately from returns in the historical volatility models, and hence theyare less restrictive and are more ready to respond to changes in volatil-ity dynamic. Studies that find historical volatility models forecast betterthan ARCH and/or SV models include Taylor (1986, 1987), Figlewski(1997), Figlewski and Green (1999), Andersen, Bollerslev, Diebold andLabys (2001) and Taylor, J. (2004). With the increased availability ofintraday data, we can expect to see research on the realized volatilityvariant of the historical model to intensify in the next few years.

3.1 MODELLING ISSUES

Unlike ARCH SV models where returns are the main input, HIS modelsdo not normally use returns information so long as the volatility estimatesare ready at hand. Take the simplest form of ARCH(1) for example,

rt = µ + εt , εt ∼ N (0, σt ) (3.1)

εt = ztσt , zt ∼ N (0, 1)

σ 2t = ω + α1ε

2t−1. (3.2)

The conditional volatility σ 2t in (3.2) is modelled as a ‘byproduct’

of the return equation (3.1). The estimation is done by maximizingthe likelihood of observing {εt} using the normal, or other chosen,density. The construction and estimation of SV models are similar tothose of ARCH, except that there is now an additional innovation termin (3.2).

31



In contrast, the HIS model is built directly on conditional volatility,e.g. an AR(1) model:

σt = γ + β1σt−1 + υt . (3.3)

The parameters γ and β1 are estimated by minimizing in-sample forecasterrors, υt , where

υt = σt − γ − β1σ t−1,

and the forecaster has the choice of reducing mean square errors, meanabsolute errors etc., as in the case of choosing an appropriate forecasterror statistic in Section 2.2.

The historical volatility estimates σt in (3.3) can be calculated assample standard deviations if there are sufficient data for each t in-terval. If there is not sufficient information, then the H -L method ofSection 1.3.2 may be used, and in the most extreme case, where onlyone observation is available for each t interval, one often resorts to usingabsolute return to proxy for volatility at t . In Section 1.3.1 we have high-lighted the danger of using daily absolute or squared returns to proxy‘actual’ daily volatility for the purpose of forecast evaluation, as thiscould lead to very misleading model ranking. The problem with the useof daily absolute return in volatility modelling is less severe providedthat long distributed lags are included (Nelson, 1992; Nelson and Foster,1995). With the increased availability of intraday data, historical volatil-ity estimates can be calculated quite accurately as realized volatilityfollowing Section 1.3.3.

3.2 TYPES OF HISTORICAL VOLATILITY MODELS

There are now two major types of HIS models: the single-state and theregime-switching models. All the HIS models differ by the number of lagvolatility terms included in the model and the weights assigned to them,reflecting the choice on the tradeoff between increasing the amount ofinformation and more updated information.

3.2.1 Single-state historical volatility models

The simplest historical price model is the random walk model, wherethe difference between consecutive period volatility is modelled as arandom noise;

σt = σt−1 + vt ,


Historical Volatility Models 33

So the best forecast for tomorrow’s volatility is today’s volatility:

σ t+1 = σt ,

where σt alone is used as a forecast for σt+1.In contrast, the historical average method makes a forecast based on

the entire history

σ t+1 = 1

t(σt + σt−1 + · · · + σ1) .

The simple moving average method below,

σ t+1 = 1

τ(σt + σt−1 + · · · + σt−τ−1) ,

is similar to the historical average method, except that older informationis discarded. The value of τ (i.e. the lag length to past informationused) could be subjectively chosen or based on minimizing in-sampleforecast error, ςt+1 = σt+1 − σ t+1. The multi-period forecasts σ t+τ forτ > 1 will be the same as the one-step-ahead forecast σ t+1 for all threemethods above.

The exponential smoothing method below,

σt = (1 − β) σt−1 + βσ t−1 + ξt and 0 ≤ β ≤ 1,

σ t+1 = (1 − β) σt + βσ t ,

is similar to the historical method, but more weight is given to the recentpast and less weight to the distant past. The smoothing parameter β isestimated by minimizing the in-sample forecast errors ξt .

The exponentially weighted moving average method (EWMA) belowis the moving average method with exponential weights:

σ t+1 =τ∑

i=1

β i σt−i−1

/ τ∑i=1

β i .

Again the smoothing parameter β is estimated by minimizing the in-sample forecast errors ξt . The JP Morgan RiskmetricsTM model is aprocedure that uses the EWMA method.

All the historical volatility models above have a fixed weightingscheme or a weighting scheme that follows some declining pattern. Othertypes of historical model have weighting schemes that are not prespec-ified. The simplest of such models is the simple regression method,

σt = γ + β1 σt−1 + β2 σt−2 + · · · + βn σt−n + υt ,

σ t+1 = γ + β1 σt + β2 σt−1 + · · · + βn σt−n+1,



which expresses volatility as a function of its past values and an errorterm.

The simple regression method is principally autoregressive. If pastvolatility errors are also included, one gets the ARMA model

σ t+1 = β1 σt + β2 σt−1 + · · · + γ1 υt + γ2 υt−1 + · · · .Introducing a differencing order I(d), we get ARIMA when d = 1 andARFIMA when d < 1.

3.2.2 Regime switching and transition exponential smoothing

In this section, we have the threshold autoregressive model from Caoand Tsay (1992):

σt = φ(i)0 + φ

(i)1 σt−1 + · · · + φ(i)

p σt−p + vt , i = 1, 2, . . . , k

σ t+1 = φ(i)0 + φ

(i)1 σt + · · · + φ(i)

p σt+1−p,

where the thresholds separate volatility into states with independentsimple regression models and noise processes in each state. The predic-tion σ t+1 could be based solely on current state information i assumingthe future will remain on current state. Alternatively it could be based oninformation of all states weighted by the transition probability for eachstate. Cao and Tsay (1992) found the threshold autoregressive modeloutperformed EGARCH and GARCH in forecasting of the 1- to 30-month volatility of the S&P value-weighted index. EGARCH providedbetter forecasts for the S&P equally weighted index, possibly becausethe equally weighted index gives more weights to small stocks wherethe leverage effect could be more important.

The smooth transition exponential smoothing model is from Taylor,J. (2004):

σ t = αt−1ε2t−1 + (1 − αt−1) σ 2

t−1 + vt , (3.4)

where

αt−1 = 1

1 + exp (β + γ Vt−1),

and Vt−1 = aεt−1 + b |εt−1| is the transition variable. The smoothingparameter αt−1 varies between 0 and 1, and its value depends on thesize and the sign of εt−1. The dependence on εt−1 means that multi-step-ahead forecasts cannot be made except through simulation. (The samewould apply to many nonlinear ARCH and SV models as we will showin the next few chapters.)


Historical Volatility Models 35

One-day-ahead forecasting results show that the smooth transition ex-ponential smoothing model performs very well against several ARCHcounterparts and even outperformed, on a few occasions, the realizedvolatility forecast. But these rankings were not tested for statistical sig-nificance, so it is difficult to come to a conclusion given the closenessof many error statistics reported.

3.3 FORECASTING PERFORMANCE

Taylor (1987) was one of the earliest to test time-series volatility fore-casting models before ARCH/GARCH permeated the volatility litera-ture. Taylor (1987) used extreme value estimates based on high, low andclosing prices to forecast 1 to 20 days DM/$ futures volatility and founda weighted average composite forecast performed best. Wiggins (1992)also gave support to extreme-value volatility estimators.

In the pre-ARCH era, there were many studies that covered a widerange of issues. Sometimes forecasters would introduce ‘learning’ byallowing parameters and weights of combined forecasts to be dynam-ically updated. These frequent updates did not always lead to betterresults, however. Dimson and Marsh (1990) found ex ante time-varyingoptimized weighting schemes do not always work well in out-of-sampleforecasts. Sill (1993) found S&P500 volatility was higher during reces-sion and that commercial T-Bill spread helped to predict stock-marketvolatility.

The randow walk and historical average method seems naive at first,but they seem to work very well for medium and long horizon forecasts.For forecast horizons that are longer than 6 months, low-frequency dataover a period at least as long as the forecast horizon works best. Toprovide equity volatility for investment over a 5-year period for exam-ple, Alford and Boatsman (1995) recommended, after studying a sam-ple of 6879 stocks, that volatility should be estimated from weekly ormonthly returns from the previous 5 years and that adjustment madebased on industry and company size. Figlewski (1997) analysed thevolatility of the S&P500, the long- and short-term US interest rate andthe Deutschemark–dollar exchange rate and the use of monthly dataover a long period provides the best long-horizon forecast. Alford andBoatsman (1995), Figlewski (1997) and Figlewski and Green (1999) allstressed the importance of having a long enough estimation period tomake good volatility forecasts over long horizon.


4

ARCH

Financial market volatility is known to cluster. A volatile period tends topersist for some time before the market returns to normality. The ARCH(AutoRegressive Conditional Heteroscedasticity) model proposed byEngle (1982) was designed to capture volatility persistence in inflation.The ARCH model was later found to fit many financial time series and itswidespread impact on finance has led to the Nobel Committee’s recog-nition of Rob Engle’s work in 2003. The ARCH effect has been shownto lead to high kurtosis which fits in well with the empirically observedtail thickness of many asset return distributions. The leverage effect, aphenomenon related to high volatility brought on by negative return,is often modelled with a sign-based return variable in the conditionalvolatility equation.

4.1 ENGLE (1982)

The ARCH model, first introduced by Engle (1982), has been ex-tended by many researchers and extensively surveyed in Bera andHiggins (1993), Bollerslev, Chou and Kroner (1992), Bollerslev,Engle and Nelson (1994) and Diebold and Lopez (1995). In contrast tothe historical volatility models described in the previous chapter, ARCHmodels do not make use of the past standard deviations, but formulateconditional variance, ht , of asset returns via maximum likelihood pro-cedures. (We follow the ARCH literature here by writing σ 2

t = ht .) Toillustrate this, first write returns, rt , as

rt = µ + εt ,

εt =√

ht zt , (4.1)

where zt ∼ D (0, 1) is a white noise. The distribution D is often taken asnormal. The process zt is scaled by ht , the conditional variance, whichin turn is a function of past squared residual returns. In the ARCH(q)process proposed by Engle (1982),

ht = ω +q∑

j=1

α jε2t− j (4.2)

37



with ω > 0 and α j ≥ 0 to ensure ht is strictly positive variance. Typi-cally, q is of high order because of the phenomenon of volatility per-sistence in financial markets. From the way in which volatility is con-structed in (4.2), ht is known at time t − 1. So the one-step-ahead forecastis readily available. The multi-step-ahead forecasts can be formulatedby assuming E

[ε2

t+τ

] = ht+τ .The unconditional variance of rt is

σ 2 = ω

1 −q∑

j=1

α j

.

The process is covariance stationary if and only if the sum of the autore-gressive parameters is less than one

∑qj=1 α j < 1.

4.2 GENERALIZED ARCH

For high-order ARCH(q) process, it is more parsimonious to modelvolatility as a GARCH(p, q) (generalized ARCH due to Bollerslev(1986) and Taylor (1986)), where additional dependencies are permittedon p lags of past ht as shown below:

ht = ω +p∑

i=1

βi ht−i +q∑

j=1

α jε2t− j

and ω > 0. For GARCH(1, 1), the constraints α1 ≥ 0 and β1 ≥ 0 areneeded to ensure ht is strictly positive. For higher orders of GARCH,the constraints on βi and α j are more complex (see Nelson and Cao(1992) for details). The unconditional variance equals

σ 2 = ω

1 −p∑

i=1

βi −q∑

j=1

α j

The GARCH(p, q) model is covariance stationary if and only if∑pi=1 βi + ∑q

j=1 α j < 1.Volatility forecasts from GARCH(1, 1) can be made by repeated sub-

stitutions. First, we make use of the relationship (4.1) to provide anestimate for the expected squared residuals

E[ε2

t

] = ht E[z2

t

] = ht .


Arch 39

The conditional variance ht+1 and the one-step-ahead forecast is knownat time t ,

ht+1 = ω + α1ε2t + β1ht . (4.3)

The forecast of ht+2 makes use of the fact that E[ε2

t+1

] = ht+1 and weget

ht+2 = ω + α1ε2t+1 + β1ht+1

= ω + (α1 + β1) ht+1.

Similarly,

ht+3 = ω + (α1 + β1) ht+2

= ω + ω (α1 + β1) + (α1 + β1)2 ht+1

= ω + ω (α1 + β1) + ω (α1 + β1)2 + (α1 + β1)2[α1ε

2t + β1ht

].

As the forecast horizon τ lengthens,

ht+τ = ω

1 − (α1 + β1)+ (α1 + β1)τ

[α1ε

2t + β1ht

]. (4.4)

If α1 + β1 < 1, the second term on the RHS of (4.4) dies out eventuallyand ht+τ converges to ω/[1 − (α1 + β1)], the unconditional variance.

If we write υt = ε2t − ht and substitute ht = ε2

t − υt into (4.3), weget

ε2t − υt = ω + α1ε

2t−1 + β1ε

2t−1 − β1υt−1

ε2t = ω + (α1 + β1) ε2

t−1 + υt − β1υt−1. (4.5)

Hence, ε2t , the squared residual returns follow an ARMA process with

autoregressive parameter (α1 + β1). If α1 + β1 is close to 1, the autore-gressive process in (4.5) dies out slowly.

4.3 INTEGRATED GARCH

For a GARCH(p, q) process, when∑p

i=1 αi + ∑qj=1 β j = 1, the un-

conditional variance σ 2 → ∞ is no longer definite. The series rt is notcovariance stationary, although it remains strictly stationary and ergodic.The conditional variance is then described as an integrated GARCH (de-noted as IGARCH) and there is no finite fourth moment.1

1 This is not the same as, and should not be confused with, the ‘integrated volatility’ described in Section 1.3.3.



An infinite volatility is a concept rather counterintuitive to real phe-nomena in economics and finance. Empirical findings suggest thatGARCH(1, 1) is the most popular structure for many financial timeseries. It turns out that RiskmetricsTM EWMA (exponentially weightedmoving average) is a nonstationary version of GARCH(1, 1) where thepersistence parameters, α1 and β1, sum to 1. To see the parallel, we firstmake repeated substitution of (4.3) and obtain

ht+2 = ω + αε2t+1 + βht+1

= ω + ωβ + αε2t+1 + αβε2

t + β2ht ,

ht+τ = ω

τ∑i=1

β i−1 + α

τ∑i=1

β i−1ε2t+τ−1 + βτ ht .

When τ → ∞, and provided that β < 1 we can infer that

ht = ω

1 − β+ α

∞∑i=1

β i−1ε2t−i . (4.6)

Next, we have the EWMA model for the sample standard deviations,where

σ 2t =

(1

1 + λ + λ2 + · · · + λn

) (σ 2

t−1 + λσ 2t−1 + · · · + λnσ 2

t−n

).

As n → ∞, and provided that λ < 1

σ 2t = (1 − λ)

∞∑i=1

λi−1σ 2t−i . (4.7)

If we view ε2t as a proxy for σ 2

t , (4.6) and (4.7) are both autoregressiveseries with long distributed lags, except that (4.6) has a constant termand (4.7) has not.2

While intuitively unconvincing as a volatility process because of theinfinite variance, the EWMA model has nevertheless been shown to bepowerful in volatility forecasting as it is not constrained by a mean levelof volatility (unlike e.g. the GARCH(1, 1) model), and hence it adjustsreadily to changes in unconditional volatility.

2 EWMA, a sample standard deviation model, is usually estimated based on minimizing in-sample forecasterrors. There is no volatility error in GARCH conditional variance. This is why σ 2

t in (4.7) has a hat and ht in(4.6) has not.


Arch 41

4.4 EXPONENTIAL GARCH

The exponential GARCH (denoted as EGARCH) model is due to Nelson(1991). The EGARCH(p, q) model specifies conditional variance in log-arithmic form, which means that there is no need to impose an estimationconstraint in order to avoid negative variance;

ln ht = α0 +q∑

j=1

β j ln ht− j

+p∑

k=1

[θkεt−k + γk

(|εt−k | −

√2/π

)]εt = εt

/√ht .

Here, ht depends on both the size and the sign of εt . With appropriateconditioning of the parameters, this specification captures the stylizedfact that a negative shock leads to a higher conditional variance in thesubsequent period than a positive shock. The process is covariance sta-tionary if and only if

∑qj=1 β j < 1.

Forecasting with EGARCH is a bit involved because of the logarithmictransformation. Tsay (2002) showed how forecasts can be formulatedwith EGARCH(1, 0) and gave the one-step-ahead forecast as

ht+1 = h2α1t exp [(1 − α1) α0] exp [g (ε)]

g (ε) = θεt−1 + γ(|εt−1| −

√2/π

).

For the multi-step forecast

ht+τ = h2α1t (τ − 1) exp (ω)

{exp

[0.5 (θ + γ )2

]� (θ + γ )

+ exp[0.5 (θ − γ )2

]� (θ − γ )

},

where

ω = (1 − α1) α0 − γ√

2/π

and � (·) is the cumulative density function of the standard normal dis-tribution.

4.5 OTHER FORMS OF NONLINEARITY

Models that also allow for nonsymmetrical dependencies include theGJR-GARCH (Glosten, Jagannathan and Runkle, 1993) as shown



below:

ht = ω +p∑

i=1

βi ht−i +q∑

j=1

(α jε

2t− j + δ j D j, t−1ε

2t− j

)Dt−1 =

{1 if εt−1 < 00 if εt−1 ≥ 0

,

The conditional volatility is positive when parameters satisfy α0 > 0,αi ≥ 0, αi + γi ≥ 0 and β j ≥ 0, for i = 1, · · · , p and j = 1, · · · , q.The process is covariance stationary if and only if

p∑i=1

βi +q∑

j=1

(α j + 1

2γ j

)< 1.

Take the GJR-GARCH(1, 1) case as an example. The one-step-aheadforecast is

ht+1 = ω + β1ht + α1ε2t + δ1ε

2t Dt ,

and the multi-step forecast is

ht+τ = ω +(

1

2(α1 + γ1) + β1

)ht+τ−1

and use repeated substitution for ht+τ−1.The TGARCH (threshold GARCH) model from Zakoıan (1994) is

similar to GJR-GARCH but is formulated with absolute return instead:

σt = α0 +p∑

i=1

[αi |εt−i | + γi Di, t−i |εt−i |

] +q∑

j=1

β jσt− j . (4.8)

The conditional volatility is positive when α0 > 0, αi ≥ 0, αi + γi ≥ 0and β j ≥ 0, for i = 1, · · · , p and j = 1, · · · , q. The process is covari-ance stationary, in the case p = q = 1, if and only if

β21 + 1

2

[α2

1 + (α1 + γ1)2] + 2√

2πβ1 (α1 + γ1) < 1.

QGARCH (quadratic GARCH) and various other nonlinear GARCHmodels are reviewed in Franses and van Dijk (2000). A QGARCH(1, 1)has the following structure

ht = ω + α (εt−1 − γ )2 + βht−1.


Arch 43


Although Taylor (1986) was one of the earliest studies to test the pre-dictive power of GARCH Akigray (1989) is more commonly cited inmany subsequent GARCH studies, although an earlier investigation hadappeared in Taylor (1986). In the following decade, there were no fewerthan 20 papers that test GARCH predictive power against other timeseries methods and against option implied volatility forecasts. The ma-jority of these forecast volatility of major stock indices and exchangerates.

The ARCH class models, and their variants, have many supporters.Akgiray finds GARCH consistently outperforms EWMA and RW inall subperiods and under all evaluation measures. Pagan and Schwert(1990) find EGARCH is best, especially in contrast to some nonpara-metric methods. Despite a low R2, Cumby, Figlewski and Hasbrouck(1993) conclude that EGARCH is better than RW. Figlewski (1997)finds GARCH superiority confined to the stock market and for forecast-ing volatility over a short horizon only.

In general, models that allow for volatility asymmetry come out wellin the forecasting contest because of the strong negative relationship be-tween volatility and shock. Cao and Tsay (1992), Heynen and Kat (1994),Lee (1991) and Pagan and Schwert (1990) favour the EGARCH modelfor volatility of stock indices and exchange rates, whereas Brailsfordand Faff (1996) and Taylor, J. (2004) find GJR-GARCH outperformsGARCH in stock indices. Bali (2000) finds a range of nonlinear modelswork well for forecasting one-week-ahead volatility of US T-Bill yields.Cao and Tsay (1992) find the threshold autoregressive model (TAR inthe previous chapter) provides the best forecast for large stocks andEGARCH gives the best forecast for small stocks, and they suspect thatthe latter might be due to a leverage effect.

Other studies find no clear-cut result. These include Lee (1991),West and Cho (1995), Brailsford and Faff (1996), Brooks (1998), andMcMillan, Speight and Gwilym (2000). All these studies (and manyother volatility forecasting studies) share one or more of the followingcharacteristics: (i) they test a large number of very similar models alldesigned to capture volatility persistence, (ii) they use a large numberof forecast error statistics, each of which has a very different loss func-tion, (iii) they forecast and calculate error statistics for variance andnot standard deviation, which makes the difference between forecastsof different models even smaller, (iv) they use squared daily, weekly or



monthly returns to proxy daily, weekly or monthly ‘actual’ volatility,which results in extremely noisy ‘actual’ volatility estimates. The noisein the ‘actual’ volatility estimates makes the small differences betweenforecasts of similar models indistinguishable.

Unlike the ARCH class model, the ‘simpler’ methods, including theEWMA method, do not separate volatility persistence from volatilityshocks and most of them do not incorporate volatility mean reversion.The ‘simpler’ methods tend to provide larger volatility forecasts mostof the time because there is no constraint on stationarity or convergenceto the unconditional variance, and may result in larger forecast errorsand less frequent VaR violations. The GJR model allows the volatilitypersistence to change relatively quickly when return switches sign frompositive to negative and vice versa. If unconditional volatility of allparametric volatility models is the same, then GJR will have the largestprobability of an underforecast.3 This possibly explains why GJR wasthe worst-performing model in Franses and Van Dijk (1996) because theyuse MedSE (median standard error) as their sole evaluation criterion. InBrailsford and Faff (1996), the GJR(1, 1) model outperforms the othermodels when MAE, RMSE and MAPE are used.

There is some merit in using ‘simpler’ methods, and especially mod-els that include long distributed lags. As ARCH class models assumevariance stationarity, the forecasting performance suffers when there arechanges in volatility level. Parameter estimation becomes unstable whenthe data period is short or when there is a change in volatility level. Thishas led to a GARCH convergence problem in several studies (e.g. Tse andTung (1992) and Walsh and Tsou (1998)). Taylor (1986), Tse (1991),Tse and Tung (1992), Boudoukh, Richardson and Whitelaw (1997),Walsh and Tsou (1998), Ederington and Guan (1999), Ferreira (1999),and Taylor, J, (2004) all favour some form of exponential smoothingmethod to GARCH for forecasting volatility of a wide range of assetsranging from equities, exchange rates to interest rates.

3 This characteristic is clearly evidenced in Table 2 of Brailsford and Faff (1996). The GJR(1, 1) modelunderforecasts 76 (out of 90) times. The RW model has an equal chance of underforecasts and overforecasts,whereas all the other methods overforecast more than 50 (out of 90) times.


5Linear and Nonlinear Long

Memory Models

As mentioned before, volatility persistence is a feature that many timeseries models are designed to capture. A GARCH model features anexponential decay in the autocorrelation of conditional variances. How-ever, it has been noted that squared and absolute returns of financialassets typically have serial correlations that are slow to decay, similar tothose of an I(d) process. A shock in the volatility series seems to havevery ‘long memory’ and to impact on future volatility over a long hori-zon. The integrated GARCH (IGARCH) model of Engle and Bollerslev(1986) captures this effect, but a shock in this model impacts upon futurevolatility over an infinite horizon and the unconditional variance doesnot exist for this model.

5.1 WHAT IS LONG MEMORY IN VOLATILITY?

Let ρτ denote the correlation between xt and xt−τ . The time series xt

is said to have a short memory if∑n

τ=1 ρτ converges to a constant as nbecomes large. A long memory series has autocorrelation coefficientsthat decline slowly at a hyperbolic rate. Long memory in volatility oc-curs when the effects of volatility shocks decay slowly which is oftendetected by the autocorrelation of measures of volatility, such as abso-lute or squared returns. A long memory process is covariance stationaryif∑n

τ=1 ρτ/τ2d−1, for some positive d < 1

2 , converges to a constant asn → ∞. When d ≥ 1

2 , the volatility series is not covariance stationaryalthough it is still strictly stationary. Taylor (1986) was the first to notethat autocorrelation of absolute returns, |rt |, is slow to decay comparedwith that of r2

t . The highly popular GARCH model is a short memorymodel based on squared returns r2

t . Following the work of Granger andJoyeux (1980) and Hosking (1981), where fractionally integrated se-ries was shown to exhibit long memory property described above, Ding,Granger and Engle (1993) propose a fractionally integrated model basedon |rt |d where d is a fraction. The whole issue of Journal of Economet-rics, 1996, vol. 73, no. 1, edited by Richard Baillie and Maxwell King

45



was devoted to long memory and, in particular, fractional integratedseries.

There has been a lot of research investigating whether long memory ofvolatility can help to make better volatility forecasts and explain anom-alies in option prices. Hitherto much of this research has used the frac-tional integrated models described in Section 5.3. More recently, severalstudies have showed that a number of nonlinear short memory volatilitymodels are capable of producing spurious long memory characteris-tics in volatility as well. Examples of such nonlinear models include thebreak model (Granger and Hyung, 2004), the volatility component model(Engle and Lee, 1999), and the regime-switching model (Hamilton andSusmel, 1994; Diebold and Inoue, 2001). In these three models, volatil-ity has short memory between breaks, for each volatility componentand within each regime. Without controlling for the breaks, the differentcomponents and the changing regimes, volatility will produce spuri-ous long memory characteristics. Each of these short memory nonlinearmodels provides a rich interpretation of the financial market volatil-ity structure compared with the apparently myopic fractional integratedmodel which simply requires financial market participants to remem-ber and react to shocks for a long time. Discussion of these competingmodels is provided in Section 5.4.

5.2 EVIDENCE AND IMPACT OF VOLATILITYLONG MEMORY

The long memory characteristic of financial market volatility is wellknown and has important implications for volatility forecasting and op-tion pricing. Some evidence of long memory has already been presentedin Section 1.3. In Table 5.1, we present some statistics from a widerrange of assets and through simulation that we published in the Finan-cial Analysts Journal recently. In the table, we report the sum of the first1000 autocorrelation coefficients for a number of volatility proxies fora selection of stock indices, stocks, exchange rates, interest rates andcommodities. We have also presented the statistics for GARCH(1, 1)and GJR-GARCH(1, 1) series, both simulated using high volatility per-sistence parameters. The statistics for the simulated series are in therange of 0.478 to 2.308 while the empirical statistics are much higher.As noted by Taylor (1986), the absolute return has a longer memorythan the square returns. This has been known as the ‘Taylor effect’.But, taking logs or trimming the data by capping the values in the 0.1%


Long Memory Models 47

tails often lengthens the memory. This phenomenon continues to puzzlevolatility researchers.

The impact of volatility long memory on option pricing has beenstudied in Bollerslev and Mikkelsen (1996, 1999), Taylor (2000) andOhanissian, Russel and Tsay (2003). The effect is best understood an-alytically from the stochastic volatility option pricing model which isbased on stock having the stochastic process below:

d St = µSdt + √υt Sdzs,t ,

dυt = κ [θ − υt ] dt + σν

√υt dzυ,t ,

which, in a risk-neutral option pricing framework, becomes

dυt = κ [θ − υt ] dt − λυt dt + σν

√υt dz*

υ,t

= κ*[θ* − υt

]dt + σν

√υt dz*

υ,t , (5.1)

where υt is the instantaneous variance, κ is the speed of mean reversion, θis the long run level of volatility, σν is the ‘volatility of volatility’, λ is themarket price of (volatility) risk, and κ* = κ + λ and θ* = κθ/(κ + λ).The two Wiener processes, dzs,t and dzυ,t have constant correlation ρ.Here κ* is the risk-neutral mean reverting parameter and θ* is the risk-neutral long run level of volatility. The parameter σν and ρ implicit inthe risk-neutral process are the same as that in the real volatility process.

In the risk-neutral stochastic volatility process in (5.1), a low κ (or κ*)corresponds to strong volatility persistence, volatility long memory andhigh kurtosis. A fast, mean reverting volatility will reduce the impact ofstochastic volatility. The effect of low κ (or high volatility persistence)is most pronounced when θ the long run level is low but the initial‘instantaneous’ volatility is high as shown in the table below. The tablereports kurtosis of the simulated distribution when κ = 0.1, λ = ρ = 0.When the correlation coefficient ρ is zero, the distribution is symmetricaland has zero skewness.

υt \ θ 0.05 0.1 0.15 0.02 0.25 0.3

0.1 5.90 4.45 3.97 3.73 3.58 3.480.2 14.61 8.80 6.87 5.90 5.32 4.940.3 29.12 16.06 11.71 9.53 8.22 7.350.4 49.44 26.22 18.48 14.61 12.29 10.740.5 75.56 39.28 27.19 21.14 17.51 15.09

At low mean version κ, the option pricing impact crucially de-pends on the initial volatility, however. Figure 5.1 below presents theBlack–Scholes implied volatility inverted from simulated option prices


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0. 10490

0.1

0.20.3

0.40.5

0.60.7

0.8

50 60 70 80 90 100 110 120 13 0 140 150

Strike Price, K

BS

IVk =0.01, ÷υt =0.7 k =3, ÷υt =0.7

k =0.01, ÷υt =0.15 k =3, ÷υt =0.15

Figure 5.1 Effect of kappa(S = 100, r = 0, T = 1, λ = 0, συ = 0.6, θ = 0.2)

produced from a stochastic option pricing model. The Black–Scholesmodel is used here only to get the implied volatility which gives aclearer relative pricing relationship. The Black–Scholes implied volatil-ity (BSIV) is directly proportional to option price. First we look at thehigh volatility state where υt = 0.7. The implied volatility for κ = 0.01is higher than that forκ = 3.0, which means that a long memory volatility(slow mean reversion and high volatility persistence) will lead to a higheroption price. But, in reverse, long memory volatility will result in loweroption prices, hence lower implied volatility at low volatility state, e.g.√

υt = 0.15. So unlike the conclusion in previous studies, long memoryin volatility does not always lead to higher option prices. It is conditionedon the current level of volatility vis-a-vis the long run level of volatility.

5.3 FRACTIONALLY INTEGRATED MODEL

Both the historical volatility models and the ARCH models have beentested for fractional integration. Baillie, Bollerslev and Mikkelsen (1996)fitted FIGARCH to US dollar–Deutschemark exchange rates. Bollerslevand Mikkelsen (1996, 1999) used FIEGARCH to study S&P500volatility and option pricing impact, and so did Taylor (2000). Vilasuso(2002) tested FIGARCH against GARCH and IGARCH for volatilityprediction for five major currencies. In Andersen, Bollerslev, Dieboldand Labys (2003), a vector autoregressive model with long distributedlags was built on the realized volatility of three exchange rates, which



they called the VAR-RV model. In Zumbach (2002) the weights appliedto the time series of realized volatility follow a power law, which hecalled the LM-ARCH model. Three other papers, viz. Li (2002), Martensand Zein (2004) and Pong, Shackleton, Taylor and Xu (2004), comparedlong memory volatility model forecasts with option implied volatility. Li(2002) used ARFIMA whereas the other two papers used log-ARFIMA.Hwang and Satchell (1998) studied the log-ARFIMA model also, butthey forecast Black–Scholes ‘risk-neutral’ implied volatility of the eq-uity option instead of the underlying asset.

5.3.1 FIGARCH

The FIGARCH(1, d, 1) model below:

ht = ω + [1 − β1L − (1 − φ1L)(1 − L)d]ε2t + β1ht−1

was used in Baillie, Bollerslev and Mikkelsen (1996), and all the fol-lowing specifications are equivalent:

(1 − β1L)ht = ω + [1 − β1L − (1 − φ1L)(1 − L)d]ε2t ,

ht = ω(1 − β1)−1 + (1 − β1L)−1

×[(1 − β1L) − (1 − φ1L)(1 − L)d]ε2t ,

ht = ω(1 − β1)−1 + [1 − (1 − β1L)−1(1 − φ1L)(1 − L)d]ε2t .

For the one-step-ahead forecast

ht+1 = ω(1 − β1)−1 + [1 − (1 − β1L)−1(1 − φ1L)(1 − L)d]ε2t ,

and the multi-step-ahead forecast is

hT +τ = ω(1 − β1)−1 + [1 − (1 − β1L)−1(1 − φ1L)(1 − L)d]ε2T +τ−1.

The FIGARCH model is estimated based on the approximate maxi-mum likelihood techniques using the truncated ARCH representation.We can transform the FIGARCH model to the ARCH model with infinitelags. The parameters in the lag polynomials

λ(L) = 1 − (1 − β1L)−1(1 − φ1L)(1 − L)d

may be written as

λ1 = φ1 − β1 + d,

λk = β1λk−1 + (πk − φ1πk−1) for k ≥ 2,



where

(1 − L)d =∞∑j=0

π j L j ,

π0 = 0.

In the literature, a truncation lag at J = 1000 is common.

5.3.2 FIEGARCH

Bollerslev and Mikkelsen (1996) find fractional integrated models pro-vide better fit to S&P500 returns. Specifically, they find that frac-tionally integrated models perform better than GARCH(p, q) andIGARCH(p, q), and that FIEGARCH specification is better than FI-GARCH. Bollerslev and Mikkelsen (1999) confirm that FIEGARCHbeats EGARCH and IEGARCH in pricing options of S&P500 LEAPS(Long-term Equity Anticipation Securities) contracts. SpecificallyBollerslev and Mikkelsen (1999) fitted an AR(2)-FIEGARCH(1, d, 1)as shown below:

rt = µ + (ρ1L + ρ2L2

)rt + zt , (5.2)

ln σ 2t = ωt + (1 + ψ1L) (1 − φ1L)−1 (1 − L)−d g (εt ) ,

g (εt ) = θεt−1 + γ [|εt−1| − E |εt−1|] ,

ωt = ω + ln (1 + δNt ) .

The FIEGARCH model in (5.2) is truly a model for absolute return.Since both EGARCH and FIEGARCH provide forecasts for ln σ , toinfer forecast for σ from ln σ requires adjustment for Jensen inequalitywhich is not a straightforward task without the assumption of a normaldistribution for ln σ .

5.3.3 The positive drift in fractional integrated series

As Hwang and Satchell (1998) and Granger (2001) pointed out, positiveI(d) process has a positive drift term or a time trend in volatility levelwhich is not observed in practice. This is a major weakness of the frac-tionally integrated model for it to be adopted as a theoretically soundmodel for volatility.

All fractional integrated models of volatility have a nonzero drift.In practice the estimation of fractional integrated models require anarbitrary truncation of the infinite lags and as a result the mean willbe biased. Zumbach’s (2002) LM-ARCH will not have this problembecause of the fixed number of lags and the way in which the weights are



calculated. Hwang and Satchell’s (1998) scaled-truncated log-ARFIMAmodel is mean adjusted to control for the bias that is due to this truncationand the log transformation. The FIGARCH has a positive mean in theconditional variance equation whereas FIEGARCH has no such problembecause the lag-dependent terms have zero mean.

5.3.4 Forecasting performance

Vilasuso (2002) finds FIGARCH produces significantly better 1- and10-day-ahead volatility forecasts for five major exchange rates thanGARCH and IGARCH. Zumbach (2002) produces only one-day-aheadforecasts and find no difference among model performance. Andersen,Bollerslev, Diebold and Labys (2003) find the realized volatility con-structed VAR model, i.e. VAR-RV, produces the best 1- and 10-day-ahead volatility forecasts. It is difficult to attribute this superiorperformance to the fractional integrated model alone because the VARstructure allows a cross series linkage that is absent in all other univari-ate models and we also know that the more accurate realized volatilityestimates would result in improved forecasting performance, everythingelse being equal.

The other three papers that compare forecasts from LM models withimplied volatility forecasts generally find implied volatility forecast toproduce the highest explanatory power. Martiens and Zein (2004) findlog-ARFIMA forecast beats implied in S&P500 futures but not in ¥/US$and crude oil futures. Li (2002) finds implied produces better short hori-zon forecast whereas the ARFIMA provides better forecast for a 6-monthhorizon. However, when regression coefficients are constrained to beα = 0 and β = 1, the regression R2 becomes negative at long horizon.From our discussion in Section 2.4, this suggests that volatility at the6-month horizon might be better forecast using the unconditional vari-ance instead of model-based forecasts. Pong, Shackleton, Taylor and Xu(2004) find implied volatility to outperform time series volatility modelsincluding the log-ARFIMA model in forecasting 1- to 3-month-aheadvolatility of the dollar-sterling exchange rate.

Many of the fractional integration papers were written more recentlyand used realized volatilities constructed from intraday high-frequencydata. When comparison is made with option implied volatility, the im-plied volatility is usually extracted from daily closing option prices,however. Despite the lower data frequency, implied appears to outper-form forecasts from LM models that use intraday information.



5.4 COMPETING MODELS FOR VOLATILITY LONGMEMORY

Fractionally integrated series is the simplest linear model that produceslong memory characteristics. It is also the most commonly used andtested model in the literature for capturing long memory in volatility.There are many other nonlinear short memory models that exhibit spu-rious long memory in volatility, viz. break, volatility component andregime-switching models. These three models, plus the fractional inte-grated model, have very different volatility dynamics and produce verydifferent volatility forecasts.

The volatility breaks model permits the mean level of volatility tochange in a step function through time with some weak constraint onthe number of breaks in the volatility level. It is more general than thevolatility component model and the regime switching model. In the caseof the volatility component model, the mean level is a slowly evolvingprocess. For the regime-switching model, the mean level of volatilitycould differ according to regimes the total number of which is usuallyconfined to a small number such as two or three.

5.4.1 Breaks

A break process can be written as

Vt = mt + ut ,

where ut is a noise variable and mt represents occasional level shifts.mt are controlled by qt (a zero–one indicator for the presence of breaks)and ηt (the size of jump) such that

mt = mt−1 + qtηt = m0 +t∑

i=1

qiηi ,

qt ={

0, with probability 1 − p1, with probability p

.

The expected number of breaks for a given sample is Tp where Tis the total number of observations. Provided that p converges to zeroslowly as the sample size increases, i.e. p → 0 as T → ∞, such thatlimT →∞ Tp is a nonzero constant, Granger and Hyung (2004) showedthat the integrating parameter, I (d), is a function of Tp. While d is



bounded between 0 and 1, the expected value of d is proportionate tothe number of breaks in the series.

One interesting empirical finding on the volatility break model comesfrom Aggarwal, Inclan and Leal (1999) who use the ICSS (integratedcumulative sums of squares) algorithm to identify sudden shifts in thevariance of 20 stock market indices and the duration of such shifts.They find most volatility shifts are due to local political events. Whendummy variables, indicating the location of sudden change in variance,were fitted to a GARCH(1,1) model, most of the GARCH parameters be-came statistically insignificant. The GARCH(1,1) with occasional breakmodel can be written as follows:

ht = ω1 D1 + · · · + ωR+1 DR+1 + α1ε2t−1 + β1ht−1,

where D1, · · · , DR+1 are the dummy variables taking the value 1 ineach regime of variance, and zero elsewhere. The one-step-ahead andmulti-step-ahead forecasts are

ht+1 = ωR+1 + α1ε2t + β1ht ,

ht+τ = ωR+1 + (α1 + β1)ht+τ−1.

In estimating the break points using the ICSS algorithms, a minimumlength between breaks is needed to reduce the possibility of any tempo-rary shocks in a series being mistaken as break.

5.4.2 Components model

Engle and Lee (1999) proposed the component GARCH (CGARCH)model whereby the volatility process is modelled as the sum of a perma-nent process, mt , that has memory close to a unit root, and a transitorymean reverting process, ut , that has a more rapid time decay. The modelcan be seen as an extension of the GARCH(1,1) model with the con-ditional variance mean-revert to a long term trend level, mt , instead ofa fixed position at σ . Specifically, mt is permitted to evolve slowly inan autoregressive manner. The CGARCH(1,1) model has the followingspecification:

(ht − mt ) = α(ε2

t−1 − mt−1) + β (ht−1 − mt−1) ≡ ut , (5.3)

mt = ω + ρmt−1 + ϕ(ε2

t−1 − ht−1),

where (ht − mt ) = ut represents the short-run transitory component andmt represents a time-varying trend or permanent component in volatility



which is driven by volatility prediction error(ε2

t−1 − ht−1)

and is inte-grated if ρ = 1.

For the one-step-ahead forecast

ht+1 = qt+1 + α(ε2

t − qt) + β (ht − qt ) ,

qt+1 = ω + ρqt + ϕ(ε2

t − ht),

and for the multi-step-ahead forecast

ht+τ = qt+τ − (α + β)qt+τ−1 + (α + β)ht+τ ,

qt+τ = ω + ρqt+τ−1,

where ht+τ and qt+τ−1 are calculated through repeat substitutions.This model has various interesting properties: (i) both mt and ut are

driven by(ε2

t−1 − ht−1); (ii) the short-run volatility component mean-

reverts to zero at a geometric rate of (α + β) if 0 < (α + β) < 1; (iii)the long-run volatility component evolves over time following an ARprocess and converge to a constant level defined by ω/ (1 − ρ) if 0 <

ρ < 1; (iv) it is assumed that 0 < (α + β) < ρ < 1 so that the long-runcomponent is more persistent than the short-run component.

This model was found to obey several economic and asset pricingrelationships. Many have observed and proposed that the volatility per-sistence of large jumps is shorter than shocks due to ordinary newsevents. The component model allows large shocks to be transitory. In-deed Engle and Lee (1999) establish that the impact of the October 1987crash on stock market volatility was temporary. The expected risk pre-mium, as measured by the expected amount of returns in excess of therisk-free interest rate, in the stock market was found to be related to thelong-run component of stock return volatility.1 The authors suggested,but did not test, that such pricing relationship may have fundamen-tal economic explanations. The well-documented ‘leverage effect’ (orvolatility asymmetry) in the stock market (see Black, 1976; Christie,1982; Nelson, 1991) is shown to have a temporary impact; the long-runvolatility component shows no asymmetric response to market changes.

The reduced form of Equation (5.3) can be expressed as aGARCH(2,2) process below:

ht = (1 − α − β) ω + (α + ϕ) ε2t−1 + [−ϕ (α + β) − αρ] ε2

t−2

+ (ρ + β − ϕ) ht−1 + [ϕ (α + β) − βρ] ht−2,

1 Merton (1980) and French, Schwert and Stambaugh (1987) also studied and measured the relationshipsbetween risk premium and ‘total’ volatility.



with all five parameters, α, β, ω, ϕ and ρ, constraint to be positive andreal, 0 < (α + β) < ρ < 1, and 0 < ϕ < β.

5.4.3 Regime-switching model

One approach for modelling changing volatility level and persistenceis to use a Hamilton (1989) type regime-switching (RS) model, whichlike GARCH model is strictly stationary and covariance stationary. BothARCH and GARCH models have been implemented with a Hamilton(1989) type regime-switching framework, whereby volatility persistencecan take different values depending on whether it is in high or low volatil-ity regimes. The most generalized form of regime-switching model isthe RS-GARCH(1, 1) model used in Gray (1996) and Klaassen (1998)

ht, St−1 = ωSt−1 + αSt−1ε2t−1 + βSt−1 ht−1, St−1

where St indicates the state of regime at time t .It has long been argued that the financial market reacts to large and

small shocks differently and the rate of mean reversion is faster for largeshocks. Friedman and Laibson (1989), Jones, Lamont and Lumsdaine(1998) and Ederington and Lee (2001) all provide explanations andempirical support for the conjecture that volatility adjustment in highand low volatility states follows a twin-speed process: slower adjust-ment and more persistent volatility in the low volatility state and fasteradjustment and less volatility persistence in the high volatility state.

The earlier RS applications, such as Pagan and Schwert (1990) andHamilton and Susmel (1994) are more rigid, where conditional vari-ance is state-dependent but not time-dependent. In these studies, onlyARCH class conditional variance is entertained. Recent extensions byGray (1996) and Klaassen (1998) allow GARCH-type heteroscedas-ticity in each state and the probability of switching between states tobe time-dependent. More recent advancement is to allow more flexi-ble switching probability. For example, Peria (2001) allowed the tran-sition probabilities to vary according to economic conditions with theRS-GARCH model below:

rt | �t−1 N (µi , hit ) w.p. pit ,

hit = ωi + αiε2t−1 + βi ht−1.

where i represents a particular regime, ‘w.p.’ stands for with probability,pit = Pr ( St = i | �t−1) and

∑pit = 1.



The STGARCH (smooth transition GARCH) model below was testedin Taylor, J. (2004)

ht = ω + (1 − F (εt−1)) αε2t−1 + F (εt−1) δε2

t−1 + βht−1,

where

F (εt−1) = 1

1 + exp (−θεt−1)for logistic STGARCH,

F (εt−1) = 1 + exp(−θε2

t−1

)for exponential STGARCH.

5.4.4 Forecasting performance

The TAR model used in Cao and Tsay (1992) is similar to a SV modelwith regime switching, and Cao and Tsay (1992) reported better forecast-ing performance from TAR than EGARCH and GARCH. Hamilton andSusmel (1994) find regime-switching ARCH with leverage effect pro-duces better volatility forecast than the asymmetry version of GARCH.Hamilton and Lin (1996) use a bivariate RS model and find stock marketreturns are more volatile during a period of recession. Gray (1996) fitsa RSGARCH (1,1) model to US 1-month T-Bill rates, where the rateof mean level reversion is permitted to differ under different regimes,and finds substantial improvement in forecasting performance. Klaassen(1998) also applies RSGARCH (1,1) to the foreign exchange market andfinds a superior, though less dramatic, performance.

It is worth noting that interest rates are different to the other assetsin that interest rates exhibit ‘level’ effect, i.e. volatility depends on thelevel of the interest rate. It is plausible that it is this level effect that Gray(1996) is picking up that result in superior forecasting performance. Thislevel effect also appears in some European short rates (Ferreira, 1999).There is no such level effect in exchange rates and so it is not surprisingthat Klaassen (1998) did not find similar dramatic improvement. Noother published forecasting results are available for break and componentvolatility models.


6

Stochastic Volatility

The stochastic volatility (SV) model is, first and foremost, a theoreticalmodel rather than a practical and direct tool for volatility forecast. Oneshould not overlook the developments in the stochastic volatility area,however, because of the rapid advancement in research, noticeably byOle Barndorff-Nielsen and Neil Shephard. As far as implementation isconcerned, the SV estimation still poses a challenge to many researchers.Recent publications indicate a trend towards the MCMC (Monte CarloMarkov Chain) approach. A good source of reference for the MCMCapproach for SV estimation is Tsay (2002). Here we will provide only anoverview. An early survey of SV work is Ghysels, Harvey and Renault(1996) but the subject is rapidly changing. A more recent SV book isShephard (2003). The SV models and the ARCH models are closelyrelated and many ARCH models have SV equivalence as continuous timediffusion limit (see Taylor, 1994; Duan, 1997; Corradi, 2000; Flemingand Kirby, 2003).

6.1 THE VOLATILITY INNOVATION

The discrete time SV model is

rt = µ + εt ,

εt = zt exp (0.5ht ) ,

ht = ω + βht−1 + υt ,

where υt may or may not be independent of zt . We have already seen thiscontinuous time specification in Section 5.2, and it will appear again inChapter 9 when we discuss stochastic volatility option pricing models.

The SV model has an additional innovative term in the volatility dy-namics and, hence, is more flexible than ARCH class models. It has beenfound to fit financial market returns better and has residuals closer to stan-dard normal. Modelling volatility as a stochastic variable immediatelyleads to fat tail distributions for returns. The autoregressive term in thevolatility process introduces persistence, and the correlation between

59



the two innovative terms in the volatility process and the return pro-cess produces volatility asymmetry (Hull and White, 1987, 1988). Longmemory SV models have also been proposed by allowing the volatilityprocess to have a fractional integrated order (see Harvey, 1998).

The volatility noise term makes the SV model a lot more flexible, butas a result the SV model has no closed form, and hence cannot be esti-mated directly by maximum likelihood. The quasi-maximum likelihoodestimation (QMLE) approach of Harvey, Ruiz and Shephard (1994) is in-efficient if volatility proxies are non-Gaussian (Andersen and Sorensen,1997). The alternatives are the generalized method of moments (GMM)approach through simulations (Duffie and Singleton, 1993), or analyt-ical solutions (Singleton, 2001), and the likelihood approach throughnumerical integration (Fridman and Harris, 1998) or Monte Carlo in-tegration using either importance sampling (Danielsson, 1994; Pitt andShephard, 1997; Durbin and Koopman, 2000) or Markov chain (e.g.Jacquier, Polson and Rossi, 1994; Kim, Shephard and Chib, 1998). Inthe following section, we will describe the MCMC approach only.

6.2 THE MCMC APPROACH

The MCMC approach to modelling stochastic volatility was made pop-ular by authors such as Jacquier, Polson and Rossi (1994). Tsay (2002)has a good description of how the algorithm works. Consider here thesimplest case:

rt = at ,

at =√

htεt ,

ln ht = α0 + α1 ln ht−1 + vt , (6.1)

where εt ∼ N (0, 1), vt ∼ N (0, σ 2ν ) and εt and vt are independent.

Let w = (α0, α1, σ2ν )′. Let R = (r1, · · · , rn)′ be the collection of n ob-

served returns, and H = (h1, · · · , hn)′ be the n-dimension unobservableconditional volatilities. Estimation of model (6.1) is made complicatedbecause the likelihood function is a mixture over the n-dimensional Hdistribution as follows:

f (R | w) =∫

f (R | H ) · f (H | w) d H.

The objective is still maximizing the likelihood of {at}, but the densityof R is determined by H which in turn is determined by w.


Stochastic Volatility 61

Assuming that prior distributions for the mean and the volatility equa-tions are independent, the Gibbs sampling approach to estimating model(6.1) involves drawing random samples from the following conditionalposterior distributions:

f (β | R, X, H, w), f (H | R, X, β, w) and f (w | R, X, β, H )

This process is repeated with updated information till the likelihood tol-erance or the predetermined maximum number of iterations is reached.

6.2.1 The volatility vector H

First, the volatility vector H is drawn element by element

f (ht |R, H−t , w )

∝ f (at |ht , rt ) f (ht |ht−1, w ) f (ht+1 |ht , w )

∝ h−0.5t exp

(− rt

2

2ht

)· h−1

t exp

[− (ln ht − µt )2

2σ 2

](6.2)

where

µt =(

1

1 + α21

)[α0(1 − α1) + α1(ln ht+1 + ln ht−1)] ,

σ 2 = σ 2ν

1 + α21

Equation (6.2) can be obtained using results for a missing value in anAR(1) model. To see how this works, start from the volatility equation

ln ht = α0 + α1 ln ht−1 + at ,

α0 + α1 ln ht−1︸︷︷︸yt

= 1︸︷︷︸xt

× ln ht −at︸︷︷︸bt

,

yt = xt ln ht + bt , (6.3)

and for t + 1

ln ht+1 − α0 = α1 + ln ht + at+1,

yt+1 = xt+1 ln ht + bt+1. (6.4)

Given that bt and bt+1 have the same distribution because at is alsoN (0, σ 2

ν ), ln ht can be estimated from (6.3) and (6.4) using the least



squares principle,

ln ht = xt yt + xt+1 yt+1

x2t + x2

t+1

= α0(1 − α1) + α1(ln ht+1 + ln ht−1)

1 + α21

.

This is the conditional mean µt in Equation (6.2). Moreover, ln ht isnormally distributed

ln ht ∼ N

(ln ht ,

σ 2ν

1 + α21

), or

∼ N (µt , σ2)

6.2.2 The parameter w

First partition w as α = (α0, α1)′ and σ 2ν . The conditional posterior dis-

tributions are:

(i) f (α∣∣R, H, σ 2

ν ) = f (α∣∣H, σ 2

ν ). Note that ln ht has an AR(1) struc-ture. Hence, if prior distribution of α is multivariate normal, α ∼M N (α0, C0), then posterior distribution f (α

∣∣H, σ 2ν |) is also mul-

tivariate normal with mean α* and covariance C*, where

C−1* = 1

σ 2ν

(n∑

t=2

zt z′t

)+ C−1

0 ,

α* = C*

[1

σ 2ν

(n∑

t=2

zt ln ht

)+ C−1

0 α0

], and

zt = (1, ln ht )′ . (6.5)

(ii) f (σ 2ν |R, H, α ) = f (σ 2

ν | H, α). If the prior distribution of σ 2ν is

mλ/σ 2ν ∼ χ2

m , then the conditional posterior distribution of σ 2ν is an

inverted chi-squared distribution with m + n − 1 degrees of free-dom, i.e.

1

σ 2ν

(mλ +

n∑t=2

ν2t

)∼ χ2

m+n−1, and

νt = ln ht − α0 − α1 ln ht−1 for t = 2, · · · , n. (6.6)


Stochastic Volatility 63

Tsay (2002) suggested using the ARCH model parameter estimates asthe starting value for the MCMC simulation.


In a PhD thesis, Heynen (1995) finds SV forecast is best for a numberof stock indices across several continents. There are only six other SVstudies and the view about SV forecasting performance is by no meansunanimous at the time of writing this book.

Heynen and Kat (1994) forecast volatility for seven stock indices andfive exchange rates they find SV provides the best forecast for indicesbut produces forecast errors that are 10 times larger than EGARCH’s andGARCH’s for exchange rates. Yu (2002) ranks SV top for forecastingNew Zealand stock market volatility, but the margin is very small, partlybecause the evaluation is based on variance and not standard deviation.Lopez (2001) finds no difference between SV and other time seriesforecasts using conventional error statistics. All three papers have the1987s crash in the in-sample period, and the impact of the 1987 crashon the result is unclear.

Three other studies, Bluhm and Yu (2000), Dunis, Laws and Chau-vin (2000) and Hol and Koopman (2002) compare SV and other timeseries forecasts with option implied volatility forecast. Dunis, Laws andChauvin (2000) find combined forecast is the best for six exchange ratesso long as the SV forecast is excluded. Bluhm and Yu (2000) rank SVequal to GARCH. Both Bluhm and Yu (2000) and Hol and Koopman(2002) conclude that implied is better than SV for forecasting stockindex volatility.


7

Multivariate Volatility Models

At the time of writing this book, there was no volatility forecastingcontest that is based on the multivariate volatility model. However, therehave been a number of studies that examined cross-border volatilityspillover in stock markets (Hamao, Masulis and Ng, 1989; King andWadhwani, 1990; Karolyi, 1995; Koutmos and Booth, 1995), exchangerates (Baillie, Bollerslev and Redfearn, 1993; Hong, 2001), and interestrates (Tse and Booth, 1996). The volatility spillover relationships arepotential source of information for volatility forecasting, especially inthe very short term and during global turbulent periods. Several vari-ants of multivariate ARCH models have existed for a long time whilemultivariate SV models are fewer and more recent. Truly multivariatevolatility models (i.e. beyond two or three returns variables) are not easyto implement. The greatest challenges are parsimony, nonlinear relation-ships between parameters, and keeping the variance–covariance matrixpositive definite. In the remainder of this short chapter, I will just illus-trate one of the more recent multivariate ARCH models that I use a lotin my research. It is the asymmetric dynamic covariance (ADC) modeldue to Kroner and Ng (1998). I must admit that I have not used ADCto fit more than three variables! The ADC model encompasses manyolder multivariate ARCH models as we will explain later. Readers whoare interested in multivariate SV models could refer to Liesenfeld andRichard (2003).

7.1 ASYMMETRIC DYNAMIC COVARIANCEMODEL

In implementing multivariate volatility of returns from different coun-tries, the adjustment for time zone differences is important. In this sec-tion, I rely much on my joint work with Martin Martens that was pub-lished in the Journal of Banking and Finance in 2001. The model we

65



used is presented below

rt = µ + εt + Mεt−1, εt ∼ N (0, Ht ) ,

hiit = θiit,

hijt = ρijt

√hiit

√hjjt + φijθijt,

θijt = ωijt + b′i Ht−1b j + a′

iεt−1ε′t−1a j + +g′

iηt−1η′t−1g j .

The matrix M is used for adjusting nonsynchronous returns. It has non-zero elements only in places where one market closes before another,except in the case of the USA where the impact could be delayed till thenext day. So Japan has an impact on the European markets but not theother way round. Europe has an impact on the USA market and the USAhas an impact on the Japanese market on the next day. The conditionalvariance–covariance matrix Ht has different specifications for diagonal(conditional variance hiit) and off-diagonal (conditional covariance hijt)elements.

Consider the following set of conditions:

(i) ai = αi ei and bi = βi ei ∀i , where ei is the i th column of an (n, n)identity matrix, and αi and βi , i = 1, · · · , n, are scalars.

(ii) A = α(ωλ′) and B = β(ωλ′) where A = (a1, · · · , an), B =(b1, · · · , bn), ω and λ are (n, 1) vectors and α and β are scalars.

The ADC model reduces to:

(i) a restricted VECH model of Bollerslev, Engle and Wooldridge(1988) if ρ12 = 0 and under condition (i) with the restrictions thatβi j = βiβ j ;

(ii) the constant correlation model of Bollerslev (1990) if φ12 = 0 andunder condition (i);

(iii) the BEKK model of Engle and Kroner (1995) if ρ12 = 0 andφ12 = 1;

(iv) the factor ARCH (FARCH) model of Engle, Ng and Rothschild(1990) if ρ12 = 0, φ12 = 1 and under condition (ii),

and, unlike most of its predecessors, it allows for volatility asymmetryin the spillover effect as well through the last term in θijt.


Multivariate Volatility Models 67

7.2 A BIVARIATE EXAMPLE

Take a two-variable case as an example;

εt =[

ε1t

ε2t

], ηt =

[min (0, ε1t )min (0, ε2t )

],

ai =[

a1i

a2i

], bi =

[b1i

b2i

], gi =

[g1i

g2i

].

The condition variance of, for example, first return is

h11t = ω11 + B11h11,t−1 + a′1εt−1ε

′t−1a1 + g′

1ηt−1η′t−1g1

= ω11 + B11h11,t−1 + a211ε

21t−1 + 2a11a21ε1t−1ε2t−1

+ a221ε

22t−1 + g2

11η21t−1 + 2g11g21η1t−1η2t−1 + g2

21η22t−1,

and similarly for the second return from, for example, another country.Here, we set B11h11,t−1 as a single element, although one could also haveB as a matrix, bringing in previous day conditional variance of returnsfrom the second country as B21h22,t−1. In the above specification, weonly allow spillover to permeate through ε2

2t−1 and assume that the impactwill then be passed on ‘internally’ through h11,t−1.

The conditional covariance hi jt = h12t is slightly more complex. Itaccommodates both constant and time-varying components as follows:

h12t = ρ12

√h11t

√h22t + φ12h*

12t ,

h*12t = ω12 + B12h12,t−1 + a′

1εt−1ε′t−1a2 + g′

1εt−1ε′t−1g2

= ω12 + B12h12,t−1 + a11a12ε21t−1 + a11a22ε1t−1ε2t−1

+ a12a21ε1t−1ε2t−1 + a21a22ε22t−1 + g11g12η

21t−1

+ g11g22η1t−1η2t−1 + g12g21η1t−1η2t−1 + g21g22η22t−1,

with ρ12 capturing the constant correlation and h*12t , a time-varying co-

variance weighted by φ12. In Martens and Poon (2001), we calculatetime-varying correlation as

ρ12t = h12t√h11t

√h22t

.

In the implementation, we first estimate all returns as univariate GJR-GARCH and estimate the MA parameters for synchronization correctionindependently. These parameter estimates are fed into the ADC modelas starting values.



7.3 APPLICATIONS

The future of multivariate volatility models very much depends on theiruse. Their use in long horizon forecasting is restricted unless one adoptsa more parsimonious factor approach (see Sentana, 1998; Sentana andFiorentini, 2001). For capturing volatility spillover, multivariate volatil-ity models will continue to be useful for short horizon forecast andunivariate risk management. The use of multivariate volatility mod-els for estimating conditional correlation and multivariate risk man-agement will be restrictive because correlation is a linear concept anda poor measure of dependence, especially among large values (Poon,Rockinger and Tawn, 2003, 2004). There are a lot of important de-tails in the modelling of multivariate extremes of financial asset returnsand we hope to see some new results soon. It will suffice to illustratehere some of these issues with a simple example on linear relationshipalone.

Let Yt be a stock return and Xt be the returns on the stock marketportfolio or another stock return from another country. The stock returnsregression gives

Yt = α + β Xt + εt , (7.1)

β = Cov (Yt , Xt )

V ar (Xt )= ρxyσy

σx, or

ρxy = βσx

σy.

If the factor loading, β, in (7.1) remains constant, then ρxy could increasesimply because σx/σy increases during the high-volatility state. This isthe main point in Forbes and Rigobon (2002) who claim findings inmany contagion studies are being driven by high volatility.

However, β, the factor loading, need not remain constant. One com-mon feature in financial crisis is that many returns will move togetherand jointly become more volatile. This means that indiosyncratic riskwill be small σ 2

ε → 0, and from (7.1)

σ 2y = β2σ 2

x ,

β = σy

σx, and ρxy → 1.

The difficulty in generalizing this relationship is that there are crisesthat are local to a country or a region that have no worldwide impact or


Multivariate Volatility Models 69

impact on the neighbouring country. We do not yet have a model thatwill make such a distinction, let alone one that will predict it.

The study of univariate jump risk in option pricing is a hot topicjust now. The study of the joint occurrence of jumps and multivariatevolatility models will probably ‘meet up’ in the not so-distant future.Before we understand how the large events jointly occur, the use of themultivariate volatility model on its own in portfolio risk managementwill be very dangerous. The same applies to asset allocation and portfolioformation, although the impact here is over a long horizon, and hencewill be less severely affected by joint-tail events.


8

Black–Scholes

A European-style call (put) option is a right, but not an obligation, topurchase (sell) an asset at the agreed strike price on option maturity date,T . An American-style option is a European option that can be exercisedprior to T .

8.1 THE BLACK–SCHOLES FORMULA

The Black–Scholes (BS) formula below is for pricing European call andput options:

c = S0 N (d1) − K e−rT N (d2) , (8.1)

p = K e−rT N (−d2) − S0 N (−d1) ,

d1 = ln (S0/K ) + (r + 1

2σ2)

T

σ√

T, (8.2)

d2 = d1 − σ√

T ,

N (d1) = 1√2π

∫ d1

−∞e−0.5z2

dz,

where c (p) is the price of the European call (put), S0 is the currentprice of the underlying assset, K is the strike or exercise price, r isthe continuously compounded risk-free interest rate, and T is the timeto option maturity. N (d1) is the cumulative probability distribution ofa standard normal distribution for the area below d1, and N (−d1) =1 − N (d1).

As T → 0,

d1 and d2 → ∞,

N (d1) and N (d2) → 1,

N (−d1) and N (−d2) → 0,

which means

c ≥ S0 − K , p ≥ 0 for S0 > K , (8.3)

c ≥ 0, p ≥ K − S0 for S0 < K . (8.4)

71



As σ → 0, again

N (d1) and N (d2) → 1,

N (−d1) and N (−d2) → 0.

This will lead to

c ≥ S0 − K e−rT , and (8.5)

p ≥ K e−rT − S0. (8.6)

The conditions (8.3), (8.4), (8.5) and (8.6) are the boundary conditionsfor checking option prices before using them for empirical tests. Theseconditions need not be specific to Black–Scholes. Options with marketprices (transaction or quote) violating these boundary conditions shouldbe discarded.

8.1.1 The Black–Scholes assumptions

(i) For constant µ and σ , d S = µSdt + σ Sdz.

(ii) Short sale is permitted with full use of proceeds.

(iii) No transaction costs or taxes; securities are infinitely divisible.

(iv) No dividend before option maturity.

(v) No arbitrage (i.e. market is at equilibrium).

(vi) Continuous trading (so that rebalancing of portfolio is doneinstantaneously).

(vii) Constant risk-free interest rate, r .

(viii) Constant volatility, σ .

Empirical findings suggest that option pricing is not sensitive to the as-sumption of a constant interest rate. There are now good approximatingsolutions for pricing American-style options that can be exercised earlyand options that encounter dividend payments before option maturity.The impact of stochastic volatility on option pricing is much more pro-found, an issue which we shall return to shortly. Apart from the constantvolatility assumption, the violation of any of the remaining assumptionswill result in the option price being traded within a band instead of atthe theoretical price.


Black–Scholes 73

8.1.2 Black–Scholes implied volatility

Here, we first show that

∂CBS

∂σ> 0,

which means that CBS is a monotonous function in σ and there is aone-to-one correspondence between CBS and σ .

From (8.1)

∂CBS

∂σ= S

∂ N

∂d1

∂d1

∂σ− K e−r (T −t) ∂ N

∂d2

∂d2

∂σ, (8.7)

∂ N (x)

∂x= 1√

2πe− 1

2 x2,

∂d1

∂σ= √

T − t − d1

σ,

∂d2

∂σ= √

T − t − d1

σ− √

T − t = −d1

σ.

Substitute these results into (8.7) and get

∂CBS

∂σ= S√

2πe− 1

2 d21

(√T − t − d1

σ

)+ Ke−r (T −t)

√2π

e− 12 d2

2d1

σ

= Se− 12 d2

1√

T − t√2π

+ d1

σ√

2π

×[−Se− 1

2 d21 + K e−r (T −t)e− 1

2 (d1−σ√

T −t)2]= Se− 1

2 d21√

T − t√2π

+ d1

σ√

2π

×[−Se− 1

2 d21 + K e−r (T −t)e(− 1

2 d21 +d1σ

√T −t− 1

2 σ 2(T −t))]

= Se− 12 d2

1√

T − t√2π

+ d1e− 12 d2

1

σ√

2π

×[−S + Ke(−r (T −t)+d1σ

√T −t− 1

2 σ 2(T −t))]. (8.8)

Also from (8.2)

d1σ√

T − t − 1

2σ 2 (T − t) − r (T − t) = log

(S

K

)(8.9)



and substituting this result into (8.8), we get

∂CBS

∂σ= Se− 1

2 d21√

T − t√2π

+ d1e− 12 d2

1

σ√

2π

[−S + K

S

K

]

= Se− 12 d2

1√

T − t√2π

> 0

8.1.3 Black–Scholes implied volatility smile

Given an observed European call option price Cobs for a contract withstrike price K and expiration date T , the implied volatility σiv is de-fined as the input value of the volatility parameter to the Black–Scholesformula such that

CBS (t, S; K , T ; σiv) = Cobs . (8.10)

The option implied volatility σiv is often interpreted as a market’s expec-tation of volatility over the option’s maturity, i.e. the period from t to T .We have shown in the previous section that there is a one-to-one corre-spondence between prices and implied volatilities. Since ∂CBS/∂σ > 0,the condition

Cobs = CBS (t, S; K , T ; σiv) > CBS (t, S; K , T ; 0)

means σiv > 0; i.e. implied volatility is always greater than zero. Theimplied volatilities from put and call options of the same strike priceand time to maturity are the same because of put–call parity. Tradersoften quote derivative prices in terms of σiv rather than dollar prices, theconversion to price being made through the Black–Scholes formula.

Given the true (unconditional) volatility is σ over period T . If Black–Scholes is correct, then

CBS (t, S; K , T ; σiv) = CBS (t, S; K , T ; σ )

for all strikes. That is the function (or graph) of σiv (K ) against K forfixed t, S, T and r , observed from market option prices is supposed to bea straight horizontal line. But, it is well known that the Black–Scholesσiv, differ across strikes. There is plenty of documented empirical ev-idence to suggest that implied volatilities are different across optionsof different strikes, and the shape is like a smile when we plot Black–Scholes implied volatilityσiv against strike price K , the shape is anythingbut a straight line. Before the 1987 stock market crash, σiv (K ) against


Black–Scholes 75

K was often observed to be U-shaped, with the minimum located ator near at-the-money options, K = Se−r (T −t). Hence, this gives rise tothe term ‘smile effect’. After the stock market crash in 1987, σiv (K )is typically downward sloping at and near the money and then curvesupward at high strikes. Such a shape is now known as a ‘smirk’. Thesmile/smirk usually ‘flattens’ out as T gets longer. Moreover, impliedvolatility from option is typically higher than historical volatility andoften decreases with time to maturity.

Since ∂CBS/∂σ > 0, the smile/smirk curve, tells us that there is a pre-mium charged for options at low strikes (OTM puts and ITM calls seefootnote 2) above their BS price as compared with the ATM options. Al-though the market uses Black–Scholes implied volatility, σiv, as pricingunits, the market itself prices options as though the constant volatilitylognormal model fails to capture the probabilities of large downwardstock price movements and so supplement the Black–Scholes price toaccount for this.

8.1.4 Explanations for the ‘smile’

There are at least two theoretical explanations (viz. distributional as-sumption and stochastic volatility) for this puzzle. Other explanationsthat are based on market microstructure and measurement errors (e.g.liquidity, bid–ask spread and tick size) and investor risk preference (e.g.model risk, lottery premium and portfolio insurance) have also been pro-posed. In the next chapter on option pricing using stochastic volatility,we will explain how violation of distributional assumption and stochasticvolatility could induce BS implied volatility smile. Here, we will con-centrate on understanding how Black–Scholes distributional assumptionproduces volatility smile. Before we proceed, we need to make use ofthe positive relationship between volatility and option price, and theput–call parity1

ct + K er (T −t) = pt + St (8.11)

which establishes the positive relationship between call and put optionprices. Since implied volatility is positively related to option price, Equa-tion (8.11) suggests there is also a positive relationship between impliedvolatilities derived from call and put options that have the same strikeprice and the same time to maturity.

1 The discussion here is based on Hull (2002).



As mentioned before, Black–Scholes requires stock price to follow alognormal distribution or the logarithmic stock returns to have a normaldistribution. There is now widely documented empirical evidence thatrisky financial asset returns have leptokurtic tails. In the case where thestrike price is very high, the call option is deep-out-of-the-money2 andthe probability for this option to be exercised is very low. Nevertheless,a leptokurtic right tail will give this option a higher probability, thanthat from a normal distribution, for the terminal asset price to exceed thestrike price and the call option to finish in the money. This higher prob-ability leads to a higher call price and a higher Black–Scholes impliedvolatility at high strike.

Next, we look at the case when the strike price is low. First notethat option value has two components: intrinsic value and time value.Intrinsic value reflects how deep the option is in the money. Time valuereflects the amount of uncertainty before the option expires, hence itis most influenced by volatility. A deep-in-the-money call option hashigh intrinsic value and little time value, and a small amount of bid–askspread or transaction tick size is sufficient to perturb the implied volatilityestimation. We could, however, make use of the previous argument butapply it to an out-of-the-money (OTM) put option at low strike price. AnOTM put option has a close to nil intrinsic value and the put option priceis due mainly to time value. Again because of the thicker tail on the left,we expect the probability that the OTM put option finishes in the moneyto be higher than that for a normal distribution. Hence the put optionprice (and hence the call option price through put–call parity) should begreater than that predicted by Black–Scholes. If we use Black–Scholes toinvert volatility estimates from these option prices, the Black–Scholesimplied will be higher than actual volatility. This results in volatilitysmile where implied volatility is much higher at very low and very highstrikes.

The above arguments apply readily to the currency market where ex-change rate returns exhibit thick tail distributions that are approximatelysymmetrical. In the stock market, volatility skew (i.e. low implied at highstrike but high implied at low strike) is more common than volatilitysmile after the October 1987 stock market crash. Since the distribution

2 In option terminology, an option is out of the money when it is not profitable to exercise the option. For acall option, this happens when S < X , and in the case of a put, the condition is S > X . The reverse is true for anin-the-money option. A call or a put is said to be at the money (ATM) when S = X . A near-the-money option isan option that is not exactly ATM, but close to being ATM. Sometimes, discounted values of S and X are usedin the conditions.


Black–Scholes 77

is skewed to the far left, the right tail can be thinner than the normaldistribution. In this case implied volatility at high strike will be lowerthan that expected from a volatility smile.

8.2 BLACK–SCHOLES AND NO-ARBITRAGE PRICING

8.2.1 The stock price dynamics

The Black–Scholes model for pricing European equity options assumesthe stock price has the following dynamics:

dS = µSdt + σ Sdz, (8.12)

and for the growth rate on stock:

dS

S= µdt + σdz. (8.13)

From Ito’s lemma, the logarithm of the stock price has the followingdynamics:

d ln S =(

µ − 1

2σ 2

)dt + σdz, (8.14)

which means that the stock price has a lognormal distribution or thelogarithm of the stock price has a normal distribution. In discrete time

d ln S =(

µ − 1

2σ 2

)dt + σdz,

� ln S =(

µ − 1

2σ 2

)�t + σ ε

√�t,

ln ST − ln S0 ∼ N

[(µ − 1

2σ 2

)T, σ

√T

],

ln ST ∼ N

[ln S0 +

(µ − 1

2σ 2

)T, σ

√T

]. (8.15)

8.2.2 The Black–Scholes partial differential equation

The derivation of the Black–Scholes partial differential equation (PDE)is based on the fundamental fact that the option price and the stock pricedepend on the same underlying source of uncertainty. A portfolio canthen be created consisting of the stock and the option which eliminatesthis source of uncertainty. Given that this portfolio is riskless, it must



therefore earn the risk-free rate of return. Here is how the logic works:

�S = µS�t + σ S�z, (8.16)

� f =[

∂ f

∂SµS + ∂ f

∂t+ 1

2

∂2 f

∂S2σ 2S2

]�t + ∂ f

∂Sσ S�z. (8.17)

We set up a hedged portfolio, �, consisting of ∂ f /∂S number of sharesand short one unit of the derivative security. The change in portfoliovalue is

�� = −� f + ∂ f

∂S�S

= −[

∂ f

∂SµS + ∂ f

∂t+ 1

2

∂2 f

∂S2σ 2S2

]�t − ∂ f

∂Sσ S�z

+∂ f

∂SµS�t + ∂ f

∂Sσ S�z

= −[∂ f

∂t+ 1

2

∂2 f

∂S2σ 2S2

]�t.

Note that uncertainty due to �z is cancelled out and µ, the premium forrisk (returns on S), is also cancelled out. Not only has �� no uncer-tainty, it is also preference-free and does not depend on µ, a parametercontrolled by the investor’s risk aversion.

If the portfolio value is fully hedged, then no arbitrage implies that itmust earn only a risk-free rate of return

r��t = ��,

r��t = −� f + ∂ f

∂S�S,

r

(− f + ∂ f

∂SS

)�t = −

[∂ f

∂SµS + ∂ f

∂t+ 1

2

∂2 f

∂S2σ 2S2

]�t − ∂ f

∂Sσ S�z

+∂ f

∂S[µS�t + σ S�z] ,

r (− f ) �t = −r S∂ f

∂S�t − ∂ f

∂SµS�t − ∂ f

∂t�t − 1

2

∂2 f

∂S2σ 2S2�t

−∂ f

∂Sσ S�z + ∂ f

∂SµS�t + ∂ f

∂Sσ S�z,

and finally we get the well-known Black–Scholes PDE

r f = r S∂ f

∂S+ ∂ f

∂t+ 1

2

∂2 f

∂S2σ 2S2 (8.18)


Black–Scholes 79

8.2.3 Solving the partial differential equation

There are many solutions to (8.18) corresponding to different derivatives,f , with underlying asset S. In other words, without further constraints,the PDE in (8.18) does not have a unique solution. The particular securitybeing valued is determined by its boundary conditions of the differentialequation. In the case of an European call, the value at expiry c (S, T ) =max (S − K , 0) serves as the final condition for the Black–Scholes PDE.Here, we show how BS formula can be derived using the risk-neutralvaluation relationship. We need the following facts:

(i) From (8.15),

ln S ∼ N

(ln S0 + µ − 1

2σ 2, σ

).

Under risk-neutral valuation relationship, µ = r and

ln S ∼ N

(ln S0 + r − 1

2σ 2, σ

).

(ii) If y is a normally distributed variable,∫ ∞

aey f (y) dy = N

(µy − a

σy+ σy

)eµy+ 1

2 σ 2y .

(iii) From the definition of cumulative normal distribution,∫ ∞

af (y) dy = 1 − N

(a − µy

σy

)= N

(µy − a

σy

).

Now we are ready to solve the BS formula. First, the terminal value ofa call is

cT = E [max (S − K , 0)]

=∫ ∞

K(S − K ) f (S) d S

=∫ ∞

ln Keln S f (ln S) d ln S − K

∫ ∞

ln Kf (ln S) d ln S.

Substituting facts (ii) and (iii) and using information from (i) with

µy = ln S0 + r − 1

2σ 2,

σy = σ,

a = ln K ,



we get

cT = S0er N

(ln S0 + r + 1

2σ2 − ln K

ln K

)

− K N

(ln S0 + r − 1

2σ2 − ln K

ln K

)= S0er N (d1) − K N (d2) , (8.19)

where

d1 = ln S0/K + r − 12σ

2

σ,

d2 = d1 − σ.

The present value of the call option is derived by applying e−r to bothsides. The put option price can be derived using put–call parity or usingthe same argument as above. The σ in the above formula is volatilityover the option maturity. If we use σ as the annualized volatility thenwe replace σ with σ

√T in the formula.

There are important insights from (8.19), all valid only in a ‘risk-neutral’ world:

(i) N (d2) is the probability that the option will be exercised.

(ii) Alternatively, N (d2) is the probability that call finishes in themoney.

(iii) X N (d2) is the expected payment.

(iv) S0erT N (d1) is the expected value E [ST − X ]+, where E [·]+ isexpectation computed for positive values only.

(v) In other words, S0erT N (d1) is the risk-neutral expectation of ST ,E Q [ST ] with ST > X .

8.3 BINOMIAL METHOD

In a highly simplified example, we assume a stock price can only moveup by one node or move down by one node over a 3-month period asshown below. The option is a call option for the right to purchase theshare at $21 at the end of the period (i.e. in 3 month’s time).


Black–Scholes 81

stock price = 20option price = c

��

�

��

�

stock price = 22option price = 1

stock price = 18option price = 0

Construct a portfolio consisting of � amount of shares and short onecall option. If we want to make sure that the value of this portfolio is thesame whether it is up state or down state, then

$22 × � − $1 = $18 × � + $0,

� = 0.25.

stock price = 20

port f olio value= 4.5e−0.12×3/12

= 4.367

��

�

��

�

stock price = 22port f olio value = 22 × 0.25 − 1 = 4.5

stock price = 18port f olio value = 18 × 0.25 = 4.5

Given that the portfolio’s value is $4.367, this means that

$20 × 0.25 − f = $4.367

f = $0.633.

This is the value of the option under no arbitrage.From the above simple example, we can make the following general-

ization;

S0

f�

��

��

��

��

S0ufu

S0dfd



The amount � is calculated using

S0u × � − fu = S0d × � − fd,

� = fu − fd

S0u − S0d. (8.20)

Since the terminal value of the ‘riskless’ portfolio is the same in the upstate and in the down state, we could use any one of the values (say theup state) to establish the following relationship

S0 × � − f = (S0u × � − fu) e−rT ,

f = S0 × � − (S0u × � − fu) e−rT . (8.21)

Substituting the value of � from (8.20) into (8.21), we get

f = S0 × fu − fd

S0u − S0d−

(S0u × fu − fd

S0u − S0d− fu

)e−rT

= fu − fd

u − d−

(u × fu − fd

u − d− fu

)e−rT

=(

erT ( fu − fd)

u − d− u ( fu − fd)

u − d+ u fu − d fu

u − d

)e−rT

=(

erT fu − erT fd

u − d+ u fd − d fu

u − d

)e−rT

=(

erT − d

u − dfu + u − erT

u − dfd

)e−rT .

By letting p = (erT − d)/(u − d), we get

f = e−rT [p fu + (1 − p) fd] (8.22)

and

1 − p = u − d − erT + d

u − d= u − erT

u − d.

We can see from (8.22) that although p is not the real probability dis-tribution of the stock price, it has all the characteristics of a probability


Black–Scholes 83

measure (viz. sum to one and nonnegative). Moreover, when the ex-pectation is calculated based on p, the expected terminal payoff is dis-counted using the risk-free interest. Hence, p is called the risk-neutralprobability measure.

We can verify that the underlying asset S also produces a risk-freerate of returns under this risk-neutral measure.

S0eµT =(

erT − d

u − d

)S0u +

(u − erT

u − d

)S0d,

eµT = uerT − ud + ud − derT

u − d,

= (u − d) erT

u − d= erT ,

µ = r.

The actual return of the stock is no longer needed and neither is theactual distribution of the terminal stock price. (This is a rather amazingdiscovery in the study of derivative securities!!!)

8.3.1 Matching volatility with u and d

We have already seen in the previous section and Equation (8.22) thatthe risk-neutral probability measure is set such that the expected growthrate is the risk-free rate, r .

f =(

erT − d

u − dfu + u − erT

u − dfd

)e−rT

= [p fu + (1 − p) fd] e−rT ,

p = erT − d

u − d.

This immediately leads to the question of how does one set the valuesof u and d? The key is that u and d are jointly determined such that thevolatility of the binomial process equal to σ which is given or can beestimated from prices of the asset underlying the option contract. Given



that there are two unknowns and there is only one constant σ , thereare a number of ways to specify u and d. The good or better ways arethose that guarantee the nodes recombined after an upstate followed bya downstate, and vice versa. In Cox, Ross and Rubinstein (1979), u andd are defined as follows:

u = eσ√

δt , and d = e−σ√

δt .

It is easy to verify that the nodes recombine since ud = du = 1. Soafter each up move and down move (and vice versa), the stock price willreturn to S0.

To verify that the volatility of stock returns is approximately σ√

δtunder the risk-neutral measure, we note that

Var = E[x2] − [E (x)]2 ,

ln u = σ√

δt, and ln d = −σ√

δt .

The expected stock returns is

E (x) = p lnS0u

S0+ (1 − p) ln

S0d

S0

= pσ√

δt − (1 − p) σ√

δt

= (2p − 1) σ√

δt,

and

E[x2] = p

(ln

S0u

S0

)2

+ (1 − p)

(ln

S0d

S0

)2

= pσ 2δt + (1 − p) σ 2δt

= σ 2δt.

Hence

Var = σ 2δt − (2p − 1)2 σ 2δt

= σ 2δt(1 − 4p2 + 4p − 1

)= σ 2δt × 4p (1 − p) .

It has been shown elsewhere that as δt → 0, p → 0.5 and Var → σ 2δt .


Black–Scholes 85

8.3.2 A two-step binomial tree and American-style options

S0

f

p1

1 − p1

��

��

��

��

S0ufu

p2

1 − p2

��

��

��

��

S0dfd

p2

1 − p2

��

��

��

��

S0u2

fuu

S0udfud

S0d2

fdd

The binomial tree is often constructed in such a way that the branchesrecombine. If the volatilities in period 1 and period 2 are different, then,in order to make the binomial tree recombine, p1 �= p2. (This is a moreadvanced topic in option pricing.) Here, we take the simple case wherevolatility is constant, and p1 = p2 = p. Hence, to price a Europeanoption, we simply take the expected terminal value under the risk-neutralmeasure and discount it with a risk-free interest rate, as follows:

f = e−r×2δt[

p2 fuu + 2p (1 − p) fud + (1 − p)2 fdd]. (8.23)

Note that the hedge ratio for state 2 will be different depending onwhether state 1 is an up state or a down state

�0 = fu − fd

S0u − S0d,

�1,u = fuu − fud

S0u2 − S0ud,

�1,d = fud − fdd

S0ud − S0d2.

This also means that, for such a model to work in practice, one has tobe able to continuously and costlessly rebalance the composition of theportfolio of stock and option. This is a very important assumption andshould not be overlooked.



We can see from (8.23) that the intermediate nodes are not requiredfor the pricing of European options. What are required are the range ofpossible values for the terminal payoff and the risk-neutral probabilitydensity for each node. This is not the case for the American option and allthe nodes in the intermediate stages are needed because of the possibilityof early exercise. As the number of nodes increases, the binomial treeconverges to a lognormal distribution for stock price.

8.4 TESTING OPTION PRICING MODELIN PRACTICE

Let C = f (K , S, r, σ, T − t) denote the theoretical (or model) priceof the option and f is some option pricing model, e.g. fBS denotesthe Black–Scholes formula. At any one time, we have options of manydifferent strikes, K , and maturities, T − t . (Here we use t and T as dates;t is now and T is option maturity date. So the time to maturity is T − t .)Since σt1 and σt2 need not be the same for t1 �= t2, we tend to use onlyoptions with the same maturity T − t because volatility itself has a termstructure. Assuming that there are Cobs

1 , Cobs2 and Cobs

3 observed optionprices (possibly these are market-traded option prices) associated withthree exercise prices K1, K2 and K3. To find the theoretical option priceC , we need the five parameters K , S, r, σ and T − t . Except for σ , theother four parameters K , S, r and T − t can be determined accuratelyand easily. We could estimateσ from historical stock prices. The problemwith this approach is that when C �= Cobs (i.e. the model price is not thesame as the market price), we do not know if this is because we did notestimate σ properly or because the option pricing model f (·) is wrong.A better approach is to use ‘backward induction’, i.e. use an iterativeprocedure to find the σ that minimizes the pricing errors

C1 − Cobs1 , C2 − Cobs

2 , and C3 − Cobs3 .

The above is usually done by minimizing the unsigned errors

n∑i=1

Wi

∣∣Ci − Cobsi

∣∣m (8.24)

with an optimization routine searching over all possible values of σ ; nis the number of observed option prices (three in this case), and Wi isthe weight applied to observation i .


Black–Scholes 87

In the simplest case, Wi = 1 for all i . To give the greatest weight tothe ATM option, we could set

Wi =

10,000 for S = K e−r (T −t)

1∣∣∣∣ S

Xe−r (T −t)− 1

∣∣∣∣ for S �= K e−r (T −t).

Wi = 10 000 is equivalent to an option price that is 0.0001 away frombeing at the money.

The power term, m, is the control for large pricing error. The largerthe value of m, the greater the emphasis placed on large errors for errors> 1. If a very large error is due to data error, then a large m means theentire estimation will be driven by this data error. Typically, m is setequal to 1 or 2 corresponding to ‘absolute errors’ and ‘squared errors’ .

Since the option price is much greater for an ITM option than foran OTM option, the pricing error is likely to be of a greater magnitudefor an ITM option. Hence, for an ITM option and an OTM option thatare an equal distance from being ATM, the procedure in (8.24) willplace a greater weight on pricing an ITM option correctly and pay littleor negligible attention to OTM options. One way to overcome this isto minimize their Black–Scholes implied volatilities instead. Here, weare using BS as a conversion tool. So long as ∂C BS/∂σ > 0 and thereis a one-to-one correspondence between option price and BS impliedvolatility. Such a procedure does not require the assumption that the BSmodel is correct.

To implement the new procedure, we start with an initial value σ*and get C1, C2 and C3 from f , the option pricing model that we wishto test. f could even be Black–Scholes, if it is our intention to testBlack–Scholes. Use the Black–Scholes model fBS to invert BS impliedvolatility I V1, I V2 and I V3 from the theoretical prices C1, C2 and C3

calculated in the previous step. If f in the previous step is indeed Black–Scholes, then I V1 = I V2 = I V3 = σ*. Use the Black–Scholes modelfBS to invert BS implied volatility I V1

obs , I V2obs and I V3

obs from themarket observed option prices C1

obs , C2obs and Cobs

3 .Finally, minimize the function

n∑i=1

Wi

∣∣IVi − IVobsi

∣∣musing the algorithm and logic as before.



8.5 DIVIDEND AND EARLY EXERCISE PREMIUM

As option holders are not entitled to dividends, the option price shouldbe adjusted for known dividends to be distributed during the life of theoption and the fact that the option holder may have the right to exerciseearly to receive the dividend.

8.5.1 Known and finite dividends

Assume that there is only one dividend at τ . Should the call optionholder decide to exercise the option, she will receive Sτ − K at time τ

and if she decides not to exercise the option, her option value will beworth c(Sτ − Dτ , K , r, T, σ ). The Black (1975) approximation involvesmaking such comparisons for each dividend date. If the decision is not toexercise, then the option is priced now at c

(St − Dτ e−r (τ−t), K , r, T, σ

).

If the decision is to exercise, then the option is priced according toc (St , K , r, τ, σ ). We note that if the decision is not to exercise, theAmerican call option will have the same value as the European call optioncalculated by removing the discounted dividend from the stock price.

A more accurate formula that takes into account of the probability ofearly exercise is that by Roll (1977), Geske (1979), and Whaley (1981),and presented in Hull (2002, appendix 11). These formulae (even theBlack-approximation) work quite well for American calls. In the caseof an American put, a better solution is to implement the Barone-Adesiand Whaley (1987) formula (see Section 8.5.3).

8.5.2 Dividend yield method

When the dividend is in the form of yield it can be easily ‘netted off’from the risk-free interest rate as in the case of a currency option. Tocalculate the dividend yield of an index option, the dividend yield, q, isthe average annualized yield of dividends distributed during the life ofthe option:

q = 1

tln

S +n∑

i=1Di er (t−ti )

S

where Di and ti are the amount and the timing of the i th dividend onthe index with ti should also be annualized in a similar fashion as t . The


Black–Scholes 89

dividend yield rate computed here is thus from the actual dividends paidduring the option’s life which will therefore account for the monthlyseasonality in dividend payments.

8.5.3 Barone-Adesi and Whaley quadratic approximation

Define M = 2rσ 2 and N = 2 (r − q)/σ 2, then3 for an American call op-

tion

C (S) = c (S) + A2

(S

S*

)q2

when S < S*

S − K when S ≥ S*.(8.25)

The variable S* is the critical price of the index above which the optionshould be exercised. It is estimated by solving the equation,

S* − K = c(S*

) + {1 − e−qt N

[d1

(S*

)]} S*

q2,

iteratively. The other variables are

q2 = 1

2

[1 − N +

√(N − 1)2 + 4M

1 − e−r t

],

A2 = S*

q2

{1 − e−qt N

[d1

(S*

)]},

d1(S*

) = ln(S*/K

) + (r − q + 0.5σ 2)t

σ√

t. (8.26)

To compute delta and vega for hedging purposes4:

�C = ∂C

∂S=

{e−qt N(d1 (S)) + A2 q2

S*

(S

S*

)(q2−1)when S < S*

1 when S ≥ S*,

�C = ∂C

∂σ=

{S√

t N′ (d1) e−qt when S < S*0 when S ≥ S*.

(8.27)

3 Note that in Barone-Adesi and Whaley (1987), K (t) is(1 − e−r t

), and b is (r − q).

4 Vega for the American options cannot be evaluated easily because C partly depends on S*, which itself is acomplex function of σ. The expression for vega in the case when S < S* in Equation (8.27) represents the vegafor the European component only. Vega for the American option could be derived using numerical methods.



For an American put option, the valuation formula is:

P (S) ={

p (S) + A1

(S

S**

)q1

when S > S**

K − S when S ≤ S**.(8.27a)

The variable S** is the critical index price below which the option shouldbe exercised. It is estimated by solving the equation,

K − S** = p(S**

) − {1 − e−qt N

[−d1(S**

)]} S**

q1,

iteratively. The other variables are

q1 = 1

2

[1 − N −

√(N − 1)2 + 4M

1 − e−r t

],

A1 = − S**

q1

{1 − e−qt N

[−d1(S**

)]},

d1(S**

) = ln(S**/K

) + (r − q + 0.5σ 2)t

σ√

t.

To compute delta and vega for hedging purposes:

�P = ∂ P

∂S=−e−qt N (d1 (S)) + A1 q1

S**

(S

S**

)(q1−1)

when S > S**

−1 when S ≤ S**,

�P = ∂ P

∂σ=

{∂C

∂σ= S

√t N′ (d1) e−qt when S > S**

0 when S ≤ S**.

8.6 MEASUREMENT ERRORS AND BIAS

Early studies of option implied volatility suffered many estimation prob-lems,5 such as the improper use of the Black–Scholes model for anAmerican style option, the omission of dividend payments, the optionprice and the underlying asset prices not being recorded at the sametime, or stale prices being used. Since transactions may take place at bidor ask prices, transaction prices of the option and the underlying assetsare subject to bid–ask bounce making the implied volatility estimation

5 Mayhew (1995) gives a detailed discussion on such complications involved in estimating implied volatilityfrom option prices, and Hentschel (2001) provides a discussion of the confidence intervals for implied volatilityestimates.


Black–Scholes 91

unstable. Finally, in the case of an S&P100 OEX option, the privilegeof a wildcard option is often omitted.6 In more recent studies, many ofthese measurement errors have been taken into account. Many studiesuse futures and options futures because these markets are more activethan the cash markets and hence there is a smaller risk of prices beingstale.

Conditions in the Black–Scholes model include: no arbitrage, trans-action cost is zero and continuous trading. As mentioned before, thelack of such a trading environment will result in options being tradedwithin a band around the theoretical price. This means that impliedvolatility estimates extracted from market option prices will also liewithin a band even without the complications described in Chapter 10.Figlewski (1997) shows that implied volatility estimates can differ byseveral percentage points due to bid–ask spread and discrete tick sizealone. To smooth out errors caused by bid–ask bounce, Harvey and Wha-ley (1992) use a nonlinear regression of ATM option prices, observed ina 10-minute interval before the market close, on model prices.

Indication of nonideal trading environment is usually reflected in poortrading volume. This means implied volatility of options written ondifferent underlying assets will have different forecasting power. Formost option contracts, ATM option has the largest trading volume. Thissupports the popularity of ATM implied volatility referred to later inChapter 10.

8.6.1 Investor risk preference

In the Black–Scholes world, investor risk preference is irrelevant inpricing options. Given that some of the Black–Scholes assumptions havebeen shown to be invalid, there is now a model risk. Figlewski andGreen (1999) simulate option writers positions in the S&P500, DM/$,US LIBOR and T-Bond markets using actual cash data over a 25-yearperiod. The most striking result from the simulations is that delta hedgedshort maturity options, with no transaction costs and a perfect knowledgeof realized volatility, finished with losses on average in all four markets.This is clear evidence of Black–Scholes model risk. If option writersare aware of this model risk and mark up option prices accordingly, theBlack–Scholes implied volatility will be greater than the true volatility.

6 This wildcard option arises because the stock market closes later than the option market. The option traderis given the choice to decide, before the stock market closes, whether or not to trade on an option whose price isfixed at an earlier closing time.



In some situations, investor risk preference may override the risk-neutral valuation relationship. Figlewski (1997), for example, comparesthe purchase of an OTM option to buying a lottery ticket. Investors arewilling to pay a price that is higher than the fair price because they likethe potential payoff and the option premium is so low that mispricingbecomes negligible. On the other hand, we also have fund managers whoare willing to buy comparatively expensive put options for fear of thecollapse of their portfolio value. Both types of behaviour could causethe market price of options to be higher than the Black–Scholes price,translating into a higher Black–Scholes implied volatility. Arbitrage ar-guments do not apply here because these are unique risk preferences(or aversions) associated with some groups of individuals. Franke, Sta-pleton and Subrahmanyam (1998) provide a theoretical framework inwhich such option trading behaviour may be analysed.

8.7 APPENDIX: IMPLEMENTING BARONE-ADESIAND WHALEY’S EFFICIENT ALGORITHM

The determination of S* and S** in Equations (8.25) and (8.27a) arenot exactly straightforward. We have some success in solving S* andS** using NAG routing C05NCF. Barone-Adesi and Whaley (1987),however, have proposed an efficient method for determining S*, detailsof which can be found in Barone-Adesi and Whaley (1987, hereafterreferred to as BAW) pp. 309 to 310. BAW claimed that convergence ofS* and S** can be achieved with three or fewer iterations.

American calls

The following are step-by-step procedures for implementing BAW’sefficient method for estimating S∗ of the American call.

Step 1. Make initial guess of σ and denote this initial guess as σ j withj = 1.

Step 2. Make initial guess of S*, Si (with i = 1), as follow; denotingS* at T = +∞ as S* (∞) :

S1 = X + [S* (∞) − K

] [1 − eh2

], (8.28)

where

S* (∞) = K

1 − 1

q2 (∞)

, (8.29)


Black–Scholes 93

q2 (∞) = 1

2

[1 − N +

√(N − 1)2 + 4M

], (8.30)

h2 = −(

(r − q) t + 2σ√

t){

K

S* (∞) − K

}. (8.31)

Note that the lower bound of S* is K . So if S1 < K , resetS1 = K . However, the condition S* < K rarely occurs.

Step 3. Compute the l.h.s. and r.h.s. of Equation (8.25a) as follows:

l.h.s. (Si ) = Si − X, and (8.32)

r.h.s. (Si ) = c (Si ) + {1 − e−qt N [d1 (Si )]

}Si/q2. (8.33)

Compute starting value of c (Si ) using the simple Black–ScholesEquation (8.25) and d1 (Si ) using Equation (8.26). It will beuseful to set up a function (or subroutine variable) for d1.

Step 4. Check tolerance level,

|l.h.s. (Si ) − r.h.s. (Si )| /K < 0.000 01. (8.34)

Step 5. If Equation (8.34) is not satisfied; compute the slope of Equation(8.33), bi , and the next guess of S*, Si+1, as follows:

bi = e−qt N [d1 (Si )] (1 − 1/q2)

+[1 − e−qt n[d1 (Si )]/σ

√t]/q2, (8.35)

Si+1 = [X + r.h.s. (Si ) − bi Si ] / (1 − bi ) , (8.36)

where n (.) is the univariate normal density function. Repeatfrom step 3.

Step 6. When Equation (8.34) is satisfied, compute C (S) according toEquation (8.25). If C (S) is greater than the observed Americancall price, try a smaller σ j+1, otherwise try a larger σ j+1. Repeatsteps 1 to 5 until C (S) is the same as the observed Americancall price. Step 6 could be handled by a NAG routine such asC05ADF for a quick solution.

American puts

To approximate S** for American puts, steps 2, 3 and 5 have to bemodified.

Step 1. Make initial guess of σ and denote this initial guess as σ j withj = 1.



Step 2. Make initial guess of S**, Si (with i = 1), as follows, denotingS** at T = +∞ as S** (∞):

S1 = S** (∞) + [K − S** (∞)

]eh1, (8.37)

where

S** (∞) = K

1 − 1

q1 (∞)

,

q1 (∞) = 1

2

[1 − N −

√(N − 1)2 + 4M

],

h1 =(

(r − q) t − 2σ√

t){

K

K − S** (∞)

}. (8.38)

Note that the upper bound of S** is K . So if S1 > K , resetS1 = K . Again, the condition S** > X rarely occurs. Accord-ing to Barone-Adesi and Whaley (1987, footnote 9), the influ-ence of (r − q) must be bounded in the put exponent to ensurecritical prices monotonically decrease in t , for very large val-ues of (r − q) and t . A reasonable bound on (r − q) is 0.6σ

√t ,

so the critical stock price declines with a minimum velocitye−1.4σ

√t . This check is required before computing h1, in Equa-

tion (8.38).Step 3. Compute the l.h.s. and r.h.s. of Equation (8.37) as follows:

l.h.s. (Si ) = K − Si , and

r.h.s. (Si ) = p (Si ) − {1 − e−qt N [−d1 (Si )]

}Si/q1. (8.39)

Step 4. Check tolerance level, as before,

|l.h.s. (Si ) − r.h.s. (Si )| /K < 0.00001. (8.40)

Step 5. If Equation (8.40) is not satisfied; compute the slope of Equation(8.39), bi , and the next guess of S**, Si+1, as follows:

bi = −e−qt N [−d1 (Si )] (1 − 1/q1)

−[1 + e−qt n [d1 (Si )] /σ

√t]/q1,

Si+1 = [X − r.h.s. (Si ) + bi Si ] / (1 + bi ) .

Repeat from step 3 above.


Black–Scholes 95

Step 6. When Equation (8.40) is satisfied, compute P (S) using Equa-tion (8.27a). If P (S) is greater than the observed American callprice, try a larger σ j+1, otherwise try a smaller σ j+1. Then repeatsteps 1 to 5 until P (S) is the same as the observed American putprice. Similarly the case for the American call, step 6 could behandled by a NAG routine such as C05ADF for a quick solution.


9Option Pricing with Stochastic

Volatility

If Black–Scholes (BS) is the correct option pricing model, then therecan only be one BS implied volatility regardless of the strike price ofthe option, or whether the option is a call or a put. BS implied volatilitysmile and skew are clear evidence that market option prices are not pricedaccording to the BS formula. This raises the important question aboutthe relationship between BS implied volatility and the true volatility.

The BS option price is a positive function of the volatility of theunderlying asset. If the BS model is correct, then market option priceshould be the same as the BS option price and the BS implied volatilityderived from market option price will be the same as the true volatility.If the BS price is incorrect and is lower than the market price, then BSimplied volatility overstates the true volatility. The reverse is true if theBS price is higher than the market price. The problem is complicatedby the fact that BS implied volatility differs across strike prices. All thetheories that predict the relationship between BS price and the marketoption price are all contingent on the proposed alternative option pricingmodel or the proposed alternative pricing dynamic being correct. Giventhat the BS implied volatility, despite all its shortcomings, has beenproven overwhelmingly to be the best forecast of volatility, it will beuseful to understand the links between BS implied volatility bias andthe true volatility. This is the objective of this chapter.

There have been a lot of efforts made to solve the BS anomalies.The stochastic volatility (SV) option pricing model is one of the mostimportant extensions of Black–Scholes. The SV option pricing modelis motivated by the widespread evidence that volatility is stochastic andthat the distribution of risky asset returns has tail(s) longer than that ofa normal distribution. An SV model with correlated price and volatilityinnovations can address both anomalies. The SV option pricing modelwas developed roughly over a decade with contributions from Johnsonand Shanno (1987), Wiggins (1987), Hull and White (1987, 1988), Scott(1987), Stein and Stein (1991) and Heston (1993). It was in Heston(1993) that a closed form solution was derived using the characteristic

97



function of the price distribution. Section 9.1 presents this landmarkHeston SV option pricing model while some details of the derivation arepresented in the Appendix to this chapter (Section 9.5). In Section 9.2,we simulate a series of Heston option prices from a range of parameters.Then we use these option prices as if they were the market option pricesto back out the corresponding BS implied volatilities. If market optionprices are priced according to the Heston formula, the simulations inthis section will give us some insight into the relationship between BSimplied volatility bias and the true volatility. In Section 9.3, we analysethe usefulness and practicality of the Heston model by looking at theimpact of Heston model parameters on skewness and kurtosis range andsensitivity, and some empirical tests of Heston model. Finally, Section9.4 analyses empirical findings on the the predictive power of Hestonimplied volatility as a volatility forecast.

9.1 THE HESTON STOCHASTIC VOLATILITYOPTION PRICING MODEL

Heston (1993) specifies the stock price and volatility price processes asfollows:

d St = µSdt + √υt Sdzs,t ,

dυt = κ [θ − υt ] dt + σν

√υt dzυ,t ,

where υt is the instantaneous variance, κ is the speed of mean reversion,θ is the long-run level of volatility and σν is the ‘volatility of volatility’.The two Wiener processes, dzs,t and dzυ,t have constant correlation ρ.The assumption that consumption growth has a constant correlation withspot-asset returns generates a risk premium proportional to υt . Given thevolatility risk premium, the risk-neutral volatility process can be writtenas

dυt = κ [θ − υt ] dt − λυt dt + σν

√υt dz*

υ,t

= κ*[θ* − υt

]dt + σν

√υt dz*

υ,t ,

where λ is the market price of (volatility) risk, and κ* = κ + λ andθ* = κθ/(κ + λ). Here κ* is the risk-neutral mean reverting parameterand θ* is the risk-neutral long-run level of volatility. The parameter σν

and ρ implicit in the risk-neutral process are the same as that in thereal volatility process. Given the price and the volatility dynamics, the


Option Pricing with Stochastic Volatility 99

Heston (1993) formula for pricing European calls is

c = S P1 − K e−r (T −t) P2,

Pj = 1

2+ 1

π

∫ ∞

0Re

[e−iφ ln K fi

iφ

]dφ, for j = 1, 2

fi = exp {C (T − t ; φ) + D (T − t ; φ) υ + iφx} ,

where

x = ln S, τ = T − t,

C (τ ; φ) = rφiτ + a

σ 2ν

{(b j − ρσνφi + d

)τ − 2 ln

[1 − gedτ

1 − g

]},

D (τ ; φ) = b j − ρσνφi + d

σ 2ν

[1 − edτ

1 − gedτ

],

g = b j − ρσνφi + d

b j − ρσνφi − d,

d =√(

ρσνφi − b j)2 − σ 2

ν

(2µφi − φ2

),

µ1 = 1/

2 , µ2 = −1/

2 , a = κθ = κ*θ*,

b1 = κ + λ − ρσν = κ* − ρσν, b2 = κ + λ = κ*.

9.2 HESTON PRICE AND BLACK–SCHOLES IMPLIED

In this section, we analyse possible BS implied bias by simulating aseries of Heston option prices with parameter values similar to those inBakshi, Cao and Chen (1997), Nandi (1998), Das and Sundaram (1999),Bates (2000), Lin, Strong and Xu (2001), Fiorentini, Angel and Rubio(2002) and Andersen, Benzoni and Lund (2002). For the simulations, weset the asset price as 100, interest rate as zero, time to maturity is 1 year,and strike prices ranging from 50 to 150. In most simulations, and unlessotherwise stated, the current ‘instantaneous’ volatility, σt , is set equal tothe long-run level, θ , at 20%. There are five other parameters used in theHeston formula, namely, κ , the speed of mean reversion, θ , the long-runvolatility level, λ, the market price of risk, συ , volatility of volatility, andρ, the correlation between the price and the volatility processes. If weset λ = 0, then the volatility process becomes risk-neutral, and κ and θ

become κ* and θ* respectively.The first set of simulations presented in Figure 9.1(a) involves repli-

cating the Black–Scholes prices as a special case. Here we set συ = 0.



(e) Effect of l on negatively correlated processes (S = 100, r = 0, T = 1, r = -0.5)

Skewness = +0.1623, Kurtosis = 3.7494

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

50 60 70 80 90 100 110 120 130 140 150

Strike Price, K

BS

Im

plie

d

l = -2 l = 0 l = 2

(f) Effect of l on positively correlated processes (S = 100, r = 0, T = 1, r = 0.5)

Skewness = -0.1623, Kurtosis = 3.7494

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

50 60 70 80 90 100 110 120 130 140 150

Strike Price, K

BS

Im

plie

d

l = -2 l = 0 l = 2

(c) Effect of correlation, r

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

50 60 70 80 90 100 110 120 130 140 150

Strike Price, K

BS

Im

plie

d

r = -0.95 r = -0.5

r = 0 r = 0.6

(d) Effect of k(S = 100, r = 0, T = 1, l = 0, su = 0.6, q = 0.2)

0.10490

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

50 60 70 80 90 100 110 120 130 140 150

Strike Price, K

BS

Im

plie

dk = 0.01, √υt = 0.7 k = 3, √υt = 0.7

k = 0.01, √υt = 0.15 k = 3, √υt = 0.15

(a) A Black–Scholes series(S = 100, r = 0, T = 1, l = 0)

(S = 100, r = 0, T = 1, l = 0)

Skewness = 0, Kurtosis = 3

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

50 60 70 80 90 100 110 120 130 140 150

Strike Price, K

BS

Im

plie

d(b) Effect of volatility of volatility (su, Kurtosis)

(S = 100, r = 0, T = 1, l = 0, k = 0.1)

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

50 60 70 80 90 100 110 120 130 140 150

Strike Price, K

BS

Im

plie

d

su = 0.1 , Kur = 3.7

su = 0.3, Kur = 9.53

su = 0.6, Kur = 29.1

su = 1.0, Kur = 75.6

Figure 9.1 Relationships between Heston option prices and Black–Scholes impliedvolatility

Since there is no volatility risk, λ = 0. This is a special case wherethe Heston price and the Black–Scholes price are identical and the BSimplied volatility is the same across strike prices. In this special case,BS implied volatility (at any strike price) is a perfect representation oftrue volatility.



In the second set of simulations presented in Figure 9.1(b), we al-ter συ , the volatility of volatility, and keep all the other parameters thesame and constant. The effect of an increase in συ is to increase theunconditional volatility and kurtosis of risk-neutral price distribution. Itis the risk-neutral distribution because λ, the market price of risk, is setequal to zero. As συ increases and without appropriate compensation forvolatility risk premium, ATM (at-the-money) implied volatility under-estimates true volatility while OTM (out-of-the-money) implied volatil-ity overestimates it. This is the same outcome as Hull and White (1987)where the price and the volatility processes are not correlated and thereis no risk premium for volatility risk. With appropriate adjustment forvolatility (which will require a volatility risk premium input), the ATMimplied volatility will be at the right level, but Black–Scholes will con-tinue to underprice OTM options (and OTM BS implied overestimatestrue volatility) because of the BS lognormal thin tail assumptions. As-suming that ρ = 0 or at least is constant over time, and that συ and λ arerelatively stable, a time series regression of historical ‘actual volatility’on historical ‘implied volatility’ at a particular strike will be sufficientto correct for these biases. This is basically the Ederington and Guan(1999) approach. We will show in the next section that ρ is not likely tobe stable. When ρ is not constant, the analysis below and Figure 9.1(c)show that ATM implied volatility is least affected by changing ρ. Thisexplains why ATM implied volatility is the most robust and popularchoice of volatility forecast.

In Figure 9.1(c), it is clear that changing the correlation coefficientalone has no impact on ATM implied volatility. Correlation has thegreatest impact on skewness of the price distribution and determines theshape of volatility smile or skew. Its impact on kurtosis is less markedwhen compared with συ , the volatility of volatility.

Figure 9.1(d) highlights the impact of κ , the mean reversion parameterwhich we have already briefly touched on in relation to the long memoryof volatility in Chapter 5. The higher the rate of mean reversion, the morelikely the return distribution will be normal even when the volatility ofvolatility, συ , and the initial volatility,

√νt , are both high. When this is

the case, there is no strike price bias in BS implied (i.e. there will not bevolatility smile). When κ is low, this is when the problem starts. A low κ

corresponds with volatility persistence where BS implied volatility willbe sensitive to the current state of volatility level. At high volatility state,high

√νt compensates for the lowκ and the strike price bias is less severe.

Strike price effect or the volatility smile is the most acute when initialvolatility level

√νt is low. ATM options will be overpriced vis-a-vis



OTM options.1 (Note that we have set λ = 0 in this set of simulations.)When συ = 0.6, θ = 0.2,

√νt = 0.15 and κ = 0.01, the ATM BS im-

plied is only 0.1049 much lower than any of the volatility parameters.Figure 9.1(e) and 9.1(f) can be used to infer the impacts of parameter

estimates above when the volatility risk premium λ is omitted. In theliterature, we often read ‘. . . volatility risk premium is negative reflect-ing the negative correlation between the price and the volatility dyna-mics . . . ’ (Buraschi and Jackwerth, 2001; Bakshi and Kapadia, 2003.All series in Figure 9.1(e) have correlation ρ = −0.5 and all series inFigure 9.1(f) have correlation ρ = +0.5. A negative λ (volatility riskpremium) produces higher Heston price and higher BS implied volatil-ity. The impact is the same whether the correlation ρ is negative orpositive. We will see later in the next section that empirical evidence in-dicates that Figure 9.1(f) is just as likely a scenario as Figure 9.1(e). Asκ* = κ + λ and θ* = κθ/(κ + λ), a negative λ has the effect of reduc-ing κ* (resulting in a smaller option price) and increasing θ* (resultingin a bigger option price). Simulations, not reported here, show that theprice impact of θ* is much greater than that of κ*, so the outcome willbe a higher option price due to the negative λ. Hence, a ‘negative riskpremium’ is to be expected whether the price and the volatility processesare positively or negatively correlated.2 This also means that, withoutaccounting for the volatility risk premium, the BS option price will betoo low and the BS implied will always overstate true volatility. Bothvolatility and volatility risk premium have positive impact on optionprice. The omission of volatility risk premium will cause the volatil-ity risk premium component to be ‘translated’ into higher BS impliedvolatility.

9.3 MODEL ASSESSMENT

In this section, we evaluate the Heston model using simulations. In par-ticular, we examine the skewness and kurtosis planes covered by a rangeof Heston parameter values. We have no information on the volatilityrisk premium. Hence, to avoid an additional dimension of complexity,we will evaluate the risk-neutral parameters κ* and θ* instead of κ andθ for the true volatility process.

1 When BS overprice options, the BS implied volatility will understate volatility because BS implied isinverted from market price, which is lower than the BS price.

2 This is really a misnomer: while the λ parameter is negative, it actually results in a higher option price. Sostrictly speaking the volatility risk premium is positive!



9.3.1 Zero correlation

We learn from the simulations in Section 9.2 and from Figure 9.1 that,according to the Heston model, skewness in stock returns distributionand BS implied volatility asymmetry are determined completely by thecorrelation parameter, ρ. When the correlation parameter is equal tozero, we get zero skewness and both the returns distribution and BSimplied volatility will be symmetrical. Figure 9.2 presents the kurtosisvalues produced by different combinations of κ , θ and σv. One importantpattern emerged that highlights the importance of the mean reversionparameter, κ . When κ is low we have high volatility persistence, andvice versa for high value of κ .

At high value of κ , kurtosis is close to 3, regardless of the value of θ andσv. This is, unfortunately, the less likely scenario for a financial markettime series that typically has high volatility persistence and low valueof κ . At low value of κ , the kurtosis is the highest at low level of σv, theparameter for volatility of volatility. At high level of σv, kurtosis dropsto 3 very consistently, regardless of the value of the other parameters.At low level of σv, the long-term level of volatility, θ , comes into effect.The higher the value of θ , the lower the kurtosis value, even though it isstill much greater than 3.

When skewness is zero and kurtosis is low (i.e. relatively flat BS im-plied volatility), it will be difficult to differentiate whether it is due toa high κ , a high σv or both. This also reflects the underlying prop-erty that a high κ , a high σv or both make the stochastic volatilitystructure less important and the BS model will be adequate in thiscase.

9.3.2 Nonzero correlation

In Figures 9.3 and 9.4, we illustrate skewness and kurtosis, respectively,for the case when the correlation coefficient, ρ, is greater than 0. Thecase for ρ < 0 will not be discussed here as it is the reflective imageof ρ > 0 (e.g. instead of positive skewness, we get negative skewnessetc.).

Figure 9.3 shows that skewness is first ‘triggered’ by a nonzero cor-relation coefficient, after which κ and σv combine to drive skewness.High skewness occurs when σv is high and κ is low (i.e. high volatilitypersistence). At relatively low skewness level, there is a huge range ofhigh κ , low σv or both that produce similar values of skewness. A low θ



0.05 0.1 0.15 0.2 0.25 0.3 0.1 0.2 0.3 0.4 0.5

0

10

20

30

40

50

60

70

80

Kurtosis

q

(a) k = 1, r = 0, Skewness = 0

0.10.2

0.30.4

0.5

0.10.2

0.30.4

0.5

0.00

10.00

20.00

30.00

40.00

50.00

60.00

70.00

80.00

Kurtosis

su q

(b) k = 0.1, r = 0, Skewness = 0

su

Figure 9.2 Impact of Heston parameters on kurtosis for symmetrical distribution withzero correlation and zero skewness

and high ρ produces high skewness, but at low level of σv, skewness ismuch less sensitive to these two parameters.

Figure 9.4 gives a similar pattern for kurtosis. Except when σv is veryhigh and κ is low, the plane for kurtosis is very flat and not sensitive toθ or ρ.


Option Pricing with Stochastic Volatility 1050.

05

0.2

0.1 0.

3 0.5 0.7 0.9

0

0.5

1

1.5

2

2.5

3

3.5

Skewness

q r

k = 1, su = 0.5

0.05

0.2

0.1 0.

3 0.5 0.7 0.9

0

0.5

1

1.5

2

2.5

3

3.5

Skewness

q r

k = 1, su = 0.1

0.05

0.2

0.1 0.

3 0.5 0.7 0.9

0

0.5

1

1.5

2

2.5

3

3.5

Skewness

q r

k = 0.1, su = 0.5

0.05

0.2

0.1 0.

3 0.5 0.7 0.9

0

0.5

1

1.5

2

2.5

3

3.5

Skewness

q r

k = 0.1, su = 0.1

Figure 9.3 Impact of Heston parameters on skewness

9.4 VOLATILITY FORECAST USING THEHESTON MODEL

The thick tail and nonsymmetrical distribution found empirically couldbe a result of volatility being stochastic. The simulation results in theprevious section suggest that σv, the volatility of volatility, is the maindriving force for kurtosis and skewness (if correlation is not equal tozero). At high κ , volatility mean reversion will cancel out much ofthe σv impact on kurtosis and some of that on skewness. Correlationbetween the price and the volatility processes, ρ, determines the sign ofthe skewness. But beyond that its impact on the magnitude of skewnessis much less compared with σv and κ . Correlation has negligible impacton kurtosis. The long-run volatility level, θ , has very little impact onskewness and kurtosis, except when σv is very high and κ is very low. Soa stochastic volatility pricing model is useful and will outperform Black–Scholes only when volatility is truly stochastic (i.e. high σv) and volatilityis persistent (i.e. lowκ). The difficulty with the Heston model is that, once



0.05

0.2

0.1 0.

3 0.5 0.7 0.9

0102030405060708090

Kurtosis

q r

k = 1, su = 0.50.

05

0.2

0.1 0.

3 0.5 0.7 0.9

0102030405060708090

Kurtosis

q r

k = 1, su = 0.10.

05

0.2

0.1 0.

3 0.5 0.7 0.9

0102030405060708090

Kurtosis

q r

k = 0.1, su = 0.5

0.05

0.2

0.1 0.

3 0.5 0.7 0.9

0102030405060708090

Kurtosis

q r

k = 0.1, su = 0.1

Figure 9.4 Impact of Heston parameters on kurtosis

we move away from the high σv and low κ region, a large combinationof parameter values can produce similar skewness and kurtosis. Thiscontributes to model parameter instability and convergence difficultyduring estimation.

Through simulation results we can predict the degree of Black–Scholes pricing bias as a result of stochastic volatility. In the case wherevolatility is stochastic and ρ = 0, Black–Scholes overprices near-the-money (NTM) or at-the-money (ATM) options and the degree of over-pricing increases with maturity. On the other hand, Black–Scholes un-derprices both in- and out-of-the-money options. In term of impliedvolatility, ATM implied volatility will be lower than actual volatilitywhile implied volatility of far-from-the-money options (i.e. either veryhigh or very low strikes) will be higher than actual volatility. The patternof pricing bias will be much harder to predict if ρ is not zero, when thereis a premium for bearing volatility risk, and if either or both values varythrough time.



Some of the early work on option implied volatility focuses on find-ing an optimal weighting scheme to aggregate implied volatility of op-tions across strikes. (See Bates (1996) for a comprehensive survey ofthese weighting schemes.) Since the plot of implied volatility againststrikes can take many shapes, it is not likely that a single weightingscheme will remove all pricing errors consistently. For this reason andtogether with the liquidity argument, ATM option implied volatility isoften used for volatility forecast but not implied volatilities at otherstrikes.

9.5 APPENDIX: THE MARKET PRICE OFVOLATILITY RISK

9.5.1 Ito’s lemma for two stochastic variables

Given two stochastic processes,3

d S1 = µ1 (S1, S2, t) dt + σ1 (S1, S2, t) d X1,

d S2 = µ2 (S1, S2, t) dt + σ2 (S1, S2, t) d X2,

E {d X1d X2} = ρdt,

where X1 and X2 are two related Brownian motions.From Ito’s lemma, the derivative function V (S1, S2, t) will have the

following process:

dV ={

Vt + 1

2σ 2

1 VS1 S1 + ρσ1σ2VS1 S2 + 1

2σ 2

2 VS2 S2

}dt

+VS1d S1 + VS2d S2,

where

Vt = ∂V

∂t, VS1 S1 = ∂2V

∂S21

and VS1 S2 = ∂

∂S1

(∂V

∂S2

).

9.5.2 The case of stochastic volatility

Here, we assume S1 is the underlying asset and S2 is the stochasticvolatility σ1 as follows:

d S1 = µ1 (S1, σ, t) dt + σ1 (S1, σ, t) d X1, (9.1)

dσ1 = p (S1, σ, t) dt + q (S1, σ, t) d X2,

E {d X1d X2} = ρdt,

3 I am grateful to Konstantinos Vonatsos for helping me with materials presented in this section.



and from Ito’s lemma, we get:

dV ={

Vt + 1

2σ 2

1 VS1 S1 + ρqσ1VS1σ + 1

2q2VS2σ

}dt (9.2)

+VS1d S1 + Vσ dσ.

In the following stochastic volatility derivation, µ1 = µSt is the meandrift of the stock price process. The volatility of S1, σ1 = f (σ ) S1 isstochastic and is level-dependent. The mean drift of the volatility pro-cess is more complex as volatility cannot become negative and shouldbe stationary in the long run. Hence an OU (Ornstein–Uhlenbeck)process is usually recommended with p (S1, σ, t) = α (m − σ ) andq (S1, σ, t) = β:

dσ1 = α (m − σ1) dt + βd X2.

Here, p = α (m − σ ) is the mean drift of the volatility process, m isthe long-term mean level of the volatility, and α is the speed at whichvolatility reverts to m, and β is the volatility of volatility.

9.5.3 Constructing the risk-free strategy

To value an option V (S1, σ, t) we must form a risk-free portfolio usingthe underlying asset to hedge the movement in S1 and use another optionV (S1, σ, t) to hedge the movement in σ . Let the risk-free portfolio be:

� = V − �1V − �S1.

Applying Ito’s lemma from (9.2) on the risk-free portfolio �,

d� ={

Vt + 1

2σ 2

1 VS1 S1 + 1

2q2Vσσ + ρqσ1VS1σ

}dt

−�1

{V t + 1

2σ 2

1 V S1 S1 + 1

2q2V σσ + ρqσ1V S1σ

}dt

+ {VS1 − �1V S1 − �

}d S1

+ {Vσ − �1V σ

}dσ.

To eliminate dσ , set

Vσ − �1V σ = 0,

�1 = Vσ

V σ

,



and, to eliminate d S1, set

VS1 − Vσ

V σ

V S1 − � = 0,

� = VS1 − Vσ

V σ

V S1 .

This results in:

d� ={

Vt + 1

2σ 2

1 VS1 S1 + 1


}dt

− Vσ

V σ

{V t + 1

2σ1

2V S1 S1 + 1


}dt

= r�dt

= r

{V − Vσ

V σ

V −[

VS1 − Vσ

V σ

V S1

]S1

}dt.

Dividing both sides by Vσ , we get:

1

Vσ

{Vt + 1

2σ 2

1 VS1 S1 + 1


}− 1

V σ

{V t + 1

2σ 2

1 V S1 S1 + 1


}= r

V

Vσ

− rV

V σ

− rVS1 S1

Vσ

+ rV S1 S1

V σ

.

Now we separate the two options by moving V to one side and V to theother:

1

Vσ

{Vt + 1

2σ 2

1 VS1 S1 + 1

2q2Vσσ + ρqσ1VS1σ − r V + r VS1 S1

}= 1

V σ

{V t + 1

2σ 2

1 V S1 S1 + 1

2q2V σσ + ρqσ1V S1σ − r V + r V S1 S1

}.

Each side of the equation is a function of S1, σ and t , and is independentof the other option. So we may write:

1

Vσ

{Vt + 1

2σ 2

1 VS1 S1 + 1


}= f (S1, σ, t) ,

Vt + 1

2σ 2

1 VS1 S1 + 1


= f (S1, σ, t) Vσ , (9.3)



and similarly for the RHS. In order to solve the PDE, we need to under-stand the function f (S1, σ, t) which depends on whether or not d S1 anddσ are correlated.

9.5.4 Correlated processes

If two Brownian motions d X1 and d X2 are correlated with correlationcoefficient ρ, then we may write:

d X2 = ρd X1 +√

1 − ρ2dεt ,

where dεt is the part of d X2 that is not related to d X1.Now consider the hedged portfolio,

� = V − �S1, (9.4)

where only the risk that is due to the underlying asset and its correlatedvolatility risk are hedged. Volatility risk orthogonal to d S1 is not hedged.From Ito’s lemma, we get:

d� = dV − �d S1

={

Vt + 1

2σ 2

1 VS1 S1 + ρqσ1VS1σ + 1

2q2Vσσ

}dt

+VS1d S1 + Vσ dσ − �d S1. (9.5)

Now write:

d S1 = µ1dt + σ1d X1, and

dσ = pdt + qd X2

= pdt + q[ρd X1 +

√1 − ρ2dεt

].

Substitute this result into (9.5) and get:

d� ={

Vt + 1

2σ 2

1 VS1 S1 + ρqσ1VS1σ + 1

2q2Vσσ

}dt

+VS1 {µ1dt + σ1d X1} + Vσ

{pdt + q

[ρd X1 +

√1 − ρ2dεt

]}−� {µ1dt + σ1d X1}

={

Vt + 1

2σ 2

1 VS1 S1 + ρqσ1VS1σ + 1

2q2Vσσ

}dt

+ (VS1µ1 + Vσ p − �µ1

)dt + (

VS1σ1 + Vσ qρ − �σ1)

d X1

+Vσ q√

1 − ρ2dεt . (9.6)



So to get rid of d X1, the hedge ratio should be:

VS1σ1 + Vσ qρ − �σ1 = 0,

� = VS1 + Vσ qρ

σ1.

With this hedge ratio, only the uncorrelated volatility risk,Vσ q

√1 − ρ2dεt , is left in the portfolio. If ρ = 1, the portfolio � would

be risk-free.Now substitute value of � into (9.6). We get:

d� ={

Vt + 1

2σ 2

1 VS1 S1 + ρqσ1VS1σ + 1

2q2Vσσ

}dt

+(

VS1µ1 + Vσ p − VS1µ1 − Vσ qρµ1

σ1

)dt + Vσ q

√1 − ρ2dεt

={

Vt + 1

2σ 2

1 VS1 S1 + ρqσ1VS1σ + 1

2q2Vσσ

}dt

+(

Vσ p − Vσ qρµ1

σ1

)dt + Vσ q

√1 − ρ2dεt . (9.7)

9.5.5 The market price of risk

Next we made the assumption that the partially hedged portfolio � in(9.4) will earn a risk-free return plus a premium for unhedged volatilityrisk, �, such that

d� = r�dt + �

= r (V − �S1) dt + �

= r

(V − VS1 S1 − Vσ qρ

σ1S1

)dt + �. (9.8)

Substituting d� from (9.7), we get:{Vt + 1

2σ 2

1 VS1 S1 + ρqσ1VS1σ + 1

2q2Vσσ

}dt

+(

Vσ p − Vσ qρµ1

σ1

)dt + Vσ q

√1 − ρ2dεt

= r V dt − r VS1 S1dt − rVσ qρ

σ1S1dt + �,



� ={

Vt + 1

2σ 2

1 VS1 S1 + ρqσ1VS1σ + 1

2q2Vσσ − r V + r VS1 S1

}dt

+[

Vσ p + Vσ qρ

σ1(r S1 − µ1)

]dt + Vσ q

√1 − ρ2dεt .

Now replace the {} term with f (S1, σ, t) in (9.3) we get:

� =(

f Vσ + Vσ p + Vσ qρ

σ1(r S1 − µ1)

)dt + Vσ q

√1 − ρ2dεt

= Vσ q√

1 − ρ2

{[f + p + qρ

σ1(r S1 − µ1)

q√

1 − ρ2

]dt + dεt

}.

Now define ‘market price of risk’ γ :

γ =f + p + qρ

σ1(r S1 − µ1)

q√

1 − ρ2, (9.9)

where γ is the ‘returns’ associated with each unit of risk that is due to dεt

(i.e. the unhedged volatility risk), hence, the denominator q√

1 − ρ2.From (9.9), we can get an expression for f :

f = −p + qρ

σ1(µ1 − r S1) + γ q

√1 − ρ2.

Substituting p = α (m − σ ) , q = β, µ1 = µS1, and σ1 = f (σ ) S1:

f = −α (m − σ ) + βρ

f (σ ) S1(µS1 − r S1) + γ β

√1 − ρ2.

= −α (m − σ ) + βρ (µ − r )

f (σ )+ γ β

√1 − ρ2.

We can now price the option with stochastic volatility in (9.3) using theexpression for f above and get:

0 = Vt + 1

2f 2S2

t VS1 S1 + 1

2β2Vσσ + ρβ f St VS1σ − r V + r VS1 S1

+[α (m − σ ) − β

(ρ (µ − r )

f (σ )+ γ

√1 − ρ2

)]Vσ (9.10)

Now write

� (S1, σ, t) = ρ (µ − r )

f (σ )+ γ

√1 − ρ2.



We write (9.10) as

Vt + 1

2f 2S2

1 VS1 S1 + r(VS1 S1 − V

)︸︷︷︸

Black–Scholes

+ρβ f S1VS1σ︸︷︷︸correlation

+1

2β2Vσσ + α (m − σ ) Vσ︸︷︷︸

Lou

− β�Vσ︸︷︷︸premium

= 0 (9.11)

or, on rearrangement,4

Vt + 1

2f 2S2

1 VS1 S1 + r VS1 S1︸︷︷︸Black–Scholes

+ ρβ f S1VS1σ︸︷︷︸correlation

+ 1

2β2Vσσ + α (m − σ ) Vσ︸︷︷︸

Lou

= rV︸︷︷︸risk-free return as in BS

+ β�Vσ︸︷︷︸premium for volatility risk

This analysis show that when volatility is stochastic in the form in(9.1), the option price will be higher. The additional risk premium isrelated to the correlation between volatility and the stock price processesand the mean-reverting dynamic of the volatility process.

4 This result is shown in Fouque, Papanicolaou and Sircar (2000).


10

Option Forecasting Power

Option implied volatility has always been perceived as a market’sexpectation of future volatility and hence it is a market-based volatil-ity forecast. It makes use of a richer and more up-to-date informationset, and arguably it should be superior to time series volatility forecast.On the other hand, we showed in the previous two chapters that op-tion model-based forecast requires a number of assumptions to hold forthe option theory to produce a useful volatility estimate. Moreover, op-tion implied also suffers from many market-driven pricing irregularities.Nevertheless, the volatility forecasting contests show overwhelminglythat option implied volatility has superior forecasting capability, out-performing many historical price volatility models and matching theperformance of forecasts generated from time series models that use alarge amount of high-frequency data.

10.1 USING OPTION IMPLIED STANDARDDEVIATION TO FORECAST VOLATILITY

Once an implied volatility estimate is obtained, it is usually scaled by√n to get an n-day-ahead volatility forecast. In some cases, a regression

model may be used to adjust for historical bias (e.g. Ederington andGuan, 2000b), or the implied volatility may be parameterized within aGARCH/ARFIMA model with or without its own persistence adjust-ment (e.g. Day and Lewis, 1992; Blair, Poon and Taylor, 2001; Hwangand Satchell, 1998).

Implied volatility, especially that of stock options, can be quite un-stable across time. Beckers (1981) finds taking a 5-day average improvesthe forecasting power of stock option implied. Hamid (1998) finds suchan intertemporal averaging is also useful for stock index option dur-ing very turbulent periods. On a slightly different note, Xu and Taylor(1995) find implied estimated from a sophisticated volatility term struc-ture model produces similar forecasting performance as implied fromthe shortest maturity option.

115



In contrast to time series volatility forecasting models, the use of im-plied volatility as a volatility forecast involves some extra complexities.A test on the forecasting power of option implied standard deviation(ISD) is a joint test of option market efficiency and a correct option pric-ing model. Since trading frictions differ across assets, some options areeasier to replicate and hedge than the others. It is therefore reasonableto expect different levels of efficiency and different forecasting powerfor options written on different assets.

While each historical price constitutes an observation in the sampleused in calculating volatility forecast, each option price constitutes avolatility forecast over the option maturity, and there can be many optionprices at any one time. The problem of volatility smile and volatility skewmeans that options of different strike prices produce different Black–Scholes implied volatility estimates.

The issue of a correct option pricing model is more fundamentalin finance. Option pricing has a long history and various extensionshave been made since Black–Scholes to cope with dividend payments,early exercise and stochastic volatility. However, none of the optionpricing models (except Heston (1993)) that appeared in the volatilityforecasting literature allows for a premium for bearing volatility risk.In the presence of a volatility risk premium, we expect the option priceto be higher which means implied volatility derived using an optionpricing model that assumes zero volatility risk premium (such as theBlack–Scholes model) will also be higher, and hence automatically bemore biased as a volatility forecast. Section 10.3 examines the issue ofbiasedness of ISD forecasts and evaluates the extent to which impliedbiasedness is due to the omission of volatility risk premium.

10.2 AT-THE-MONEY OR WEIGHTED IMPLIED?

Since options of different strikes have been known to produce differ-ent implied volatilities, a decision has to be made as to which of theseimplied volatilities should be used, or which weighting scheme shouldbe adopted, that will produce a forecast that is most superior. The mostcommon strategy is to choose the implied derived from an ATM op-tion based on the argument that an ATM option is the most liquid andhence ATM implied is least prone to measurement errors. The analysisin Chapter 9 shows that, omitting volatility risk premium, ATM impliedis also least likely to be biased.

If ATM implied is not available, then an NTM (nearest-to-the-money)option is used instead. Sometimes, to reduce measurement errors and


Option Forecasting Power 117

the effect of bid–ask bounce, an average is taken from a group of NTMimplied volatilities. Weighting schemes that also give greater weightto ATM implied are vega (i.e. the partial derivative of option pricew.r.t. volatility) weighted or trading volume weighted, weighted leastsquares (WLS) and some multiplicative versions of these three. TheWLS method, first appeared in Whaley (1982), aims to minimize thesum of squared errors between the market and the theoretical prices of agroup of options. Since the ATM option has the highest trading volumeand the ATM option price is the most sensitive to volatility input, allthree weighting schemes (and the combinations thereof) have the ef-fect of placing the greatest weight on ATM implied. Other less popularweighting schemes include equally weighted, and weight based on theelasticity of option price to volatility.

The forecasting power of individual and composite implied volatilitieshas been tested in Ederington and Guan (2000b), Fung, Lie and Moreno(1990), Gemmill (1986), Kroner, Kneafsey and Claessens (1995), Scottand Tucker (1989) and Vasilellis and Meade (1996). The general con-sensus is that among the weighted implied volatilities, those that favourthe ATM option such as the WLS and the vega weighted implied arebetter. The worst performing ones are equally weighted and elastic-ity weighted implied using options across all strikes. Different findingsemerged as to whether an individual implied volatility forecasts betterthan a composite implied. Beckers (1981) Feinstein (1989b), Fung, Lieand Moreno (1990) and Gemmill (1986) find evidence to support indi-vidual implied although they all prefer a different implied (viz. ATM,Just-OTM, OTM and ITM respectively for the four studies). Kroner,Kneafsey and Claessens find composite implied volatility forecasts bet-ter than ATM implied. On the other hand, Scott and Tucker (1989) con-clude that when emphasis is placed on ATM implied, which weightingscheme one chooses does not really matter.

A series of studies by Ederington and Guan have reported some inter-esting findings. Ederington and Guan (1999) report that the informationcontent of implied volatility of S&P500 futures options exhibits a frownshape across strikes with options that are NTM and have moderatelyhigh strike (i.e. OTM calls and ITM puts) possess the largest informa-tion content with R2 equal to 17% for calls and 36% for puts.

10.3 IMPLIED BIASEDNESS

Usually, forecast unbiasedness is not an overriding issue in any forecast-ing exercise. Forecast bias can be estimated and corrected if the degree



of bias remains stable through time. Testing for biasedness is usuallycarried out using the regression equation (2.3), where Xi = Xt is theimplied forecast of period t volatility. For a forecast to be unbiased, onewould require α = 0 and β = 1. Implied forecast is upwardly biased ifα > 0 and β = 1, or α = 0 and β > 1. In the case where α > 0 andβ < 1, which is the most common scenario, implied underforecasts lowvolatility and overforecasts high volatility.

It has been argued that implied bias will persist only if it is difficultto perform arbitrage trades that are needed to remove the mispricing.This is more likely in the case of stock index options and less likelyfor futures options. Stocks and stock options are traded in differentmarkets. Since trading of a basket of stocks is cumbersome, arbitragetrades in relation to a mispriced stock index option may have to bedone indirectly via index futures. On the other hand, futures and futuresoptions are traded alongside each other. Trading in these two contractsare highly liquid. Despite these differences in trading friction, impliedbiasedness is reported in both the S&P100 OEX market (Canina andFiglewski, 1993; Christensen and Prabhala, 1998; Fleming, Ostdiek andWhaley, 1995; Fleming, 1998) and the S&P500 futures options market(Feinstein, 1989b; Ederington and Guan, 1999, 2002).

Biasedness is equally widespread among implied volatilities of cur-rency options (see Guo, 1996b; Jorion, 1995; Li, 2002; Scott and Tucker,1989; Wei and Frankel, 1991). The only exception is Jorion (1996) whocannot reject the null hypothesis that the one-day-ahead forecasts fromimplied are unbiased. The five studies listed earlier use implied to fore-cast exchange rate volatility over a much longer horizon ranging fromone to nine months.

Unbiasedness of implied forecast was not rejected in the Swedish mar-ket (Frennberg and Hansson, 1996). Unbiasedness of implied forecastwas rejected for UK stock options (Gemmill, 1986), US stock options(Lamoureux and Lastrapes, 1993), options and futures options across arange of assets in Australia (Edey and Elliot, 1992) and for 35 futuresoptions contracts traded over nine markets ranging from interest rateto livestock futures (Szakmary, Ors, Kim and Davidson, 2002). On theother hand, Amin and Ng (1997) find the hypothesis that α = 0 andβ = 1 cannot be rejected for the Eurodollar futures options market.

Where unbiasedness was rejected, the bias in all but two cases wasdue to α > 0 and β < 1. These two exceptions are Fleming (1998) whoreports α = 0 and β < 1 for S&P100 OEX options, and Day and Lewis(1993) who find α > 0 and β = 1 for distant-term oil futures optionscontracts.


Option Forecasting Power 119

Christensen and Prabhala (1998) argue that implied is biased becauseof error-in-variable caused by measurement errors. Using last periodimplied and last period historical volatility as instrumental variables tocorrect for these measurement errors, Christensen and Prabhala (1998)find unbiasedness cannot be rejected for implied volatility of the S&P100OEX option. Ederington and Guan (1999, 2002) find bias in S&P500futures options implied also disappeared when similar instrument vari-ables were used.

10.4 VOLATILITY RISK PREMIUM

It has been suggested that implied biasedness could not have been causedby model misspecification or measurement errors because this has rel-atively small effects for ATM options, used in most of the studies thatreport implied biasedness. In addition, the clientele effect cannot explainthe bias either because it only affects OTM options. The volatility riskpremium analysed in Chapter 9 is now often cited as an explanation.

Poteshman (2000) finds half of the bias in S&P500 futures optionsimplied was removed when actual volatility was estimated with a moreefficient volatility estimator based on intraday 5-minute returns. Theother half of the bias was almost completely removed when a moresophisticated and less restrictive option pricing model, i.e. the Heston(1993) model, was used. Further research on option volatility risk pre-mium is currently under way in Benzoni (2001) and Chernov (2001).Chernov (2001) finds, similarly to Poteshman (2000), that when impliedvolatility is discounted by a volatility risk premium and when the errors-in-variables problems in historical and realized volatility are removed,the unbiasedness of the S&P100 index option implied volatility cannotbe rejected over the sample period from 1986 to 2000. The volatility riskpremium debate continues if we are able to predict the magnitude andthe variations of the volatility premium and if implied from an optionpricing model that permits a nonzero market price of risk will outperformtime series models when all forecasts (including forecasts of volatilityrisk premium) are made in an ex ante manner.

Ederington and Guan (2000b) find that using regression coefficientsproduced from in-sample regression of forecast against realized volatil-ity is very effective in correcting implied forecasting bias. They alsofind that after such a bias correction, there is little to be gained fromaveraging implied across strikes. This means that ATM implied togetherwith a bias correction scheme could be the simplest, and yet the best,way forward.


11

Volatility Forecasting Records

11.1 WHICH VOLATILITY FORECASTING MODEL?

Our JEL survey has concentrated on two questions: is volatility fore-castable? If it is, which method will provide the best forecasts? To con-sider these questions, a number of basic methodological viewpoints needto be discussed, mostly about the evaluation of forecasts. What exactly isbeing forecast? Does the time interval (the observation interval) matter?Are the results similar for different speculative markets? How does onemeasure predictive performance?

Volatility forecasts are classified in this section as belonging in oneof the following four categories:

� HISVOL: for historical volatility, which include random walk, histor-ical averages of squared returns, or absolute returns. Also includedin this category are time series models based on historical volatilityusing moving averages, exponential weights, autoregressive models,or even fractionally integrated autoregressive absolute returns, for ex-ample. Note that HISVOL models can be highly sophisticated. Themultivariate VAR realized volatility model in Andersen, Bollerslev,Diebold and Labys (2001) is classified here as a ‘HISVOL’ model. Allmodels in this group model volatility directly, omitting the goodnessof fit of the returns distribution or any other variables such as optionprices.

� GARCH: any member of the ARCH, GARCH, EGARCH and so forthfamily is included.

� SV: for stochastic volatility model forecasts.� ISD: for option implied standard deviation, based on the Black–

Scholes model and various generalizations.

The survey of papers includes 93 studies, but 25 of them did notinvolve comparisons between methods from at least two of these groups,and so were not helpful for comparison purposes.

Table 11.1 involves just pairwise comparisons. Of the 66 studies thatwere relevant, some compared just one pair of forecasting techniques,

121



Table 11.1 Pair-wise comparisons of forecasting performanceof various volatility models

Number of studies Studies percentage

HISVOL > GARCH 22 56%GARCH > HISVOL 17 44%

HISVOL > ISD 8 24%ISD > HISVOL 26 76%

GARCH > ISD 1 6%ISD > GARCH 17 94%

SV > HISVOL 3SV > GARCH 3GARCH > SV 1ISD > SV 1

Note: “A > B” means model A’s forecasting performance is better thanthat of model B’s

other compared several. For those involving both HISVOL and GARCHmodels, 22 found HISVOL better at forecasting than GARCH (56% ofthe total), and 17 found GARCH superior to HISVOL (44%).

The combination of forecasts has a mixed picture. Two studies find itto be helpful but another does not.

The overall ranking suggests that ISD provides the best forecastingwith HISVOL and GARCH roughly equal, although possibly HISVOLdoes somewhat better in the comparisons. The success of the impliedvolatility should not be surprising as these forecasts use a larger, andmore relevant, information set than the alternative methods as they useoption prices. They are also less practical, not being available for allassets.

Among the 93 papers, 17 studies compared alternative version ofGARCH. It is clear that GARCH dominates ARCH. In general, mod-els that incorporate volatility asymmetry such as EGARCH and GJR-GARCH, perform better than GARCH. But certain specialized specifi-cations, such as fractionally integrated GARCH (FIGARCH) and regimeswitching GARCH (RSGARCH) do better in some studies. However,it seems clear that one form of study that is included is conducted justto support a viewpoint that a particular method is useful. It might nothave been submitted for publication if the required result had not beenreached. This is one of the obvious weaknesses of a comparison such asthis: the papers being reported have been prepared for different reasons


Volatility Forecasting Records 123

and use different data sets, many kinds of assets, various intervals anda variety of evaluation techniques. Rarely discussed is if one methodis significantly better than another. Thus, although a suggestion can bemade that a particular method of forecasting volatility is the best, nostatement is available about the cost–benefit from using it rather thansomething simpler or how far ahead the benefits will occur.

Financial market volatility is clearly forecastable. The debate is onhow far ahead one can accurately forecast and to what extent volatilitychanges can be predicted. This conclusion does not violate market effi-ciency since accurate volatility forecast is not in conflict with underlyingasset and option prices being correct. The option implied volatility, beinga market-based volatility forecast, has been shown to contain most in-formation about future volatility. The supremacy among historical timeseries models depends on the type of asset being modelled. But, as arule of thumb, historical volatility methods work equally well comparedwith more sophisticated ARCH class and SV models. Better rewardcould be gained by making sure that actual volatility is measured accu-rately. These are broad-brush conclusions, omitting the fine details thatwe outline in this book. Because of the complex issues involved and theimportance of volatility measure, volatility forecasting will continue toremain a specialist subject and to be studied vigorously.

11.2 GETTING THE RIGHT CONDITIONALVARIANCE AND FORECAST WITH THE

‘WRONG’ MODELS

Many of the time series volatility models, including the GARCH mod-els, can be thought of as approximating a deeper time-varying volatilityconstruction, possibly involving several important economic explana-tory variables. Since time series models involve only lagged returns itseems likely that they will provide an adequate, possibly even a verygood, approximation to actuality for long periods but not at all times.This means that they will forecast well on some occasions, but less wellon others, depending on fluctuations in the underlying driving variables.

Nelson (1992) proves that if the true process is a diffusion or near-diffusion model with no jumps, then even when misspecified, appropri-ately defined sequences of ARCH terms with a large number of laggedresiduals may still serve as consistent estimators for the volatility ofthe true underlying diffusion, in the sense that the difference betweenthe true instantaneous volatility and the ARCH estimates converges to



zero in probability as the length of the sampling frequency diminishes.Nelson (1992) shows that such ARCH models may misspecify both theconditional mean and the dynamic of the conditional variance; in fact themisspecification may be so severe that the models make no sense as data-generating processes, they could still produce consistent one-step-aheadconditional variance estimates and short-term forecasts.

Nelson and Foster (1995) provide further conditions for such mis-specified ARCH models to produce consistent forecasts over the mediumand long term. They show that forecasts by these misspecified modelswill converge in probability to the forecast generated by the true diffusionor near-diffusion process, provided that all unobservable state variablesare consistently estimated and that the conditional mean and conditionalcovariances of all state variables are correctly specified. An exampleof a true diffusion process given by Nelson and Foster (1995) is thestochastic volatility model described in Chapter 6.

These important theoretical results confirm our empirical observa-tions that under normal circumstances, i.e. no big jumps in prices, theremay be little practical difference in choosing between volatility mod-els, provided that the sampling frequency is small and that, whichevermodel one has chosen, it must contain sufficiently long lagged residuals.This might be an explanation for the success of high-frequency and longmemory volatility models (e.g. Blair, Poon and Taylor, 2001; Andersen,Bollerslev, Diebold and Labys, 2001).

11.3 PREDICTABILITY ACROSS DIFFERENT ASSETS

Early studies that test the forecasting power of option ISD are fraughtwith many estimation deficiencies. Despite these complexities, optionISD has been found empirically to contain a significant amount of in-formation about future volatility and it often beats volatility forecastsproduced by sophisticated time series models. Such a superior perfor-mance appears to be common across assets.

11.3.1 Individual stocks

Latane and Rendleman (1976) were the first to discover the forecast-ing capability of option ISD. They find actual volatilities of 24 stockscalculated from in-sample period and extended partially into the futureare more closely related to implied than historical volatility. Chiras andManaster (1978) and Beckers (1981) find prediction from implied can



explain a large amount of the cross-sectional variations of individualstock volatilities. Chiras and Manaster (1978) document an R2 of 34–70% for a large sample of stock options traded on CBOE whereasBeckers (1981) reports an R2 of 13–50% for a sample that varies from62 to 116 US stocks over the sample period. Gemmill (1986) producesan R2 of 12–40% for a sample of 13 UK stocks. Schmalensee andTrippi (1978) find implied volatility rises when stock price falls and thatimplied volatilities of different stocks tend to move together. From atime series perspective, Lamoureux and Lastrapes (1993) and Vasilellisand Meade (1996) find implied volatility could also predict time seriesvariations of equity volatility better than forecasts produced from timeseries models.

The forecast horizons of this group of studies that forecast equityvolatility are usually quite long, ranging from 3 months to 3 years. Stud-ies that examine incremental information content of time series fore-casts find volatility historical average provides significant incrementalinformation in both cross-sectional (Beckers, 1981; Chiras and Man-aster, 1978; Gemmill, 1986) and time series settings (Lamoureux andLastrapes, 1993) and that combining GARCH and implied volatilityproduces the best forecast (Vasilellis and Meade, 1996). These findingshave been interpreted as an evidence of stock option market inefficiencysince option implied does not subsume all information. In general, stockoption implied volatility exhibits instability and suffers most from mea-surement errors and bid–ask spread because of the lower liquidity.

11.3.2 Stock market index

There are 22 studies that use index option ISD to forecast stock indexvolatility; seven of these forecast volatility of S&P100, ten forecastvolatility of S&P500 and the remaining five forecast index volatilityof smaller stock markets. The S&P100 and S&P500 forecasting resultsmake an interesting contrast as almost all studies that forecast S&P500volatility use S&P500 futures options which is more liquid and lessprone to measurement errors than the OEX stock index option writtenon S&P100. We have dealt with the issue of measurement errors in thediscussion of biasness in Section 10.3.

All but one study (viz. Canina and Figlewski, 1993) conclude thatimplied volatility contains useful information about future volatility.Blair, Poon and Taylor (2001) and Poteshman (2000) record the highestR2 for S&P100 and S&P500 respectively. About 50% of index volatility



is predictable up to a 4-week horizon when actual volatility is estimatedmore accurately using very high-frequency intraday returns.

Similar, but less marked, forecasting performance emerged from thesmaller stock markets, which include the German, Australian, Canadianand Swedish markets. For a small market such as the Swedish market,Frennberg and Hansson (1996) find seasonality to be prominent and thatimplied volatility forecast cannot beat simple historical models such asthe autoregressive model and random walk. Very erratic and unstableforecasting results were reported in Brace and Hodgson (1991) for theAustralian market. Doidge and Wei (1998) find the Canadian Torontoindex is best forecast with GARCH and implied volatility combined,whereas Bluhm and Yu (2000) find VDAX, the German version of VIX,produces the best forecast for the German stock index volatility.

A range of forecast horizons were tested among this group of stud-ies, though the most popular choice is 1 month. There is evidence thatthe S&P implied contains more information after the 1987 crash (seeChristensen and Prabhala (1998) for S&P100 and Ederington and Guan(2002) for S&P500). Some described this as the ‘awakening’ of the S&Poption markets.

About half of the papers in this group test if there is incrementalinformation contained in time series forecasts. Day and Lewis (1992),Ederington and Guan (1999, 2004), and Martens and Zein (2004) findARCH class models and volatility historical average add a few percent-age points to the R2, whereas Blair, Poon and Taylor (2001), Christensenand Prabhala (1998), Fleming (1998), Fleming, Ostdiek and Whaley(1995), Hol and Koopman (2001) and Szakmary, Ors, Kim and Davidson(2002) all find option implied dominates time series forecasts.

11.3.3 Exchange rate

The strong forecasting power of implied volatility is again confirmedin the currency markets. Sixteen papers study currency options for anumber of major currencies, the most popular of which are DM/US$and ¥/US$. Most studies find implied volatility to contain informationabout future volatility for a short horizon up to 3 months. Li (2002) andScott and Tucker (1989) find implied volatility forecast well for up to a6–9-month horizon. Both studies register the highest R2 in the region of40–50%.

A number of studies in this group find implied volatility beats timeseries forecasts including volatility historical average (see Fung, Lie and



Moreno, 1990; Wei and Frankel, 1991) and ARCH class models (seeGuo, 1996a, 1996b; Jorion, 1995, 1996; Martens and Zein, 2004; Pong,Shackleton, Taylor and Xu, 2002; Szakmary, Ors, Kim and Davidson,2002; Xu and Taylor, 1995). Some studies find combined forecast is thebest choice (see Dunis, Law and Chauvin, 2000; Taylor and Xu, 1997).

Two studies find high-frequency intraday data can produce more ac-curate time series forecast than implied. Fung and Hsieh (1991) findone-day-ahead time series forecast from a long-lag autoregressive modelfitted to 15-minutes returns is better than implied volatility. Li (2002)finds the ARFIMA model outperformed implied in long-horizon fore-casts while implied volatility dominates over shorter horizons. Impliedvolatility forecasts were found to produce higher R2 than other longmemory models, such as the Log-ARFIMA model in Martens and Zein(2004) and Pong, Shackleton, Taylor and Xu (2004). All these long mem-ory forecasting models are more recent and are built on volatility com-piled from high-frequency intraday returns, while the implied volatilityremains to be constructed from less frequent daily option prices.

11.3.4 Other assets

The forecasting power of implied volatility from interest rate options wastested in Edey and Elliot (1992), Fung and Hsieh (1991) and Amin andNg (1997). Interest rate option models are very different from otheroption pricing models because of the need to price the whole termstructure of interest rate derivatives consistently all at the same time inorder to rule out arbitrage opportunities. Trading in interest rate instru-ments is highly liquid as trading friction and execution cost are negli-gible. Practitioners are more concerned about the term structure fit thanthe time series fit, as millions of pounds of arbitrage profits could changehands instantly if there is any inconsistency in contemporaneous prices.

Earlier studies such as Edey and Elliot (1992) and Fung and Hsieh(1991) use the Black model (a modified version of Black–Scholes) thatprices each interest rate option without cross-referencing to prices ofother interest rate derivatives. The single factor Heath–Jarrow–Mortonmodel, used in Amin and Ng (1997) and fitted to short rate only, worksin the same way, although the authors have added different constraintsto the short-rate dynamics as the main focus of their paper is to comparedifferent variants of short-rate dynamics. Despite the complications, allthree studies find significant forecasting power is implied of interest rate(futures) options. Amin and Ng (1997) in particular report an R2 of 21%



for 20-day-ahead volatility forecasts, and volatility historical averageadds only a few percentage points to the R2.

Implied volatilities from options written on nonfinancial assets wereexamined in Day and Lewis (1993, crude oil), Kroner, Kneafsey andClaessens (1995, agriculture and metals), Martens and Zein (2004,crude oil) and a recent study (Szakmary, Ors, Kim and Davidson, 2002)that covers 35 futures options contracts across nine markets includingS&P500, interest rates, currency, energy, metals, agriculture and live-stock futures. All four studies find implied volatility dominates timeseries forecasts although Kroner, Kneafsey and Claessens (1995) findcombining GARCH and implied produces the best forecast.


12Volatility Models in Risk

Management

The volatility models described in this book are useful for estimatingvalue-at-risk (VaR), a measure introduced by the Basel Committee in1996. In many countries, it is mandatory for banks to hold a minimumamount of capital calculated as a function of VaR. Some financial in-stitutions other than banks also use VaR voluntarily for internal riskmanagement. So volatility modelling and forecasting has a very impor-tant role in the finance and banking industries. In Section 12.1, we givea brief background of the Basel Committee and the Basel Accords. InSection 12.2, we define VaR and explain how the VaR estimate is testedaccording to regulations set out in the Basel Accords. Section 12.3 de-scribes how volatility models can be combined with extreme value theoryto produce, hitherto the most accurate, VaR estimate. The content in thissection is largely based on McNeil and Frey (2000) in the context wherethere is only one asset (or one risk factor). A multivariate extension ispossible but is still under development. Section 12.4 describes variousways to evaluate the VaR model based on Lopez (1998).

Market risk and VaR represent only one of the many types of riskdiscussed in the Basel Accords. We have specifically omitted credit riskand operational risk as volatility models have little use in predictingthese risks. Readers who are interested in risk management in a broadercontext could refer to Jorion (2001) or Banks (2004).

12.1 BASEL COMMITTEE AND BASELACCORDS I & II

The Basel Accords have been in place for a number of years. They set outan international standard for minimum capital requirement among in-ternational banks to safeguard against credit, market and operationalrisks. The Bank for International Settlements (BIS) based at Basel,Switzerland, hosts the Basel Committee who in turn set up the BaselAccords. While Basel Committee members are all from the G10 coun-tries and have no formal supranational supervisory authority, the Basel

129



Accords have been adopted by almost all countries that have activeinternational banks. Many financial institutions that are not regulatedby the national Banking Acts also pay attention to the risk managementprocedures set out in the Basel Accords for internal risk monitoringpurposes. The IOSCO (International Organization of Securities Com-missions), for example, has issued several parallel papers containingguidelines similar to the Basel Accords for the risk management ofderivative securities.

The first Basel Accord, which was released in 1988 and which becameknown as the Capital Accord, established a minimum capital standardat 8% for assets subject to credit risk:

Liquidity-weighted assets

Risk-weighted assets≥ 8%. (12.1)

Detailed guidelines were set for deriving the denominator according tosome predefined risk weights; typically a very risky loan will be givena 100% weight. The numerator consists of bank capital weighted theliquidity of the assets according to a list of weights published by theBasel Committee.

In April 1995, an amendment was issued to include capital chargefor assets that are vulnerable to ‘market risk’, which is defined as therisk of loss arising from adverse changes in market prices. Specifically,capital charges are to be supplied: (i) to the current market value ofopen positions (including derivative positions) in interest rate relatedinstruments and equities in banks’ trading books, and (ii) to banks’ totalcurrency and commodities position in respect of foreign exchange andcommodities risk respectively. A detailed ‘Standardised MeasurementMethod’ was prescribed by the Basel Committee for calculating thecapital charge for each market risk category.

If we rearrange Equation (12.1) such that

Liquidity-weighted assets ≥ 8% × Risk-weighted assets, (12.2)

then the market risk related capital charge is added to credit risk related‘Liquidity-weighted assets’ in the l.h.s. of Equation (12.2). This effec-tively increases the ‘Risk-weighted assets’ in the r.h.s. by 12.5 times theadditional market risk related capital charge.

In January 1996, another amendment was made to allow banks to usetheir internal proprietary model together with the VaR approach for cal-culating market risk related risk capital. This is the area where volatilitymodels could play an important role because, by adopting the internal


Volatility Models in Risk Management 131

approach, the banks are given the flexibility to specify model parametersand to take into consideration the correlation (and possible diversifica-tion) effects across as well as within broad risk factor categories. Thecondition for the use of the internal model is that it is subject to regularbacktesting procedures using at least one year’s worth of historical data.More about VaR estimation and backtesting will be provided in the nextsections.

In June 2004, Basel II was released with two added dimensions, viz.supervisory review of an institution’s internal assessment process andcapital adequacy, and market discipline through information disclosure.Basel II also saw the introduction of operational risk for the first timein the calculation of risk capital to be included in the denominator inEquation (12.1). ‘Operational risk’ is defined as the risk of losses result-ing from inadequate or failed internal processes, people and systems, orexternal events. The Basel Committee admits that assessments of oper-ational risk are imprecise and it will accept a crude approximation thatis based on applying a multiplicative factor to the bank’s gross income.

12.2 VaR AND BACKTEST

In this section, we discuss market risk related VaR only, since this is thearea where volatility models can play an important role. The computa-tion of VaR is needed only if the bank chooses to adopt its own internalmodel for calculating market risk related capital requirement.

12.2.1 VaR

‘Value-at-risk’ (VaR) is defined as the 1% quantile of the lower taildistribution of the trading book held over a 10-day period.1 The capitalcharge will then be the higher of the previous day’s VaR or three times theaverage daily VaR of the preceding 60 business days. The multiplicativefactor of three was used as a cushion for cumulative losses arising fromadverse market conditions and to account for potential weakness in themodelling process.2 Given that today’s portfolio value is known, theprediction of losses over a 10-day period amounts to predicting the rateof change (or portfolio returns) over the 10-day period (Figure 12.1).

1 A separate VaR will be calculated for each risk factor. So there will be separate VaR for interest rate relatedinstruments, equity, foreign exchange risk and commodities risk. If correlations among the four risk factors arenot considered, then the total VaR will be the sum of the four VaR estimates.

2 While one may argue that such a multiplicative factor is completely arbitrary, it is nevertheless mandatory.



Figure 12.1 Returns distribution and VaR

The VaR estimate for tomorrow’s trading position is then calculated astoday’s portfolio value times the 1% quantile value if it is a negativereturn. (A positive return will not attract any risk capital.)

The discovery of stochastic volatility has led to the common practice ofmodelling returns distribution conditioned on volatility level at a specificpoint in time. Volatility dynamic has been extensively studied since theseminal work of Engle (1982). It is now well known that a volatile periodin the financial markets tends to be followed by another volatile period,whereas a tranquil period tends to be followed by another tranquil period.VaR as defined by the Basel Committee is a short-term forecast. Hence agood VaR model should fully exploit the dynamic of volatility structure.

12.2.2 Backtest

For banks who decide to use their own internal models, they have toperform a backtest procedure at least at a quarterly interval. The back-test procedure involves comparing the bank’s daily profits and losseswith model-generated VaR measures in order to gauge the quality andaccuracy of their risk measurement systems. Specifically, this is done bycounting from the record of the last 12 months (or 250 trading days) thenumber of times when actual losses are greater than the predicted riskmeasure. The proportion actually covered can then be checked to see ifit is consistent with a 99% level of confidence.

In the 1996 backtest document, the Basel Committee was unclearabout whether the exceptional losses should take into account fee incomeand changes in portfolio position. Hence the long discussion about thechoice of 10-day, 1-day or intraday intervals for calculating exceptional



losses, and the discussion of whether actual or simulated trading resultsshould be tested. Before we discuss these two issues, it is importantto note that (i) actual VaR violations could be due to an inadequate(volatility) model or a bad decision (e.g. a decision to change portfoliocomposition at the wrong time); (ii) financial market volatility often doesnot obey the scaling law, i.e. variance of 10-day return is not equal to1-day variance times

√10.

To test model adequacy, the backtest should be based on a simulatedportfolio assuming that the bank has been holding the same portfolio forthe last 12 months. This will help to separate a bad model from a baddecision. Given that the VaR used for calculating the capital requirementis for a 10-day holding period, the backtest should also be performedusing a 10-day window to accumulate profits/losses. The current rules,which require the VaR test to be calculated for a 1-day profits/losses,fail to recognize the volatility dynamics over the 10-day period is vastlydifferent from the 1-day volatility dynamic.

The Basel Committee recommends that backtest also be conducted onactual trading outcomes in addition to the simulated portfolio position.Specifically, it recommends a comprehensive approach that involves adetailed attribution of income by source, including fees, spreads, marketmovements and intraday trading results. This is very useful for uncov-ering risks that are not captured in the volatility model.

12.2.3 The three-zone approach to backtest evaluation

The Basel Committee then specifies a three-zone approach to evaluatethe outcome of the backtest. To understand the rationale of the three-zoneapproach, it is important to recognize that all appropriately implementedcontrol systems are subject to random errors. On the other hand, thereare cases where the control system is a bad one and yet there are nofailures. The objective of the backtest is to distinguish the two situationswhich are known as type I (rejecting a system when it is working) andtype II (accepting a bad system) errors respectively.

Given that the confidence level set for the VaR measure is 99%, thereis a 1% chance of exceptional losses that are tolerated. For 250 trad-ing days, this translates into 2.5 occurrences where the VaR estimatewill be violated. Figure 12.2 shows the outcome of simulations involv-ing a system with 99% coverage and 95% coverage as reported in theBasel document (January 1996, Table 1). We can see that the numberof exceptions under the true 99% coverage can range from 0 to 9, with



Figure 12.2 Type I and II backtest errors

most of the instances centred around 2. If the true coverage is 95%, wemight get four or more exceptions with the mean centred around 12. Ifonly these two coverages are possible then one may conclude that if thenumber of exceptions is four or below, then the 99% coverage is true. Ifthere are ten or more exceptions it is more likely that the 99% coverageis not true. If the number of exceptions is between five and nine, then itis possible that the true coverage might be from either the 99% or the95% population and it is impossible to make a conclusion.

The difficulty one faces in practice is that there could potentially be anendless range of possible coverages (i.e. 98%, 97%, 96%, . . . , etc). Sowhile we are certain about the range of type I errors (because we knowthat our objective is to have a 99% coverage), we cannot be precise aboutthe possible range of type II errors (because we do not know the truecoverage). The three-zone approach and the recommended increase inscaling factors (see Table 12.1) is the Basel Committee’s effort to seeka compromise in view of this statistical uncertainty. From Table 12.1,if the backtest reveals that in the last trading year there are 10 or moreVaR violations, for example, then the capital charge will be four timesVaR, instead of three times VaR.



Table 12.1 Three-zone approach to internal backtesting of VaR

No. of Increase in CumulativeZones exceptions scaling factor probability

Green 0–4 0 8.11 to 89.22%

5 +0.40 95.88%6 +0.50 98.63%

Yellow 7 +0.65 99.60%8 +0.75 99.89%9 +0.85 99.97%

Red 10 or more +1 99.99%

Note: The table is based on a sample of 250 observations. The cumulativeprobability is the probability of obtaining a given number of or fewer ex-ceptions in a sample of 250 observations when the true coverage is 99%.(Source: Basel 1996 Amendment, Table 2).

When the backtest signals a red zone, the national supervisory bodywill investigate the reasons why the bank’s internal model produced sucha large number of misses, and may demand the bank to begin workingon improving its model immediately.

12.3 EXTREME VALUE THEORY ANDVaR ESTIMATION

The extreme value approach to VaR estimation is a response to the find-ing that standardized residuals of many volatility models have longertails than the normal distribution. This means that a VaR estimate pro-duced from the standard volatility model without further adjustment willunderestimate the 1% quantile. Using GARCH-t , where the standard-ized residuals are assumed to follow a Student-t distribution, partiallyalleviates the problem, but GARCH-t is inadequate when the left tailand the right tail are not symmetrical. The EVT-GARCH method pro-posed by McNeil and Frey (2000) is to model conditional volatility andmarginal distribution of the left tail separately. We need to model onlythe left tail since the right tail is not relevant as far as VaR computationis concerned.

Tail event is by definition rare and a long history of data is required touncover the tail structure. For example, one would not just look into thelast week’s or the last year’s data to model and forecast the next earth-quake or the next volcano eruption. The extreme value theory (EVT) isbest suited for studying rare extreme events of this kind where sound



statistical theory for maximal has been well established. There is a prob-lem, however, in using a long history of data to produce a short-termforecast. The model is not sensitive enough to current market condition.Moreover, one important assumption of EVT is that the tail events areindependent and identically distributed (iid). This assumption is likelyto be violated because of stochastic volatility and volatility persistencein particular. Volatility persistence suggests one tail event is likely tobe followed by another tail event. One way to overcome this violationof the iid assumption is to filter the data with a volatility model andstudy the volatility filtered iid residuals using EVT. This is exactly whatMcNeil and Frey (2000) proposed. In order to produce a VaR estimate inthe original return scale to conform with the Basel requirement, we thentrace back the steps by first producing the 1% quantile estimate for thevolatility filtered return residuals, and convert the 1% quantile estimateto the original return scale using the conditional volatility forecast forthe next day.

12.3.1 The model

Following the GARCH literature, let us write the return process as

rt = µ + zt

√ht . (12.3)

In the process above, we assume there is no serial correlation in dailyreturn. (Otherwise, an AR(1) or an MA(1) term could be added to ther.h.s. of (12.3).) Equation (12.3) is estimated with some appropriatespecification for the volatility process, ht (see Chapter 4 for details). Forstock market returns, ht typically follows an EGARCH(1,1) or a GJR-GARCH(1,1) process. The GARCH model in (12.3) is estimated usingquasi-maximum likelihood with a Gaussian likelihood function, eventhough we know that zt is not normally distributed. QME estimators areunbiased and since the standardized residuals will be modelled using anextreme value distribution, such a procedure is deemed appropriate.

The standardized residuals zt are obtained by rearranging (12.3)

zt = rt − µ√ht

.

Since our main concern is about losses, we could multiply zt by −1 sothat we are always working with positive values for convenience. Thezt variable is then ranked in descending order, such that z(1) ≥ z(2) ≥· · · ≥ z(n) where n is the number of observations.



The next stage involves estimating the generalized Pareto distribu-tion (GPD) to all z that are greater than a high threshold u. The GPDdistribution has density function

f (z) =

1 −[

1 + ξ (z − u)

β

]−1/ξfor ξ �= 0

1 − e−(z−u)/β for ξ = 0

.

The parameter ξ is called the tail index and β is the scale parameter.The estimation of the model parameters (the tail index ξ in particular)

and the choice of the threshold u are not independent processes. Asu becomes larger, there will be fewer and fewer observations beingincluded in the GPD estimation. This makes the estimation of ξ veryunstable with a large standard error. But, as u decreases, the chance of anobservation that does not belong to the tail distribution being included inthe GPD estimation increases. This increases the risk of the ξ estimatebeing biased. The usual advice is to estimate ξ (and β) at differentlevels of u. Then starting from the highest value of u, a lower value ofu is preferred unless there is a change in the level of ξ estimate, whichindicates there may be a possible bias caused by the inclusion of toomany observations in the GPD estimation.

Once the parameters ξ and β are estimated, the 1% quantile is obtainedby inverting the cumulative density function

F (z) = 1 − k

n

[1 + ξ

(z − u

β

)]−1/ξ,

zq = u + β

ξ

[(1 − q

k/n

)−ξ

− 1

],

where q = 0.01 and k is the number of z exceeding the threshold u.We are now ready to calculate the VaR estimate using (12.3) and the

volatility forecast for the next day:

VaRt+1 = current position ×(

µ − zq

√ht+1

).

12.3.2 10-day VaR

It is well-known that the variance of a Gaussian variable follows a simplescaling law and the Basel Committee, in its 1996 Amendment, states that



it will accept a simple√

T scaling of 1-day VaR for deriving the 10-dayVaR required for calculating the market risk related risk capital.

The stylized facts of financial market volatility and research findingshave repeatedly shown that a 10-day VaR is not likely to be the same as√

10 × 1-day VaR. First, the dynamic of a stationary volatility processsuggests that if the current level of volatility is higher than unconditionalvolatility, the subsequent daily volatility forecasts will decline and con-verge to unconditional volatility, and vice versa for the case where theinitial volatility is lower than the unconditional volatility. The rate ofconvergence depends on the degree of volatility persistence. In the casewhere initial volatility is higher than unconditional volatility, the scal-ing factor will be less than

√10. In the case where initial volatility is

lower than unconditional volatility, the scaling factor will be more than√10. In practice, due to volatility asymmetry and other predictive vari-

ables that might be included in the volatility model, it is always bestto calculate ht+1, ht+2, · · · , ht+10 separately. The 10-day VaR is thenproduced using the 10-day volatilty estimate calculated from the sum ofht+1, ht+2, · · · , ht+10.

Secondly, financial asset returns are not normally distributed.Danielsson and deVries (1997) show that the scaling parameter forquantile derived using the EVT method increases at the approximaterate of T ξ , which is typically less than the square-root-of-time adjust-ment. For a typical value of ξ (= 0.25) , T ξ = 1.778, which is less than100.5 (= 3.16). McNeil and Frey (2000) on the other hand dispute thisfinding and claim the exponent to be greater than 0.5. The scaling fac-tor of 100.5 produced far too many VaR violations in the backtest of fivefinancial series, except for returns on gold. In view of the conflicting em-pirical findings, one possible solution is to build models using 10-dayreturns data. This again highlights the difficulty due to the inconsistencyin the rule applies to VaR for calculating risk capital and that applies toVaR for backtesting.

12.3.3 Multivariate analysis

The VaR computation described above is useful for the single asset caseand cases where there is only one risk factor. The cases for multi-assetand multi-risk-factor are a lot more complex which require multivariateextreme theories and a better understanding of the dependence struc-ture between the variables of interest. Much research in this area isstill ongoing. But it is safe to say that correlation coefficient, the key



measure used in portfolio diversification, can produce very misleadinginformation about the dependence structure of extreme events in finan-cial markets (Poon, Rockinger and Tawn, 2004). The VaR of a portfoliois not a simple function of the weighted sum of the VaR of the individ-ual assets. Detailed coverage of multivariate extreme value theories andapplications is beyond the scope of this book. The simplest solution wecould offer here is to treat portfolio returns as a univariate variable andapply the procedures above. Such an approach does not provide insightabout the tail relationship between assets and that between risk factors,but it will at least produce a sensible estimate of portfolio VaR.

12.4 EVALUATION OF VaR MODELS

In practice, there will be many different models for calculating VaR,many of which will satisfy Basel’s backtest requirement. The importantquestions are ‘Which model should one use?’ and ‘If there are excep-tions, how do we know if the model is malfunctioning?’. Lopez (1998)proposes two statistical tests and a supplementary evaluation that is basedon the user specifying a loss function.

The first statistical test involves modelling the number of exceptionsas independent draws from a binomial distribution with a probabilityof occurrence equal to 1%. Let x be the actual number of exceptionsobserved for a sample of 250 trading outcomes. The probability of ob-serving x exceptions from a 99 % coverage is

Pr (x) = C250x × 0.01x × 0.99250−x .

The likelihood ratio statistic for testing if the actual unconditional cov-erage α = x/250 = 0.01 is

LRuc = 2[log

(αx × (1 − α)250−x

) − log(0.01x × 0.99250−x

)].

The LRuc test statistic has an asymptotic χ2 distribution with one degreeof freedom.

The second test makes use of the fact that VaR is the interval forecastof the lower 1% tail of the one-step-ahead conditional distribution of re-turns. So given a set of VaRt , the indicator variable It+1 is constructed as

It+1 ={

1 for rt+1 ≤ VaRt

0 for rt+1 > VaRt.



If VaRt provides correct conditional coverage, It+1 must equal un-conditional coverage, and It+1 must be serially independent. The LRcc

test is a joint test of these two properties. The relevant test statistic is

LRcc = LRuc + LRind,

which has an asymptotic χ2 distribution with two degrees of freedom.The LRind statistic is the likelihood ratio statistic for the null hypothesisof serial independence against first-order serial dependence.

The LRuc test and the LRcc test are formal statistical tests for thedistribution of VaR exceptions. It is useful to supplement these formaltests with some numerical scores that are based on the loss function ofthe decision maker. The loss fuction is specified as the cost of variousoutcomes below:

Ct+1 ={

f (rt+1, VaRt ) for rt+1 ≤ VaRt

g (rt+1, VaRt ) for rt+1 > VaRt.

Since this is a cost function and the prevention of VaR exception is ofparamount importance, f (x, y) ≥ g (x, y) for a given y. The best VaRmodel is one that provides the smallest total cost, Ct+1.

There are many ways to specify f and g depending on the con-cern of the decision maker. For example, for the regulator, the concernis principally about VaR exception where rt+1 ≤ VaRt and not whenrt+1 > VaRt . So the simplest specification for f and g will be f = 1and g = 0 as follows:

Ct+1 ={

1 for rt+1 ≤ VaRt

0 for rt+1 > VaRt.

If the exception as well as the magnitude of the exception are bothimportant, one could have

Ct+1 ={

1 + (rt+1 − VaRt )2 for rt+1 ≤ VaRt

0 for rt+1 > VaRt.

The expected shortfall proposed by Artzner, Delbaen, Eber and Heath(1997, 1999) is similar in that the magnitude of loss above VaR isweighted by the probability of occurrence. This is equivalent to

Ct+1 ={ |rt+1 − VaRt | for rt+1 ≤ VaRt

0 for rt+1 > VaRt.

For banks who implement the VaR model and has to set aside capitalreserves, g = 0 is not appropriate because liquid assets do not provide



good returns. So one cost function that will take into account the oppor-tunity cost of money is

Ct+1 ={ |rt+1 − VaRt |γ for rt+1 ≤ VaRt

|rt+1 − VaRt | × i for rt+1 > VaRt,

where γ reflects the seriousness of large exception and i is a function ofinterest rate.


13

VIX and Recent Changes in VIX

The volatility index (VIX) compiled by the Chicago Board of OptionExchange has always been shown to capture financial turmoil and pro-duce good forecast of S&P100 volatility (Fleming, Ostdiek and Whaley,1995; Ederington and Guan, 2000a; Blair, Poon and Taylor, 2001; Holand Koopman, 2002). It is compiled on a real-time basis aiming to re-flect the volatility over the next 30 calendar days. In September 2003,the CBOE revised the way in which VIX is calculated and in March2004 it started futures trading on VIX. This is to be followed by optionson VIX and another derivative product may be variance swap. The oldversion of VIX, now renamed as VXO, continued to be calculated andreleased during the transition period.

13.1 NEW DEFINITION FOR VIX

There are three important differences between VIX and VXO:

(i) The new VIX uses information from out-of-the-money call and putoptions of a wide range of strike prices, whereas VXO uses eightat- and near-the-money options.

(ii) The new VIX is model-free whereas VXO is a weighted average ofBlack–Scholes implied volatility.

(iii) The new VIX is based on S&P500 index options whereas VXO isbased on S&P100 index options.

The VIX is calculated as the aggregate value of a weighted strip ofoptions using the formula below:

σ 2vi x = 2

T

∑i

�Ki

K 2i

erT Q (Ki ) − 1

T

[F

K0− 1

]2

, (13.1)

F = K0 + erT (c0 − p0) , (13.2)

�Ki = Ki+1 + Ki−1

2, (13.3)

143



where r is the continuously compounded risk-free interest rate to expi-ration, T is the time to expiration (in minutes!), F is the forward price ofthe index calculated using put–call parity in (13.2), K0 is the first strikejust below F , Ki is the strike price of i th out-of-the-money options (i.e.call if Ki > F and put if Ki < F), Q (Ki ) is the midpoint of the bid–askspread for option at strike price Ki , �Ki in (13.3) is the interval betweenstrike prices. If i is the lowest (or highest) strike, then �Ki = Ki+1 − Ki

(or �Ki = Ki − Ki−1).Equations (13.1) to (13.3) are applied to two sets of options contracts

for the near term T1 and the next near term T2 to derive a constant 30-dayvolatility index VIX:

VIX = 100 ×√{

T1σ21

(NT2 − N30

NT2 − NT1

)+ T2σ

22

(N30 − NT1

NT2 − NT1

)}× N365

N30,

where Nτ is the number of minutes (N30 = 30 × 1400 = 43,200 andN365 = 365 × 1400 = 525,600).

13.2 WHAT IS THE VXO?

VXO, the predecessor of VIX, was released in 1993 and replaced bythe new VIX in September 2003. VXO is an implied volatility com-posite compiled from eight options written on the S&P100. It is con-structed in such a way that it is at-the-money (by combining just-in-and just-out-of-the-money options) and has a constant 28 calendar daysto expiry (by combining the first nearby and second nearby optionsaround the targeted 28 calendar days to maturity). Eight option prices areused, including four calls and four puts, to reduce any pricing bias andmeasurement errors caused by staleness in the recorded index level.Since options written on S&P100 are American-style, a cash-dividendadjusted binomial model was used to capture the effect of early ex-ercise. The mid bid–ask option price is used instead of traded pricebecause transaction prices are subject to bid–ask bounce. (See Whaley(1993) and Fleming, Ostdiek and Whaley (1995) for further details.)Owing to the calendar day adjustment, VIX is about 1.2 times (i.e.√

365/252) greater than historical volatility computed using trading-daydata.


VX

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1990

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13.1

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evo

latil

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s

145



13.3 REASON FOR THE CHANGE

There are many reasons for the change; if nothing else the new volatilityindex is hedgeable and the old one is not. The new VIX can be replicatedwith a static portfolio of S&P500 options or S&P500 futures. Hence, itallows hedging and, more importantly, the corrective arbitrage options ofVIX derivatives if prices are not correct. The CBOE argues that the newVIX reflect information in a broader range of options rather than just thefew at-the-money options. More importantly, the new VIX is aiming tocapture the information in the volatility skew. It is linked to the broader-based S&P500 index instead of the S&P100 index. The S&P500 is theprimary index for most portfolio benchmarking so derivative productsthat are more closely linked to S&P500 will facilitate risk management.

Although the two volatility indices are compiled very differently, theirstatistical properties are very similar. Figures 13.1(a) and 13.1(b) showthe time series plots of VIX and VXO over the period 2 January 1990to 28 June 2004, and Figure 13.1(c) provides a scatterplot showing therelationship between the two. The new VIX has a smaller mean and ismore stable than the old VXO. There is no doubt that researchers arealready investigating the new index and all the issues that it has broughtabout, such as the pricing and hedging of derivatives written on thenew VIX.


14

Where Next?

The volatility forecasting literature is still very active. Many more newresults are expected in the near future. There are several areas wherefuture research could seek to make improvements. First is the issueabout forecast evaluation and combining forecasts of different models.It would be useful if statistical tests were conducted to test whether theforecast errors from Model A are significantly smaller, in some sense,than those from Model B, and so on for all pairs. Even if Model A is foundto be better than all the other models, the conclusion is NOT that oneshould henceforth forecast volatility with Model A and ignore the othermodels as it is very likely that a linear combination of all the forecastsmight be superior. To find the weights one can either run a regressionof empirical volatility (the quantity being forecast) on the individualforecasts, or as an approximation just use equal weights. Testing theeffectiveness of a composite forecast is just as important as testing thesuperiority of the individual models, but this has not been done veryoften or across different data sets.

A mere plot of any measure of volatility against time will show thefamiliar ‘volatility clustering’ which indicates some degree of forecast-ability. The biggest challenge lies in predicting changes in volatility. Ifimplied volatility is agreed to be the best performing forecast, on average,this is in agreement with the general forecast theory, which emphasizesthe use of a wider information set than just the past of the processbeing forecast. Implied volatility uses option prices and so potentiallythe information set is richer. What needs further consideration is if allof its information is now being extracted and if it could still be widenedto further improve forecast accuracy especially that of long horizonforecasts. To achieve this we need to understand better the cause ofvolatility (both historical and implied). Such an understanding will helpto improve time series methods, which are the only viable methods whenoptions, or market-based forecast, are not available.

Closely related to the above is the need to understand the source ofvolatility persistence and the volume-volatility research appears to bepromising in providing a framework in which volatility persistence may

147



be closely scrutinized. The mixture of distribution hypothesis (MDH)proposed by Clark (1973), the link between volume-volatility and mar-ket trading mechanism in Tauchen and Pitts (1983), and the empiricalfindings of the volume-volatility relationship surveyed in Karpoff (1987)are useful starting points. Given that Lamoureux and Lastrapes (1990)find volume to be strongly significant when it is inserted into the ARCHvariance process, while returns shocks become insignificant, and thatGallant, Rossi and Tauchen (1993) find conditioning on lagged vol-ume substantially attenuates the ‘leverage’ effect, the volume-volatilityresearch may lead to a new and better way for modelling returns distri-butions. To this end, Andersen (1996) puts forward a generalized frame-work for the MDH where the joint dynamics of returns and volume areestimated, and reports a significant reduction in the estimated volatilitypersistence. Such a model may be useful for analysing the economicfactors behind the observed volatility clustering in returns, but such aline of research has not yet been pursued vigorously.

There are many old issues that have been around for a long time.These include consistent forecasts of interest rate volatilities that sat-isfies the no-arbitrage relationship across all interest rate instruments,more tests on the use of absolute returns models in comparison withsquared returns models in forecasting volatility, a multivariate approachto volatility forecasting where cross-correlation and volatility spillovermay be accommodated, etc.

There are many new adventures that are currently under way as well.1

These include the realized volatility approach, noticeably driven byAndersen, Bollerslev, Diebold and various co-authors, the estimationand forecast of volatility risk premium, the use of spot and option pricedata simultaneously (e.g. Chernov and Ghysels, 2000), and the use ofBayesian and other methods to estimate stochastic volatility models (e.g.Jones, 2001), etc.

It is difficult to envisage in which direction volatility forecasting re-search will flourish in the next five years. If, within the next five years,we can cut the forecast error by half and remove the option pricing bias inex ante forecast, this will be a very good achievement indeed. Producingby then forecasts of large events will mark an important milestone.

1 We thank a referee for these suggestions.

JWBK021-apx JWBK021-Poon March 15, 2005 14:16 Char Count= 0

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to60

%by

allo

win

gfo

ras

ymm

etri

es,

leve

leff

ecta

ndch

angi

ngvo

latil

ity

CK

LS:

Cha

n,K

arol

yi,L

ongs

taff

and

Sand

ers

(199

2)

Con

tinu

ed

151


Dat

aD

ata

Fore

cast

ing

Fore

cast

ing

Eva

luat

ion

Aut

hor(

s)A

sset

(s)

peri

odfr

eque

ncy

met

hods

and

rank

hori

zon

and

R-s

quar

edC

omm

ents

8.B

ecke

rs(1

981)

62to

116

stoc

ksop

tions

28/4

/75–

21/1

0/77

DFB

SDIm

plie

d AT

Mca

ll,5

days

ave

Impl

ied v

ega

call,

5da

ysav

e

RW

last

quar

ter

(ran

ked,

both

impl

ieds

are

5-da

yav

erag

ebe

caus

eof

larg

eva

riat

ions

inda

ilyst

ock

impl

ied)

Ove

rop

tion’

sm

atur

ity(3

mon

ths)

,10

non-

over

lapp

ing

cycl

es.U

sesa

mpl

eSD

ofda

ilyre

turn

sov

erop

tion

mat

urity

topr

oxy

‘act

ual

vol.’

MPE

,MA

PE.

Cro

ssse

ctio

nal

R2

rang

esbe

twee

n34

and

70%

acro

ssm

odel

san

dex

piry

cycl

es.

FBSD

appe

ars

tobe

leas

tbia

sed

with

α=

0,β

=1.

α>

0,β

<1

for

the

othe

rtw

oim

plie

ds

FBSD

:Fis

her

Bla

ck’s

optio

npr

icin

gse

rvic

eta

kes

into

acco

unts

tock

vol.

tend

tom

ove

toge

ther

,mea

nre

vert

,le

vera

geef

fect

and

impl

ied

can

pred

ict

futu

re.A

TM

,bas

edon

vega

WL

S,ou

tper

form

sve

gaw

eigh

ted

impl

ied,

and

isno

tsen

sitiv

eto

adho

cdi

vide

ndad

just

men

t.In

crem

enta

linf

orm

atio

nfr

omal

lmea

sure

ssu

gges

tsop

tion

mar

ket

inef

ficie

ncy.

Mos

tfo

reca

sts

are

upw

ardl

ybi

ased

asac

tual

vol.

was

ona

decr

easi

ngtr

end

50st

ock

optio

ns4

date

s:18

/10/

76,

24/1

/77,

18/4

/77,

18/7

/77

D(f

rom

Tic

k)T

ISD

vega

Impl

ied A

TM

call,

1da

ysav

e

(ran

ked)

ditto

Cro

ss-s

ectio

nal

R2

rang

esbe

twee

n27

and

72%

acro

ssm

odel

san

dex

piry

cycl

es

TIS

D:S

ingl

ein

trad

aytr

ansa

ctio

nda

tath

atha

sth

ehi

ghes

tveg

a.T

hesu

peri

ority

ofT

ISD

over

impl

ied

ofcl

osin

gop

tion

pric

essu

gges

tsig

nific

ant

non-

sim

ulta

neity

and

bid–

ask

spre

adpr

oble

ms

152


9.B

era

and

Hig

gins

(199

7)

Dai

lySP

500,

Wee

kly

$/£,

Mon

thly

US

Ind.

Prod

.

SP1/

1/88

–28

/5/9

3$/

£12

/12/

85–

28/2

/91

Ind.

Prod

.1/

60–3

/93

D W M

GA

RC

HB

iline

arm

odel

(ran

ked)

One

step

ahea

d.R

eser

ve90

%of

data

for

estim

atio

n

Cox

ML

ER

MSE

(LE

:log

arith

mic

erro

r)

Con

side

rif

hete

rosc

edas

ticity

isdu

eto

bilin

ear

inle

vel.

Fore

cast

ing

resu

ltssh

owst

rong

pref

eren

cefo

rG

AR

CH

10.

Bla

ir,

Poon

and

Tayl

or(2

001)

S&P1

00(V

XO

)2/

1/87

–31

/12/

99

Out

:4/1

/93–

31/2

/99

Tic

kIm

plie

d VX

O

GJR

HIS

100

(ran

ked)

1,5,

10an

d20

days

ahea

des

timat

edus

ing

aro

lling

sam

ple

of10

00da

ys.D

aily

actu

alvo

latil

ityis

calc

ulat

edfr

om5-

min

retu

rns

1-da

y-ah

ead

R2

is45

%fo

rV

XO

,an

d50

%fo

rco

mbi

ned.

VX

Ois

dow

nwar

dbi

ased

inou

t-of

-sam

ple

peri

od

Usi

ngsq

uare

dre

turn

sre

duce

sR

2to

36%

for

both

VX

Oan

dco

mbi

ned.

Impl

ied

vola

tility

has

itsow

npe

rsis

tenc

est

ruct

ure.

GJR

has

noin

crem

enta

lin

form

atio

nth

ough

inte

grat

edH

ISvo

l.ca

nal

mos

tmat

chIV

fore

cast

ing

pow

er

11.

Blu

hman

dY

u(2

000)

Ger

man

DA

Xst

ock

inde

xan

dV

DA

Xth

eD

AX

vola

tility

inde

x

In:1

/1/8

8–28

/6/9

6O

ut:1

/7/6

6–30

/6/9

9

DIm

plie

d VD

AX

GA

RC

H(-

M),

SVE

WM

A,E

GA

RC

H,

GJR

,H

IS(a

ppro

x.ra

nked

)

45ca

lend

arda

ys,

1,10

and

180

trad

ing

days

.‘A

ctua

l’is

the

sum

ofda

ilysq

uare

dre

turn

s

MA

PE,L

INE

XR

anki

ngva

ries

alo

tde

pend

onfo

reca

stho

rizo

nsan

dpe

rfor

man

cem

easu

res

Con

tinu

ed

153


Dat

aD

ata

Fore

cast

ing

Fore

cast

ing

Eva

luat

ion

Aut

hor(

s)A

sset

(s)

peri

odfr

eque

ncy

met

hods

and

rank

hori

zon

and

R-s

quar

edC

omm

ents

12.

Bou

douk

h,R

icha

rd-

son

and

Whi

tela

w(1

997)

3-m

onth

US

T-B

ill19

83–1

992

DE

WM

A,M

DE

GA

RC

H(1

,1),

HIS

(ran

ked)

1da

yah

ead

base

don

150-

day

rolli

ngpe

riod

estim

atio

n.R

ealiz

edvo

latil

ityis

the

daily

squa

red

chan

ges

aver

aged

acro

sst+

1to

t+

5

MSE

and

regr

essi

on.M

DE

has

the

high

est

R2

whi

leE

WM

Aha

sth

esm

alle

stM

SE

MD

Eis

mul

tivar

iate

dens

ityes

timat

ion

whe

revo

latil

ityw

eigh

tsde

pend

onin

tere

stra

tele

vela

ndte

rmsp

read

.EW

MA

and

MD

Eha

veco

mpa

rabl

epe

rfor

man

cean

dar

ebe

tter

than

HIS

and

GA

RC

H

13.

Bra

cean

dH

odgs

on(1

991)

Futu

res

optio

non

Aus

tral

ian

Stoc

kIn

dex

(mar

king

tom

arke

tis

need

edfo

rth

isop

tion)

1986

–87

DH

IS5,

20,65

days

Impl

ied N

TM

call,

20–7

5da

ys

(ran

ked)

20da

ysah

ead.

Use

daily

retu

rns

toca

lcul

ate

stan

dard

devi

atio

ns

Adj

R2

are

20%

(HIS

),17

%(H

IS+

impl

ied)

.All

α>

0an

dsi

g.so

me

unir

egr

coef

far

esi

gne

gativ

e(f

orbo

thH

ISan

dim

plie

d)

Lar

geflu

ctua

tions

ofR

2

from

mon

thto

mon

th.

Res

ults

coul

dbe

due

toth

edi

fficu

ltyin

valu

ing

futu

res

styl

eop

tions

14.

Bra

ilsfo

rdan

dFa

ff(1

996)

Aus

tral

ian

Stat

ex-

Act

uari

esA

ccum

ulat

ion

Inde

xfo

rto

p56

1/1/

74–

30/6

/93

In:J

an74

–D

ec85

Out

:Jan

86–

Dec

93(i

nclu

de87

’scr

ash

peri

od)

DG

JR,

Reg

r,H

IS,G

AR

CH

,M

A,E

WM

A,

RW

,ES

(ran

kse

nsiti

veto

erro

rst

atis

tics)

1m

onth

ahea

d.M

odel

ses

timat

edfr

oma

rolli

ng12

-yea

rw

indo

w

ME

,MA

E,

RM

SE,M

APE

,an

da

colle

ctio

nof

asym

met

ric

loss

func

tions

Tho

ugh

the

rank

sar

ese

nsiti

ve,s

ome

mod

els

dom

inat

eot

hers

;M

A12

>M

A5

and

Reg

r.>

MA

>E

WM

A>

ES.

GJR

cam

eou

tqui

tew

ell

buti

sth

eon

lym

odel

that

alw

ays

unde

rpre

dict

s

154


15.

Bro

oks

(199

8)D

JC

ompo

site

17/1

1/78

–30

/12/

88

Out

:17

/10/

86–

30/1

2/88

DR

W,H

IS,M

A,E

S,E

WM

A,A

R,G

AR

CH

,E

GA

RC

H,G

JR,

Neu

raln

etw

ork

(all

abou

tthe

sam

e)

1da

yah

ead

squa

red

retu

rns

usin

gro

lling

2000

obse

rvat

ions

for

estim

atio

n

MSE

,MA

Eof

vari

ance

,%ov

erpr

edic

t.R

2

isar

ound

4%in

crea

ses

to24

%fo

rpre

-cra

shda

ta

Sim

ilar

perf

orm

ance

acro

ssm

odel

ses

peci

ally

whe

n87

’scr

ash

isex

clud

ed.

Soph

istic

ated

mod

els

such

asG

AR

CH

and

neur

alne

tdi

dno

tdom

inat

e.V

olum

edi

dno

thel

pin

fore

cast

ing

vola

tility

16.

Can

ina

and

Figl

ewsk

i(1

993)

S&P1

00(O

EX

)15

/3/8

3–28

/3/8

7(p

re-c

rash

)

DH

IS60

cale

ndar

days

Impl

ied B

inom

ialC

all

(ran

ked)

Impl

ied

in4

mat

urity

gp,e

ach

subd

ivid

edin

to8

intr

insi

cgp

7to

127

cale

ndar

days

mat

chin

gop

tion

mat

urity

,ov

erla

ppin

gfo

reca

sts

with

Han

sen

std

erro

r.U

sesa

mpl

eSD

ofda

ilyre

turn

sto

prox

y‘a

ctua

lvo

l.’

Com

bine

dR

2is

17%

with

little

cont

ribu

tion

from

impl

ied.

All

αim

plie

d>

0,β

impl

ied

<1

with

robu

stSE

Impl

ied

has

noco

rrel

atio

nw

ithfu

ture

vola

tility

and

does

noti

ncor

pora

tein

fo.

cont

aine

din

rece

ntly

obse

rved

vola

tility

.Res

ults

appe

arto

bepe

culia

rfo

rpr

e-cr

ash

peri

od.T

ime

hori

zon

of‘a

ctua

lvol

.’ch

ange

sda

yto

day.

Dif

fere

ntle

velo

fim

plie

dag

greg

atio

npr

oduc

essi

mila

rre

sults

17.

Cao

and

Tsa

y(1

992)

Exc

ess

retu

rns

for

S&P,

VW

EW

indi

ces

1928

–198

9M

TAR

EG

AR

CH

(1,0

)A

RM

A(1

,1)

GA

RC

H(1

,1)

(ran

ked)

1to

30m

onth

s.E

stim

atio

npe

riod

rang

esfr

om68

4to

743

mon

ths

Dai

lyre

turn

sus

edto

cons

truc

t‘a

ctua

lvol

.’

MSE

,MA

ETA

Rpr

ovid

esbe

stfo

reca

sts

for

larg

est

ocks

.E

GA

RC

Hgi

ves

best

long

-hor

izon

fore

cast

sfo

rsm

alls

tock

s(m

aybe

due

tole

vera

geef

fect

).D

iffe

renc

ein

MA

Eca

nbe

asla

rge

as38

%

Con

tinu

ed

155


Dat

aD

ata

Fore

cast

ing

Fore

cast

ing

Eva

luat

ion

Aut

hor(

s)A

sset

(s)

peri

odfr

eque

ncy

met

hods

and

rank

hori

zon

and

R-s

quar

edC

omm

ents

18.

Chi

ras

and

Man

aste

r(1

978)

All

stoc

kop

tions

from

CB

OE

23m

onth

sfr

omJu

n73

toA

pr75

MIm

plie

d(w

eigh

ted

bypr

ice

elas

ticity

)H

IS20

mon

ths

(ran

ked)

20m

onth

sah

ead.

Use

SDof

20m

onth

lyre

turn

sto

prox

y‘a

ctua

lvo

l.’

Cro

ss-s

ectio

nal

R2

ofim

plie

dra

nges

13–5

0%ac

ross

23m

onth

s.H

ISad

ds0–

15%

toR

2

Impl

ied

outp

erfo

rmed

HIS

espe

cial

lyin

the

last

14m

onth

s.Fi

ndim

plie

din

crea

ses

and

bette

rbe

have

afte

rdi

vide

ndad

just

men

tsan

dev

iden

ceof

mis

pric

ing

poss

ibly

due

toth

eus

eE

urop

ean

pric

ing

mod

elon

Am

eric

anst

yle

optio

ns

19.

Chr

iste

nsen

and

Prab

hala

(199

8)

S&P1

00(O

EX

)

Mon

thly

expi

rycy

cle

Nov

83–

May

95M

Impl

ied B

SA

TM

1-m

onth

Cal

lH

IS18

days

(ran

ked)

Non

-ove

rlap

ping

24ca

lend

ar(o

r18

trad

ing)

days

.U

seSD

ofda

ilyre

turn

sto

prox

y‘a

ctua

lvol

.’

R2

oflo

gva

rar

e39

%(i

mpl

ied)

,32

%(H

IS)

and

41%

(com

bine

d).

α<

0(b

ecau

seof

log)

,β<

1w

ithro

bust

SE.

Impl

ied

ism

ore

bias

edbe

fore

the

cras

h

Not

adj.

for

divi

dend

and

earl

yex

erci

se.I

mpl

ied

dom

inat

esH

IS.H

ISha

sno

addi

tiona

linf

orm

atio

nin

subp

erio

dan

alys

is.

Prov

edth

atre

sults

inC

anin

aan

dFi

glew

ski

(199

3)is

due

topr

e-cr

ash

char

acte

rist

ics

and

high

degr

eeof

data

over

lap

rela

tive

totim

ese

ries

leng

th.I

mpl

ied

isun

bias

edaf

ter

cont

rolli

ngfo

rm

easu

rem

ente

rror

sus

ing

impl

ied t

−1an

dH

ISt−

1

156


20.

Chr

isto

ffer

sen

and

Die

bold

(200

0)

4st

kin

dice

s4

exra

tes

US

10-y

ear

T-B

ond

1/1/

73–

1/5/

97D

No

mod

el.

(no

rank

;eva

luat

evo

latil

ityfo

reca

stab

ility

(or

pers

iste

nce)

bych

ecki

ngin

terv

alfo

reca

sts)

1to

20da

ysR

unte

sts

and

Mar

kov

tran

sitio

nm

atri

xei

genv

alue

s(w

hich

isba

sica

lly1s

t-or

der

seri

alco

effic

ient

ofth

ehi

tseq

uenc

ein

the

run

test

)

Equ

ityan

dFX

:fo

reca

stab

ility

decr

ease

rapi

dly

from

1to

10da

ys.B

ond:

may

exte

ndas

long

as15

to20

days

.E

stim

ate

bond

retu

rns

from

bond

yiel

dsby

assu

min

gco

upon

equa

lto

yiel

d

21.

Cum

by,

Figl

ewsk

ian

dH

asbr

ouck

(199

3)

¥/$,

stoc

ks(¥

,$),

bond

s(¥

,$)

7/77

to9/

90W

EG

AR

CH

HIS

(ran

ked)

1w

eek

ahea

d,es

timat

ion

peri

odra

nges

from

299

to68

9w

eeks

R2

vari

esfr

om0.

3%to

10.6

%.

EG

AR

CH

isbe

tter

than

naiv

ein

fore

cast

ing

vola

tility

thou

ghR

-squ

ared

islo

w.

Fore

cast

ing

corr

elat

ion

isle

sssu

cces

sful

22.

Day

and

Lew

is(1

992)

S&P1

00O

EX

optio

nO

ut:

11/1

1/83

–31

/12/

89

WIm

plie

d BS

Cal

l(s

hort

est

but>

7da

ys,

volu

me

WL

S)H

IS1

wee

k

GA

RC

HE

GA

RC

H(r

anke

d)

1w

eek

ahea

des

timat

edfr

oma

rolli

ngsa

mpl

eof

410

obse

rvat

ions

.U

sesa

mpl

eva

rian

ceof

daily

retu

rns

topr

oxy

wee

kly

‘act

ual

vol.’

R2

ofva

rian

cere

gr.a

re2.

6%(i

mpl

ied)

and

3.8%

(enc

omp.

).A

llfo

reca

sts

add

mar

gina

lin

fo.H

0:

αim

plie

d=

0,β

impl

ied

=1

cann

otbe

reje

cted

with

robu

stSE

Om

itea

rly

exer

cise

.E

ffec

tof

87’s

cras

his

uncl

ear.

Whe

nw

eekl

ysq

uare

dre

turn

sw

ere

used

topr

oxy

‘act

ual

vol,’

R2

incr

ease

and

was

max

for

HIS

cont

rary

toex

pect

atio

n(9

%co

mpa

red

with

3.7%

for

impl

ied)

Rec

onst

ruct

edS&

P100

In:2

/1/7

6–11

/11/

83

Con

tinu

ed

157


Dat

aD

ata

Fore

cast

ing

Fore

cast

ing

Eva

luat

ion

Aut

hor(

s)A

sset

(s)

peri

odfr

eque

ncy

met

hods

and

rank

hori

zon

and

R-s

quar

edC

omm

ents

23.

Day

and

Lew

is(1

993)

Cru

deoi

lfu

ture

sop

tions

14/1

1/86

–18/

3/91

8/4/

83–1

8/3/

91co

inci

dew

ithK

uwai

tinv

asio

nby

Iraq

inse

cond

half

ofsa

mpl

e

DIm

plie

d Bin

omia

lAT

MC

all

HIS

fore

cast

hori

zon

GA

RC

H-M

EG

AR

CH

-AR

(1)

(ran

ked)

Opt

ion

mat

urity

of4

near

byco

ntra

cts,

(ave

rage

13.9

,32.

5,50

.4an

d68

trad

ing

days

tom

atur

ity).

ME

,RM

SE,

MA

E.

R2

ofva

rian

cere

gr.a

re72

%(s

hort

mat

.)an

d49

%(l

ong

mat

urity

).W

ithro

bust

SEα

>0

for

shor

tand

α=

0fo

rlo

ng,

β=

1fo

ral

lm

atur

ity

Impl

ied

perf

orm

edex

trem

ely

wel

l.Pe

rfor

man

ceof

HIS

and

GA

RC

Har

esi

mila

r.E

GA

RC

Hm

uch

infe

rior

.Bia

sad

just

edan

dco

mbi

ned

fore

cast

sdo

not

perf

orm

asw

ella

sun

adju

sted

impl

ied.

GA

RC

Hha

sno

incr

emen

tal

info

rmat

ion.

Res

ult

likel

yto

bedr

iven

byK

uwai

tinv

asio

nby

Iraq

Cru

deoi

lfu

ture

sE

stim

ated

from

rolli

ng50

0ob

serv

atio

ns

24.

Dim

son

and

Mar

sh(1

990)

UK

FTA

llSh

are

1955

–89

QE

S,R

egre

ssio

nR

W,H

A,M

A(r

anke

d)

Nex

tqua

rter

.Use

daily

retu

rns

toco

nstr

uct‘

actu

alvo

l.’

MSE

,RM

SE,

MA

E,R

MA

ER

ecom

men

dex

pone

ntia

lsm

ooth

ing

and

regr

essi

onm

odel

usin

gfix

edw

eigh

ts.

Find

exan

tetim

e-va

ryin

gop

timiz

atio

nof

wei

ghts

does

notw

ork

wel

lex

post

158


25.

Doi

dge

and

Wei

(199

8)To

ront

o35

stoc

kin

dex

&E

urop

ean

optio

ns

In:2

/8/8

8–31

/12/

91O

ut:1

/92–

7/95

DC

ombi

ne3

Com

bine

2G

AR

CH

EG

AR

CH

HIS

100

days

Com

bine

1Im

plie

d BS

Cal

l+Pu

t

(All

mat

uriti

es>

7da

ys,v

olum

eW

LS)

(ran

ked)

1m

onth

ahea

dfr

omro

lling

sam

ple

estim

atio

n.N

om

entio

non

how

‘act

ual

vol.’

was

deri

ved

MA

E,M

APE

,R

MSE

Com

bine

1eq

ual

wei

ghtf

orG

AR

CH

and

impl

ied

fore

cast

s.C

ombi

ne2

wei

ghs

GA

RC

Han

dim

plie

dba

sed

onth

eir

rece

ntfo

reca

stac

cura

cy.

Com

bine

3pu

tsim

plie

din

GA

RC

Hco

nditi

onal

vari

ance

.Com

bine

3w

ases

timat

edus

ing

full

sam

ple

due

toco

nver

genc

epr

oble

m;

sono

trea

llyou

t-of

-sam

ple

fore

cast

26.

Dun

is,L

aws

and

Cha

uvin

(200

0)

DM

/¥,£

/DM

,£/

$,$/

CH

F,$/

DM

,$/¥

In:2

/1/9

1–27

/2/9

8O

ut:2

/3/9

8–31

/12/

98

DG

AR

CH

(1,1

)A

R(1

0)-S

qre

turn

sA

R(1

0)-A

bsre

turn

sSV

(1)

inlo

gfo

rmH

IS21

or63

trad

ing

days

1-&

3-M

forw

ard

Impl

ied A

TM

quot

es

Com

bine

Com

bine

(exc

ept

SV)

(ran

kch

ange

sac

ross

curr

enci

esan

dfo

reca

stho

rizo

ns)

1an

d3

mon

ths

(21

and

63tr

adin

gda

ys)

with

rolli

nges

timat

ion.

Act

ual

vola

tility

isca

lcul

ated

asth

eav

erag

eab

solu

tere

turn

over

the

fore

cast

hori

zon

RM

SE,M

AE

,M

APE

,The

il-U

,C

DC

(Cor

rect

Dir

ectio

nal

Cha

nge

inde

x)

No

sing

lem

odel

dom

inat

esth

ough

SVis

cons

iste

ntly

wor

st,

and

impl

ied

alw

ays

impr

oves

fore

cast

accu

racy

.Rec

omm

end

equa

lwei

ghtc

ombi

ned

fore

cast

excl

udin

gSV

Con

tinu

ed

159


Dat

aD

ata

Fore

cast

ing

Fore

cast

ing

Eva

luat

ion

Aut

hor(

s)A

sset

(s)

peri

odfr

eque

ncy

met

hods

and

rank

hori

zon

and

R-s

quar

edC

omm

ents

27.

Ede

ring

ton

and

Gua

n(1

999)

‘Fro

wn’

S&P5

00fu

ture

sop

tions

1Jan

88–3

0Apr

98D

Impl

ied B

K16

Cal

ls16

Puts

HIS

40da

ys

(ran

ked)

Ove

rlap

ping

10to

35da

ysm

atch

ing

mat

urity

ofne

ares

tto

expi

ryop

tion.

Use

SDof

daily

retu

rns

topr

oxy

‘act

ual

vol.’

Pane

lR

219

%an

din

divi

dual

R2

rang

es6–

17%

(cal

ls)

and

15–3

6%(p

uts)

.Im

plie

dis

bias

edan

din

effic

ient

,α

impl

ied

>0

and

βim

plie

d<

1w

ithro

bust

SE

Info

rmat

ion

cont

ento

fim

plie

dac

ross

stri

kes

exhi

bita

frow

nsh

ape

with

optio

nsth

atar

eN

TM

and

have

mod

erat

ely

high

stri

kes

poss

ess

larg

est

info

rmat

ion

cont

ent.

HIS

typi

cally

adds

2–3%

toth

eR

2an

dno

nlin

ear

impl

ied

term

sad

dan

othe

r2–

3%.I

mpl

ied

isun

bias

edan

def

ficie

ntw

hen

mea

sure

men

ter

ror

isco

ntro

lled

usin

gIm

plie

d t−1

and

HIS

t−1.

160


28.

Ede

ring

ton

and

Gua

n(2

000a

)‘F

orec

astin

gvo

latil

ity’

5D

JSt

ocks

S&P5

003m

Eur

o$ra

te10

yrT-

Bon

dyi

eld

DM

/$

2/7/

62–3

0/12

/94

2/7/

62–2

9/12

/95

1/1/

73–2

0/6/

972/

1/62

–13/

6/97

1/1/

71–3

0/6/

97

DG

WM

AD

GW

STD

,G

AR

CH

,EG

AR

CH

AG

AR

CH

HIS

MA

D,n,

HIS

STD

,n

(ran

ked,

erro

rst

atis

tics

are

clos

e;G

WM

AD

lead

sco

nsis

tent

lyth

ough

with

only

smal

lmar

gin)

n=

10,2

0,40

,80

and

120

days

ahea

des

timat

edfr

oma

1260

-day

rolli

ngw

indo

w;

para

met

ers

re-e

stim

ated

ever

y40

days

.U

seda

ilysq

uare

dde

viat

ion

topr

oxy

‘act

ual

vol.’

RM

SE,M

AE

GW

:geo

met

ric

wei

ght,

MA

D:m

ean

abso

lute

devi

atio

n,ST

D:s

tand

ard

devi

atio

n.V

olat

ility

aggr

egat

edov

era

long

erpe

riod

prod

uces

abe

tter

fore

cast

.Abs

olut

ere

turn

sm

odel

sge

nera

llype

rfor

mbe

tter

than

squa

rere

turn

sm

odel

s(e

xcep

tGA

RC

H>

AG

AR

CH

).A

sho

rizo

nle

ngth

ens,

nopr

oced

ure

dom

inat

es.

GA

RC

Han

dE

GA

RC

Hes

timat

ions

wer

eun

stab

leat

times

Con

tinu

ed

161


Dat

aD

ata

Fore

cast

ing

Fore

cast

ing

Eva

luat

ion

Aut

hor(

s)A

sset

(s)

peri

odfr

eque

ncy

met

hods

and

rank

hori

zon

and

R-s

quar

edC

omm

ents

29.

Ede

ring

ton

and

Gua

n(2

000b

)‘A

vera

ging

’

S&P5

00fu

ture

sop

tions

In:4

/1/8

8–31

/12/

91D

Impl

ied:

∗99

%,V

XO

HIS

40tr

adin

gda

ys

Impl

ied:

VX

O>

Eq4

Impl

ied:

WL

S>

vega

>E

q32

>el

astic

ity(r

anke

d)

Ove

rlap

ping

10to

35da

ysm

atch

ing

mat

urity

ofne

ares

tto

expi

ryop

tion.

Use

SDof

daily

retu

rns

topr

oxy

‘act

ual

vol.’

RM

SE,M

AE

,M

APE

VX

O:2

calls

+2pu

ts,

NT

Mw

eigh

ted

toge

tA

TM

.Eq4

/32:

calls

+put

seq

ually

wei

ghte

d.W

LS,

vega

and

elas

ticity

are

othe

rw

eigh

ting

sche

me.

99%

mea

ns1%

ofre

gr.

erro

rus

edin

wei

ghtin

gal

lim

plie

ds.O

nce

the

bias

edne

ssha

sbe

enco

rrec

ted

usin

gre

gr.,

little

isto

bega

ined

byan

yav

erag

ing

insu

cha

high

lyliq

uid

S&P5

00fu

ture

sm

arke

t

Out

:2/

1/92

–31/

12/9

2‘*

’in

dica

tes

indi

vidu

alim

plie

dsw

ere

corr

ecte

dfo

rbi

ased

ness

first

befo

reav

erag

ing

usin

gin

-sam

ple

regr

.on

real

ized

162


30.

Ede

ring

ton

and

Gua

n(2

002)

‘Effi

cien

tpr

edic

tor’

S&P5

00fu

ture

sop

tions

1/1/

83–1

4/9/

95D

Impl

ied B

lack

4NT

M

GA

RC

H,H

IS40

days

(ran

ked)

Ove

rlap

ping

optio

nm

atur

ity7–

90,9

1–18

0,18

1–36

5an

d7–

365

days

ahea

d.U

sesa

mpl

eSD

over

fore

cast

hori

zon

topr

oxy

‘act

ual

vol.’

R2

rang

es22

–12%

from

shor

tto

long

hori

zon.

Post

87’s

cras

hR

2ne

arly

doub

led.

Impl

ied

isef

ficie

ntbi

ased

;α

impl

ied

>0

and

βim

plie

d<

1w

ithro

bust

SE

GA

RC

Hpa

ram

eter

sw

ere

estim

ated

usin

gw

hole

sam

ple.

GA

RC

Han

dH

ISad

dlit

tleto

7–90

day

R2.

Whe

n87

’scr

ash

was

excl

uded

HIS

add

sig.

expl

anat

ory

pow

erto

181–

365

day

fore

cast

.W

hen

mea

sure

men

ter

rors

wer

eco

ntro

lled

usin

gim

plie

d t−5

and

impl

ied t

+5as

inst

rum

entv

aria

bles

impl

ied

beco

mes

unbi

ased

for

the

who

lepe

riod

butr

emai

nsbi

ased

whe

ncr

ash

peri

odw

asex

clud

ed

Con

tinu

ed

163


Dat

aD

ata

Fore

cast

ing

Fore

cast

ing

Eva

luat

ion

Aut

hor(

s)A

sset

(s)

peri

odfr

eque

ncy

met

hods

and

rank

hori

zon

and

R-s

quar

edC

omm

ents

31.

Ede

yan

dE

lliot

(199

2)Fu

ture

sop

tions

onA

$90

d-B

ill,

10yr

bond

,Sto

ckin

dex

Futu

res

optio

ns:

ince

ptio

nto

12/8

8

WIm

plie

d BK

NT

M,c

all

Impl

ied B

KN

TM

,pu

t

Opt

ion

mat

urity

upto

3M.U

sesu

mof

(ret

urn

squa

repl

usIm

plie

d t+1

)as

‘act

ualv

ol.’

Reg

ress

ion

(see

com

men

t).I

nm

ostc

ases

αim

plie

d>

0an

dβ

impl

ied

<1

with

robu

stSE

.For

stoc

kin

dex

optio

nβ

impl

ied

=1

cann

otbe

reje

cted

usin

gro

bust

SE

R2

cann

otbe

com

pare

dw

ithot

her

stud

ies

beca

use

ofth

ew

ay‘a

ctua

l’is

deri

ved

and

lagg

edsq

uare

sre

turn

sw

ere

adde

dto

the

RH

S

A$/

US$

optio

nsA

$/U

S$op

tion:

12/8

4–12

/87

W

(No

rank

,1ca

llan

d1

put,

sele

cted

base

don

high

estt

radi

ngvo

lum

e)

Con

stan

t1M

.U

sesu

mof

wee

kly

squa

red

retu

rns

topr

oxy

‘act

ualv

ol.’

32.

Eng

le,N

gan

dR

oths

child

(199

0)

1to

12m

onth

sT-

Bill

retu

rns,

VW

inde

xof

NY

SE&

AM

SEst

ocks

Aug

64–N

ov85

M1-

fact

orA

RC

HU

niva

riat

eA

RC

H-M

(ran

ked)

1m

onth

ahea

dvo

latil

ityan

dri

skpr

emiu

mof

2to

12m

onth

sT-

Bill

s

Mod

elfit

Equ

ally

wei

ghte

dbi

llpo

rtfo

liois

effe

ctiv

ein

pred

ictin

g(i

.e.i

nan

expe

ctat

ion

mod

el)

vola

tility

and

risk

prem

iaof

indi

vidu

alm

atur

ities

164


33.

Fein

stei

n(1

989b

)S&

P500

futu

res

optio

ns(C

ME

)Ju

n83–

Dec

88O

ptio

nex

piry

cycl

eIm

plie

d:A

tlant

a>

aver

age

>ve

ga>

elas

ticity

Just

-O

TM

Cal

l>P+

C>

Put

HIS

20da

ys

(ran

ked,

note

pre-

cras

hra

nkis

very

diff

eren

tand

erra

tic)

23no

n-ov

erla

ppin

gfo

reca

sts

of57

,38

and

19da

ysah

ead.

Use

sam

ple

SDof

daily

retu

rns

over

the

optio

nm

atur

ityto

prox

y‘a

ctua

lvo

l.’

MSE

,MA

E,M

E.

T-te

stin

dica

tes

allM

E>

0(e

xcep

tHIS

)in

the

post

-cra

shpe

riod

whi

chm

eans

impl

ied

was

upw

ardl

ybi

ased

Atla

nta:

5-da

yav

erag

eof

Just

-OT

Mca

llim

plie

dus

ing

expo

nent

ialw

eigh

ts.

Inge

nera

lJus

t-O

TM

Impl

ied c

allis

the

best

34.

Ferr

eira

(199

9)Fr

ench

&G

erm

anin

terb

ank

1Mm

id-r

ate

In:

Jan8

1–D

ec89

Out

:Ja

n90–

Dec

97

(ER

Mcr

ises

:Se

p92–

Sep9

3)

WE

S,H

IS26

,52

,al

l

GA

RC

H(-

L)

(E)G

JR(-

L)

(ran

kva

ries

betw

een

Fren

chan

dG

erm

anra

tes,

sam

plin

gm

etho

dan

der

ror

stat

istic

s)

1w

eek

ahea

d.U

seda

ilysq

uare

dra

tech

ange

sto

prox

yw

eekl

yvo

latil

ity

Reg

ress

ion,

MPE

,MA

PE,

RM

SPE

.R

2is

41%

for

Fran

cean

d3%

for

Ger

man

y

L:i

nter

estr

ate

leve

l,E

:exp

onen

tial.

Fren

chra

tew

asve

ryvo

latil

edu

ring

ER

Mcr

ises

.Ger

man

rate

was

extr

emel

yst

able

inco

ntra

st.

Alth

ough

ther

ear

elo

tsof

diff

eren

ces

betw

een

the

two

rate

s,be

stm

odel

sar

eno

npar

amet

ric;

ES

(Fre

nch)

and

sim

ple

leve

leff

ect

(Ger

man

).Su

gges

tadi

ffer

enta

ppro

ach

isne

eded

for

fore

cast

ing

inte

rest

rate

vola

tility C

onti

nued

165


Dat

aD

ata

Fore

cast

ing

Fore

cast

ing

Eva

luat

ion

Aut

hor(

s)A

sset

(s)

peri

odfr

eque

ncy

met

hods

and

rank

hori

zon

and

R-s

quar

edC

omm

ents

35.

Figl

ewsk

i(1

997)

S&P5

003m

US

T-B

ill20

yrT-

Bon

dD

M/$

1/47

–12/

951/

47–1

2/95

1/50

–7/9

31/

71–1

1/95

MH

IS6,

12,24

,36

,48

,60

m

GA

RC

H(1

,1)

for

S&P

and

bond

yiel

d.(r

anke

d)

6,12

,24,

36,

48,6

0m

onth

s.U

seda

ilyre

turn

sto

com

pute

‘act

ual

vol.’

RM

SEFo

reca

stof

vola

tility

ofth

elo

nges

tho

rizo

nis

the

mos

tac

cura

te.H

ISus

esth

elo

nges

tes

timat

ion

peri

odis

the

best

exce

ptfo

rsh

ortr

ate

S&P5

003m

US

T-B

ill20

yrT-

Bon

dD

M/$

2/7/

62–

29/1

2/95

2/1/

62–

29/1

2/95

2/1/

62–

29/1

2/95

4/1/

71–

30/1

1/95

DG

AR

CH

(1,1

)H

IS1,

3,6,

12,24

,60

mon

ths

(S&

P’s

rank

,rev

erse

for

the

othe

rs)

1,3,

6,12

,24

mon

ths

RM

SEG

AR

CH

isbe

stfo

rS&

Pbu

tgav

ew

orst

perf

orm

ance

inal

lth

eot

her

mar

kets

.In

gene

ral,

asou

t-of

-sam

ple

hori

zon

incr

ease

s,th

ein

-sam

ple

leng

thsh

ould

also

incr

ease

166


36.

Figl

ewsk

iand

Gre

en(1

999)

S&P5

00U

SL

IBO

R10

yrT-

Bon

dyi

eld

DM

/$

1/4/

71–

12/3

1/96

Out

:Fro

mFe

b96

DH

is3,

12,60

mon

ths

ES

(ran

kva

ries

)1,

3,12

mon

ths

for

daily

data

RM

SEE

Sw

orks

best

for

S&P

(1–3

mon

th)

and

shor

trat

e(a

llth

ree

hori

zons

).H

ISw

orks

best

for

bond

yiel

d,ex

chan

gera

tean

dlo

ngho

rizo

nS&

Pfo

reca

st.T

helo

nger

the

fore

cast

hori

zon,

the

long

erth

ees

timat

ion

peri

od

1/4/

71–

12/3

1/96

Out

:Fro

mJa

n92

MH

is26

,60

,al

lmon

ths

ES

(ran

ked)

24an

d60

mon

ths

for

mon

thly

data

For

S&P,

bond

yiel

dan

dD

M/$

,iti

sbe

stto

use

alla

vaila

ble

‘mon

thly

’da

ta.5

year

’sw

orth

ofda

taw

orks

best

for

shor

tra

te

Con

tinu

ed

167


Dat

aD

ata

Fore

cast

ing

Fore

cast

ing

Eva

luat

ion

Aut

hor(

s)A

sset

(s)

peri

odfr

eque

ncy

met

hods

and

rank

hori

zon

and

R-s

quar

edC

omm

ents

37.

Flem

ing

(199

8)S&

P100

(OE

X)

10/8

5–4/

92(a

llob

serv

atio

nsth

atov

erla

pw

ith87

’scr

ash

wer

ere

mov

ed)

DIm

plie

d FW

AT

Mca

lls

Impl

ied F

WA

TM

puts

(bot

him

plie

dsar

eW

LS

usin

gal

lA

TM

optio

nsin

the

last

10m

inut

esbe

fore

mar

ket

clos

e)A

RC

H/G

AR

CH

HIS

H-L

28da

ys

(ran

ked)

Opt

ion

mat

urity

(sho

rtes

tbut

>15

days

,av

erag

e30

cale

ndar

days

),1

and

28da

ysah

ead.

Use

daily

squa

rere

turn

devi

atio

nsto

prox

y‘a

ctua

lvo

l.’

R2

is29

%fo

rm

onth

lyfo

reca

stan

d6%

for

daily

fore

cast

.All

αim

plie

d=

0,β

impl

ied

<1

with

robu

stSE

for

the

last

two

fixed

hori

zon

fore

cast

s

Impl

ied

dom

inat

es.

All

othe

rva

riab

les

rela

ted

tovo

latil

itysu

chas

stoc

kre

turn

s,in

tere

stra

tean

dpa

ram

eter

sof

GA

RC

Hdo

not

poss

ess

info

rmat

ion

incr

emen

talt

oth

atco

ntai

ned

inim

plie

d

38.

Flem

ing,

Kir

byan

dO

stdi

ek(2

000)

S&P5

00,

T-B

ond

and

gold

futu

res

3/1/

83–

31/1

2/97

DE

xpon

entia

llyw

eigh

ted

var-

cov

mat

rix

Dai

lyre

bala

nced

port

folio

Shar

pera

tio(p

ortf

olio

retu

rnov

erri

sk)

Effi

cien

tfro

ntie

rof

vola

tility

timin

gst

rate

gypl

otte

dab

ove

that

offix

edw

eigh

tpor

tfol

io

168


39.

Flem

ing,

Ost

diek

and

Wha

ley

(199

5)

S&P1

00(V

XO

)Ja

n86–

Dec

92D

,WIm

plie

dV

XO

HIS

20da

ys

(ran

ked)

28ca

lend

ar(o

r20

trad

ing)

day.

Use

sam

ple

SDof

daily

retu

rns

topr

oxy

‘act

ual

vol.’

R2

incr

ease

dfr

om15

%to

45%

whe

ncr

ash

isex

clud

ed.

αV

XO

=0,

βV

XO

<1

with

robu

stSE

VX

Odo

min

ates

HIS

,but

isbi

ased

upw

ard

upto

580

basi

spo

ints

.O

rtho

gona

lity

test

reje

cts

HIS

whe

nV

XO

isin

clud

ed.

Adj

ustV

XO

fore

cast

sw

ithav

erag

efo

reca

ster

rors

ofth

ela

st25

3da

yshe

lps

toco

rrec

tfo

rbi

ased

ness

whi

lere

tain

ing

impl

ied’

sex

plan

ator

ypo

wer

40.

Fran

ses

and

Ghi

jsel

s(1

999)

Dut

ch,G

erm

an,

Span

ish

and

Ital

ian

stoc

km

arke

tret

urns

1983

–94

WA

O-G

AR

CH

(GA

RC

Had

just

edfo

rad

ditiv

eou

tlier

sus

ing

the

‘les

s-on

e’m

etho

d)G

AR

CH

GA

RC

H-t

(ran

ked)

1w

eek

ahea

des

timat

edfr

ompr

evio

us4

year

s.U

sew

eekl

ysq

uare

dde

viat

ions

topr

oxy

‘act

ual

vol.’

MSE

and

Med

SEFo

reca

stin

gpe

rfor

man

cesi

gnifi

cant

lyim

prov

edw

hen

para

met

eres

timat

esar

eno

tinfl

uenc

edby

‘out

liers

’.Pe

rfor

man

ceof

GA

RC

H-t

isco

nsis

tent

lym

uch

wor

se.S

ame

resu

ltsfo

ral

lfou

rst

ock

mar

kets

Con

tinu

ed

169


Dat

aD

ata

Fore

cast

ing

Fore

cast

ing

Eva

luat

ion

Aut

hor(

s)A

sset

(s)

peri

odfr

eque

ncy

met

hods

and

rank

hori

zon

and

R-s

quar

edC

omm

ents

41.

Fran

ses

and

Van

Dijk

(199

6)

Stoc

kin

dice

s(G

erm

any,

Net

herl

ands

,Sp

ain,

Ital

y,Sw

eden

)

1986

–94

WQ

GA

RC

HR

WG

AR

CH

GJR

(ran

ked)

1w

eek

ahea

des

timat

edfr

omro

lling

4ye

ars.

Use

wee

kly

squa

red

devi

atio

nsto

prox

y‘a

ctua

lvo

l.’

Med

SEQ

GA

RC

His

best

ifda

taha

sno

extr

emes

.RW

isbe

stw

hen

87’s

cras

his

incl

uded

.GJR

cann

otbe

reco

mm

ende

d.R

esul

tsar

elik

ely

tobe

influ

ence

dby

Med

SEth

atpe

naliz

eno

nsym

met

ry.

Bra

ilsfo

rdan

dFa

ff(1

996)

supp

ortG

JRas

best

mod

elal

thou

ghit

unde

rpre

dict

sov

er70

%of

the

time

170


42.

Fren

nber

gan

dH

anss

on(1

996)

VW

Swed

ish

stoc

km

arke

tre

turn

s

Inde

xop

tion

(Eur

opea

nst

yle)

In:

1919

–197

6O

ut:1

977–

82,

1983

–90

Jan8

7–D

ec90

MA

R12

(AB

S)-S

RW

,Im

plie

d BS

AT

MC

all

(opt

ion

mat

urity

clos

estt

o1

mon

th)

GA

RC

H-S

,AR

CH

-S(r

anke

d)

Mod

els

that

are

not

adj.

for

seas

onal

itydi

dno

tper

form

asw

ell

1m

onth

ahea

des

timat

edfr

omre

curs

ivel

yre

-est

imat

edex

pand

ing

sam

ple.

Use

daily

ret.

toco

mpi

lem

onth

lyvo

l.,ad

just

edfo

rau

toco

rrel

atio

n

MA

PE,

R2

is2–

7%in

first

peri

odan

d11

–24%

inse

cond

,mor

evo

latil

epe

riod

.H

0:α

impl

ied

=0

and

βim

plie

d=

1ca

nnot

bere

ject

edw

ithro

bust

SE

S:se

ason

ality

adju

sted

.RW

mod

else

ems

tope

rfor

mre

mar

kabl

yw

elli

nsu

cha

smal

lsto

ckm

arke

twhe

rere

turn

sex

hibi

tst

rong

seas

onal

ity.

Opt

ion

was

intr

oduc

edin

86an

dco

vere

d87

’scr

ash;

outp

erfo

rmed

byR

W.A

RC

H/

GA

RC

Hdi

dno

tper

form

asw

elli

nth

em

ore

vola

tile

seco

ndpe

riod

43.

Fung

,Lie

and

Mor

eno

(199

0)

£/$,

C$/

$,FF

r$,D

M/$

,¥/

$&

SrFr

/$op

tions

onPH

LX

1/84

–2/8

7(p

re-c

rash

)D

Impl

ied O

TM

>A

TM

Impl

ied v

ega,

elas

ticity

Impl

ied e

qual

wei

ght

HIS

40da

ys,I

mpl

ied I

TM

(ran

ked,

alli

mpl

ied

are

from

calls

)

Opt

ion

mat

urity

;ov

erla

ppin

gpe

riod

s.U

sesa

mpl

eSD

ofda

ilyre

turn

sov

erop

tion

mat

urity

topr

oxy

‘act

ual

vol.’

RM

SE,M

AE

ofov

erla

ppin

gfo

reca

sts

Eac

hda

y,5

optio

nsw

ere

stud

ied;

1A

TM

,2ju

stin

and

2ju

stou

t.D

efine

AT

Mas

S=

X,

OT

Mm

argi

nally

outp

erfo

rmed

AT

M.

Mix

edto

geth

erim

plie

dof

diff

eren

tco

ntra

ctm

onth

s

Con

tinu

ed

171


Dat

aD

ata

Fore

cast

ing

Fore

cast

ing

Eva

luat

ion

Aut

hor(

s)A

sset

(s)

peri

odfr

eque

ncy

met

hods

and

rank

hori

zon

and

R-s

quar

edC

omm

ents

44.

Fung

and

Hsi

eh(1

991)

S&P5

00,D

M$

US

T-bo

nd

Futu

res

and

futu

res

optio

ns

3/83

–7/8

9(D

M/$

futu

res

from

26Fe

b85

)

D(1

5m

in)

RV

-AR

(n)

Impl

ied B

AW

NT

MC

all/P

ut

RV

,RW

(C-t

-C)

HL

(ran

ked,

som

eof

the

diff

eren

ces

are

smal

l)

1da

yah

ead.

Use

15-m

inda

tato

cons

truc

t‘a

ctua

lvol

.’

RM

SEan

dM

AE

oflo

gσ

RV

:Rea

lized

vol.

from

15-m

inre

turn

s.A

R(n

):au

tore

gres

sive

lags

ofor

der

n.R

W(C

-t-C

):ra

ndom

wal

kfo

reca

stba

sed

oncl

ose

tocl

ose

retu

rns.

HL

:Pa

rkin

son’

sda

ilyhi

gh-l

owm

etho

d.Im

pact

of19

87cr

ash

does

not

appe

arto

bedr

astic

poss

ibly

due

tota

king

log.

Inge

nera

l,hi

gh-f

requ

ency

data

impr

oves

fore

cast

ing

pow

ergr

eatly

172


45.

Gem

mill

(198

6)13

UK

stoc

ksLT

OM

optio

ns.

Stoc

kpr

ice

May

78–J

ul83

Jan

78–N

ov83

M D

Impl

ied I

TM

Impl

ied A

TM

,veg

aW

LS

Impl

ied e

qual

,OT

M,e

last

icity

HIS

20W

eeks

(ran

ked,

alli

mpl

ied

are

from

calls

)

13–2

1no

n-ov

erla

ppin

gop

tion

mat

urity

(eac

hav

erag

e19

wee

ks).

Use

sam

ple

SDof

wee

kly

retu

rns

over

optio

nm

atur

ityto

prox

y‘a

ctua

lvo

l.’

ME

,RM

SE,

MA

Eag

greg

ated

acro

ssst

ocks

and

time.

R2

are

6–12

%(p

oole

d)an

d40

%(p

anel

with

firm

spec

ific

inte

rcep

ts).

All

α>

0,β

<1

Add

ing

HIS

incr

ease

sR

2fr

om12

%to

15%

.But

exan

teco

mbi

ned

fore

cast

from

HIS

and

Impl

ied I

TM

turn

edou

tto

bew

orse

than

indi

vidu

alfo

reca

sts.

Suff

ered

smal

lsa

mpl

ean

dno

nsyn

chro

neity

prob

lem

san

dom

itted

divi

dend

s

Con

tinu

ed

173


Dat

aD

ata

Fore

cast

ing

Fore

cast

ing

Eva

luat

ion

Aut

hor(

s)A

sset

(s)

peri

odfr

eque

ncy

met

hods

and

rank

hori

zon

and

R-s

quar

edC

omm

ents

46.

Gra

y(1

996)

US

1mT-

Bill

1/70

–4/9

4W

RSG

AR

CH

with

time

vary

ing

prob

abili

tyG

AR

CH

Con

stan

tvar

ianc

e(r

anke

d)

1w

eek

ahea

d(m

odel

not

re-e

stim

ated

).U

sew

eekl

ysq

uare

dde

viat

ion

topr

oxy

vola

tility

R2

calc

ulat

edw

ithou

tcon

stan

tte

rm,i

s4

to8%

for

RSG

AR

CH

,ne

gativ

efo

rso

me

CV

and

GA

RC

H.

Com

para

ble

RM

SEan

dM

AE

betw

een

GA

RC

Han

dR

SGA

RC

H

Vol

atili

tyfo

llow

sG

AR

CH

and

CIR

squa

rero

otpr

oces

s.In

tere

stra

teri

sein

crea

ses

prob

abili

tyof

switc

hing

into

high

-vol

atili

tyre

gim

e

Low

-vol

atili

type

rsis

tenc

ean

dst

rong

rate

leve

lm

ean

reve

rsio

nat

high

-vol

atili

tyst

ate.

Atl

ow-v

olat

ility

stat

e,ra

teap

pear

sra

ndom

wal

kan

dvo

latil

ityis

high

lype

rsis

tent

174


47.

Guo

(199

6a)

PHL

XU

S$/¥

optio

nsJa

n91–

Mar

93D

Impl

ied H

esto

n

Impl

ied H

W

Impl

ied B

S

GA

RC

HH

IS60

(ran

ked)

Info

rmat

ion

not

avai

labl

eR

egre

ssio

nw

ithro

bust

SE.N

oin

form

atio

non

R2

and

fore

cast

bias

edne

ss

Use

mid

ofbi

d–as

kop

tion

pric

eto

limit

‘bou

nce’

effe

ct.

Elim

inat

e‘n

onsy

nchr

onei

ty’

byus

ing

sim

ulta

neou

sex

chan

gera

tean

dop

tion

pric

e.H

ISan

dG

AR

CH

cont

ain

noin

crem

enta

lin

form

atio

n.Im

plie

d Hes

ton

and

Impl

ied H

War

eco

mpa

rabl

ean

dar

em

argi

nally

bette

rth

anIm

plie

d BS.

Onl

yha

veac

cess

toab

stra

ct

Con

tinu

ed

175


Dat

aD

ata

Fore

cast

ing

Fore

cast

ing

Eva

luat

ion

Aut

hor(

s)A

sset

(s)

peri

odfr

eque

ncy

met

hods

and

rank

hori

zon

and

R-s

quar

edC

omm

ents

48.

Guo

(199

6b)

PHL

XU

S$/¥,

US$

/DM

optio

ns

Spot

rate

Jan8

6–Fe

b93

Tic

k

D

Impl

ied H

W(W

LS,

0.8

<S/

X<

1.2,

20<

T<

60da

ys)

GA

RC

H(1

,1)

HIS

60da

ys

(ran

ked)

60da

ysah

ead.

Use

sam

ple

vari

ance

ofda

ilyre

turn

sto

prox

yac

tual

vola

tility

US$

/DM

R2

is4,

3,1%

for

the

thre

em

etho

ds.

(9,4

,1%

for

US$

/¥)

All

fore

cast

sar

ebi

ased

α>

0,β

<1

with

robu

stSE

Con

clus

ion

sam

eas

Guo

(199

6a).

Use

Bar

one-

Ade

si/W

hale

yap

prox

imat

ion

for

Am

eric

anop

tions

.N

ori

skpr

emiu

mfo

rvo

latil

ityva

rian

ceri

sk.G

AR

CH

has

noin

crem

enta

lin

form

atio

n.V

isua

lin

spec

tion

offig

ures

sugg

ests

impl

ied

fore

cast

sla

gged

actu

al

49.

Ham

id(1

998)

S&P5

00fu

ture

sop

tions

3/83

–6/9

3D

13sc

hem

es(i

nclu

ding

HIS

,im

plie

dcr

oss-

stri

keav

erag

ean

din

tert

empo

ral

aver

ages

)(r

anke

d,se

eco

mm

ent)

Non

-ov

erla

ppin

g15

,35

and

55da

ysah

ead

RM

SE,M

AE

Impl

ied

isbe

tter

than

hist

oric

alan

dcr

oss-

stri

keav

erag

ing

isbe

tter

than

inte

rtem

pora

lav

erag

ing

(exc

ept

duri

ngve

rytu

rbul

entp

erio

ds)

176


50.

Ham

ilton

and

Lin

(199

6)E

xces

sst

ock

retu

rns

(S&

P500

min

usT-

Bill

)&

Ind.

Prod

uctio

n

1/65

–6/9

3M

Biv

aria

teR

SAR

CH

Uni

vari

ate

RSA

RC

HG

AR

CH

+LA

RC

H+L

AR

(1)

(ran

ked)

1m

onth

ahea

d.U

sesq

uare

dm

onth

lyre

sidu

alre

turn

sto

prox

yvo

latil

ity

MA

EFo

und

econ

omic

rece

ssio

nsdr

ive

fluct

uatio

nsin

stoc

kre

turn

svo

latil

ity.‘

L’de

note

sle

vera

geef

fect

.RS

mod

elou

tper

form

edA

RC

H/G

AR

CH

+L51

.H

amilt

onan

dSu

smel

(199

4)

NY

SEV

Wst

ock

inde

x3/

7/62

–29

/12/

87W

RSA

RC

H+L

GA

RC

H+L

AR

CH

+L(r

anke

d)

1,4

and

8w

eeks

ahea

d.U

sesq

uare

dw

eekl

yre

sidu

alre

turn

sto

prox

yvo

latil

ity

MSE

,MA

E,

MSL

E,M

AL

E.

Err

ors

calc

ulat

edfr

omva

rian

cean

dlo

gva

rian

ce

Allo

win

gup

to4

regi

mes

with

tdi

stri

butio

n.R

SAR

CH

with

leve

rage

(L)

prov

ides

best

fore

cast

.Stu

dent

-tis

pref

erre

dto

GE

Dan

dG

auss

ian

52.

Har

vey

and

Wha

ley

(199

2)

S&P1

00(O

EX

)O

ct85

–Jul

89D

Impl

ied A

TM

calls

+put

s

(Am

eric

anbi

nom

ial,

shor

test

mat

urity

>15

days

)(p

redi

ctch

ange

sin

impl

ied)

1da

yah

ead

impl

ied

for

use

inpr

icin

gne

xtda

yop

tion

R2

is15

%fo

rca

llsan

d4%

for

puts

(exc

ludi

ng19

87cr

ash)

Impl

ied

vola

tility

chan

ges

are

stat

istic

ally

pred

icta

ble,

but

mar

ketw

asef

ficie

nt,

assi

mul

ated

tran

sact

ions

(NT

Mca

llan

dpu

tand

delta

hedg

edus

ing

futu

res)

did

not

prod

uce

profi

t

Con

tinu

ed

177


Dat

aD

ata

Fore

cast

ing

Fore

cast

ing

Eva

luat

ion

Aut

hor(

s)A

sset

(s)

peri

odfr

eque

ncy

met

hods

and

rank

hori

zon

and

R-s

quar

edC

omm

ents

53.

Hey

nen

and

Kat

(199

4)7

stoc

kin

dice

san

d5

exch

ange

rate

s

1/1/

80–

31/1

2/92

In:8

0–87

Out

:88–

92

(87’

scr

ash

incl

uded

inin

-sam

ple)

DSV

(?)

EG

AR

CH

GA

RC

HR

W(r

anke

d,se

eal

soco

mm

ent)

Non

-ov

erla

ppin

g5,

10,1

5,20

,25,

50,7

5,10

0da

ysho

rizo

nw

ithco

nsta

ntup

date

ofpa

ram

eter

ses

timat

es.U

sesa

mpl

est

anda

rdde

viat

ions

ofda

ilyre

turn

sto

prox

y‘a

ctua

lvo

l.’

Med

SESV

appe

ars

todo

min

ate

inin

dex

butp

rodu

ces

erro

rsth

atar

e10

times

larg

erth

an(E

)GA

RC

Hin

exch

ange

rate

.The

impa

ctof

87’s

cras

his

uncl

ear.

Con

clud

eth

atvo

latil

itym

odel

fore

cast

ing

perf

orm

ance

depe

nds

onth

eas

set

clas

s

54.

Hol

and

Koo

pman

(200

2)

S&P1

00(V

XO

)2/

1/86

–29

/6/2

001

DSI

VSV

X+

SV (ran

ked)

1,2,

5,10

,15

and

20da

ysah

ead.

Use

10-m

inre

turn

sto

cons

truc

t‘a

ctua

lvol

.’

R2

rang

esbe

twee

n17

and

33%

,MSE

,M

edSE

,MA

E.

αan

dβ

not

repo

rted

.All

fore

cast

sun

dere

stim

ate

actu

als

SVX

isSV

with

impl

ied V

XO

asan

exog

enou

sva

riab

lew

hile

SVX

+is

SVX

with

pers

iste

nce

adju

stm

ent.

SIV

isst

ocha

stic

impl

ied

with

pers

iste

nce

para

met

erse

tequ

alto

zero

Out

:Ja

n97–

Jun0

1

178


55.

Hw

ang

and

Satc

hell

(199

8)

LIF

FEst

ock

optio

ns23

/3/9

2–7/

10/9

6

240

daily

out-

of-s

ampl

efo

reca

sts.

DL

og-A

RFI

MA

-RV

Scal

edtr

unca

ted

Det

rend

edU

nsca

led

trun

cate

dM

Aop

tn=2

0-I

VA

djM

Aop

tn=2

0-R

VG

AR

CH

-RV

(ran

ked,

fore

cast

impl

ied)

1,5,

10,2

0,..

.,90

,100

,120

days

ahea

dIV

estim

ated

from

aro

lling

sam

ple

of77

8da

ilyob

serv

atio

ns.

Dif

fere

ntes

timat

ion

inte

rval

sw

ere

test

edfo

rro

bust

ness

MA

E,M

FEFo

reca

stim

plie

d AT

MB

Sof

shor

test

mat

urity

optio

n(w

ithat

15tr

adin

gda

ysto

mat

urity

).B

uild

MA

inIV

and

AR

IMA

onlo

g(I

V).

Err

orst

atis

tics

for

all

fore

cast

sar

ecl

ose

exce

ptth

ose

for

GA

RC

Hfo

reca

sts.

The

scal

ing

inL

og-A

RFI

MA

-RV

isto

adju

stfo

rJe

nsen

ineq

ualit

y

56.

Jori

on(1

995)

DM

/$,¥

/$,

SrFr

/$fu

ture

sop

tions

onC

ME

1/85

–2/9

27/

86–2

/92

3/85

–2/9

2

DIm

plie

d AT

MB

Sca

ll+pu

t

GA

RC

H(1

,1),

MA

20

(ran

ked)

1da

yah

ead

and

optio

nm

atur

ity.

Use

squa

red

retu

rns

and

aggr

egat

eof

squa

rere

turn

sto

prox

yac

tual

vola

tility

R2

is5%

(1-d

ay)

or10

–15%

(opt

ion

mat

urity

).W

ithro

bust

SE,

αim

plie

d>

0an

dβ

impl

ied

<1

for

long

hori

zon

and

isun

bias

edfo

r1-

day

fore

cast

s

Impl

ied

issu

peri

orto

the

hist

oric

alm

etho

dsan

dle

ast

bias

ed.M

Aan

dG

AR

CH

prov

ide

only

mar

gina

lin

crem

enta

lin

form

atio

n Con

tinu

ed

179


Dat

aD

ata

Fore

cast

ing

Fore

cast

ing

Eva

luat

ion

Aut

hor(

s)A

sset

(s)

peri

odfr

eque

ncy

met

hods

and

rank

hori

zon

and

R-s

quar

edC

omm

ents

57.

Jori

on(1

996)

DM

/$

futu

res

optio

nson

CM

E

Jan8

5–Fe

b92

DIm

plie

d Bla

ck,A

TM

GA

RC

H(1

,1)

(ran

ked)

1da

yah

ead,

use

daily

squa

red

topr

oxy

actu

alvo

latil

ity

R2

abou

t5%

.H

0:α

impl

ied=

0,β

impl

ied=

1ca

nnot

bere

ject

edw

ithro

bust

SE

R2

incr

ease

sfr

om5%

to19

%w

hen

unex

pect

edtr

adin

gvo

lum

eis

incl

uded

.Im

plie

dvo

latil

itysu

bsum

edin

form

atio

nin

GA

RC

Hfo

reca

st,

expe

cted

futu

res

trad

ing

volu

me

and

bid–

ask

spre

ad

58.

Kar

olyi

(199

3)74

stoc

kop

tions

13/1

/84–

11/1

2/85

MB

ayes

ian

impl

ied C

all

Impl

ied C

all

HIS

20,6

0

(Pre

dict

optio

npr

ice

not‘

actu

alvo

l.’)

20da

ysah

ead

vola

tility

MSE

Bay

esia

nad

just

men

tto

impl

ied

toin

corp

orat

ecr

oss-

sect

iona

lin

form

atio

nsu

chas

firm

size

,lev

erag

ean

dtr

adin

gvo

lum

eus

eful

inpr

edic

ting

next

peri

odop

tion

pric

e

180


59.

Kla

asse

n(1

998)

US$

/£,

US$

/D

Man

dU

S$/¥

3/1/

78–

23/7

/97

Out

:20/

10/8

7–23

/7/9

7

DR

SGA

RC

HR

SAR

CH

GA

RC

H(1

,1)

(ran

ked)

1an

d10

days

ahea

d.U

sem

ean

adju

sted

1-an

d10

-day

retu

rnsq

uare

sto

prox

yac

tual

vola

tility

MSE

ofva

rian

ce,

regr

essi

onth

ough

R2

isno

tre

port

ed

GA

RC

H(1

,1)

fore

cast

sar

em

ore

vari

able

than

RS

mod

els.

RS

prov

ides

stat

istic

ally

sign

ifica

ntim

prov

emen

tin

fore

cast

ing

vola

tility

for

US$

/D

Mbu

tnot

the

othe

rex

chan

gera

tes

60.

Kro

ner,

Kne

afse

yan

dC

laes

sens

(199

5)

Futu

res

optio

nson

coco

a,co

tton,

corn

,go

ld,s

ilver

,su

gar,

whe

at

Jan8

7–D

ec90

(kep

tlas

t40

obse

rvat

ions

for

out-

of-

sam

ple

fore

cast

)

DG

R>

CO

MB

Impl

ied B

AW

Cal

l(W

LS

>A

VG

>A

TM

)H

IS7

wee

ks>

GA

RC

H(r

anke

d)

225

cale

ndar

days

(160

wor

king

days

)ah

ead,

whi

chis

long

erth

anav

erag

e

MSE

,ME

GR

:Gra

nger

and

Ram

anat

han

(198

4)’s

regr

essi

onw

eigh

ted

com

bine

dfo

reca

st,C

OM

B:l

agim

plie

din

GA

RC

Hco

nditi

onal

vari

ance

equa

tion.

Com

bine

dm

etho

dis

best

sugg

ests

optio

nm

arke

tine

ffici

ency

Futu

res

pric

esJa

n87–

Jul9

1

Con

tinu

ed

181


Dat

aD

ata

Fore

cast

ing

Fore

cast

ing

Eva

luat

ion

Aut

hor(

s)A

sset

(s)

peri

odfr

eque

ncy

met

hods

and

rank

hori

zon

and

R-s

quar

edC

omm

ents

61.L

amou

reux

and

Las

trap

es(1

993)

Stoc

kop

tions

for

10no

n-di

vide

nd-

payi

ngst

ocks

(CB

OE

)

19/4

/82–

31/3

/84

DIm

plie

d Hul

l-W

hite

NT

MC

all

(int

erm

edia

tete

rmto

mat

urity

,WL

S)H

ISup

date

dex

pand

ing

estim

ate

GA

RC

H(r

anke

d,ba

sed

onre

gres

sion

resu

lt)

90to

180

days

mat

chin

gop

tion

mat

urity

estim

ated

usin

gro

lling

300

obse

rvat

ions

and

expa

ndin

gsa

mpl

e.U

sesa

mpl

eva

rian

ceof

daily

retu

rns

topr

oxy

‘act

ualv

ol.’

ME

,MA

E,

RM

SE.

Ave

rage

impl

ied

islo

wer

than

actu

alfo

ral

lst

ocks

.R

2on

vari

ance

vari

esbe

twee

n3

and

84%

acro

ssst

ocks

and

mod

els

Impl

ied

vola

tility

isbe

stbu

tbia

sed.

HIS

prov

ides

incr

emen

tal

info

.to

impl

ied

and

has

the

low

estR

MSE

.W

hen

allt

hree

fore

cast

sar

ein

clud

ed;

α>

0,1

>β

impl

ied

>0,

βG

AR

CH

=0,

βH

IS<

0w

ithro

bust

SE.P

laus

ible

expl

anat

ions

incl

ude

optio

ntr

ader

sov

erre

actt

ore

cent

vola

tility

shoc

ks,a

ndvo

latil

ityri

skpr

emiu

mis

nonz

ero

and

time-

vary

ing

182


62.L

atan

ean

dR

endl

eman

(197

6)

24st

ock

optio

nsfr

omC

BO

E5/

10/7

3–28

/6/7

4W

Impl

ied

vega

wei

ghte

dH

IS4

year

s(r

anke

d)

In-s

ampl

efo

reca

stan

dfo

reca

stth

atex

tend

part

ially

into

the

futu

re.U

sew

eekl

yan

dm

onth

lyre

turn

sto

calc

ulat

eac

tual

vola

tility

ofva

riou

sho

rizo

ns

Cro

ss-s

ectio

nco

rrel

atio

nbe

twee

nvo

latil

ityes

timat

esfo

r38

wee

ksan

da

2-ye

arpe

riod

Use

dE

urop

ean

mod

elon

Am

eric

anop

tions

and

omitt

eddi

vide

nds.

‘Act

ual’

ism

ore

corr

elat

ed(0

.686

)w

ith‘I

mpl

ied’

than

HIS

vola

tility

(0.4

63)

Hig

hest

corr

elat

ion

isth

atbe

twee

nim

plie

dan

dac

tual

stan

dard

devi

atio

nsw

hich

wer

eca

lcul

ated

part

ially

into

the

futu

re

63.L

ee(1

991)

$/D

M,$

/£,

$/¥,

$/FF

r,$/

C$

(Fed

.Res

.B

ulle

tin)

7/3/

73–

4/10

/89

W (Wed

,12

pm)

Ker

nel(

Gau

ssia

n,tr

unca

ted)

Inde

x(c

ombi

ning

AR

MA

and

GA

RC

H)

EG

AR

CH

(1,1

)G

AR

CH

(1,1

)

1w

eek

ahea

d(4

51ob

serv

atio

nsin

sam

ple

and

414

obse

rvat

ions

out-

of-s

ampl

e)

RM

SE,M

AE

.It

isno

tcle

arho

wac

tual

vola

tility

was

estim

ated

Non

linea

rm

odel

sar

e,in

gene

ral,

bette

rth

anlin

ear

GA

RC

H.K

erne

lmet

hod

isbe

stw

ithM

AE

.But

mos

tof

the

RM

SEan

dM

AE

are

very

clos

e.O

ver

30ke

rnel

mod

els

wer

efit

ted,

buto

nly

thos

ew

ithsm

alle

stR

MSE

and

MA

Ew

ere

repo

rted

.Iti

sno

tcle

arho

wth

eno

nlin

ear

equi

vale

nce

was

cons

truc

ted.

Mul

ti-st

epfo

reca

stre

sults

wer

em

entio

ned

butn

otsh

own

Out

:21

Oct

81–1

1O

ct89

.

IGA

RC

Hw

ithtr

end

(ran

kch

ange

sse

eco

mm

entf

orge

nera

las

sess

men

t)

183


Dat

aD

ata

Fore

cast

ing

Fore

cast

ing

Eva

luat

ion

Aut

hor(

s)A

sset

(s)

peri

odfr

eque

ncy

met

hods

and

rank

hori

zon

and

R-s

quar

edC

omm

ents

64.L

i(20

02)

$/D

M,$

/£,

$/¥

3/12

/86–

30/1

2/99

In:1

2/8/

86–

11/5

/95

Tic

k(5

min

)

D D

Impl

ied G

KO

TC

AT

MA

RFI

MA

real

ised

1,2,

3an

d6

mon

ths

ahea

d.Pa

ram

eter

sno

tre-

estim

ated

.U

se5-

min

retu

rns

toco

nstr

uct‘

actu

alvo

l.’

MA

E.

R2

rang

es0.

3–51

%(i

mpl

ied)

,7.

3–47

%(L

M),

16–5

3%(e

ncom

pass

).Fo

rbo

thm

odel

s,H

0:

α=

0,β

=1

are

reje

cted

and

typi

cally

β<

1w

ithro

bust

SE

Bot

hfo

reca

sts

have

incr

emen

tal

info

rmat

ion

espe

cial

lyat

long

hori

zon.

Forc

ing:

α=

0,β

=1

prod

uce

low

/neg

ativ

eR

2

(esp

ecia

llyfo

rlo

ngho

rizo

n).M

odel

real

ized

stan

dard

devi

atio

nas

AR

FIM

Aw

ithou

tlo

gtr

ansf

orm

atio

nan

dw

ithno

cons

tant

,whi

chis

awkw

ard

asa

theo

retic

alm

odel

for

vola

tility

(im

plie

dbe

tter

atsh

orte

rho

rizo

nan

dA

RFI

MA

bette

rat

long

hori

zon)

OT

CA

TM

optio

ns$/

£,$/

¥$/

DM

19/6

/94–

13/6

/99

19/6

/94–

30/1

2/98

184


65.L

opez

(200

1)C

$/U

S$,

DM

/U

S$,

¥/U

S$,U

S$/£

1980

–199

5D

SV-A

R(1

)-no

rmal

GA

RC

H-g

evE

WM

A-n

orm

al

1da

yah

ead

and

prob

abili

tyfo

reca

sts

for

four

‘eco

nom

icev

ents

’,vi

z.cd

fof

spec

ific

regi

ons.

Use

daily

squa

red

resi

dual

sto

prox

yvo

latil

ity.U

seem

piri

cald

istr

ibut

ion

tode

rive

cdf

MSE

,MA

E,

LL

,HM

SE,

GM

LE

and

QPS

(qua

drat

icpr

obab

ility

scor

es)

LL

isth

elo

gari

thm

iclo

ssfu

nctio

nfr

omPa

gan

and

Schw

ert(

1990

),H

MSE

isth

ehe

tero

sced

astic

ity-a

dj.

MSE

from

Bol

lers

lev

and

Ghy

sels

(199

6)an

dG

ML

Eis

the

Gau

ssia

nqu

asi-

ML

func

tion

from

Bol

lers

lev,

Eng

lean

dN

elso

n(1

994)

.For

ecas

tsfr

omal

lmod

els

are

indi

stin

guis

habl

e.Q

PSfa

vour

sSV

-n,G

AR

CH

-gan

dE

WM

A-n

In:1

980–

1993

Out

:19

94–1

995

GA

RC

H-n

orm

al,-

tE

WM

A-t

AR

(10)

-Sq,

-Abs

Con

stan

t(a

ppro

x.ra

nk,s

eeco

mm

ents

)

66.L

oudo

n,W

atta

ndY

adav

(200

0)

FTA

llSh

are

Jan7

1–O

ct97

DE

GA

RC

H,G

JR,

TS-

GA

RC

H,

TG

AR

CH

NG

AR

CH

,V

GA

RC

H,

GA

RC

H,

MG

AR

CH

(no

clea

rra

nk,

fore

cast

GA

RC

Hvo

l.)

Para

met

ers

estim

ated

inpe

riod

1(o

r2)

used

topr

oduc

eco

nditi

onal

vari

ance

sin

peri

od2

(or

3).U

seG

AR

CH

squa

red

resi

dual

sas

‘act

ual’

vola

tility

RM

SE,

regr

essi

onon

log

vola

tility

and

alis

tof

diag

nost

ics.

R2

isab

out4

%in

peri

od2

and

5%in

peri

od3

TS-

GA

RC

His

anab

solu

tere

turn

vers

ion

ofG

AR

CH

.All

GA

RC

Hsp

ecifi

catio

nsha

veco

mpa

rabl

epe

rfor

man

ceth

ough

nonl

inea

r,as

ymm

etri

cve

rsio

nsse

emto

fare

bette

r.M

ultip

licat

ive

GA

RC

Hap

pear

sw

orst

,fo

llow

edby

NG

AR

CH

and

VG

AR

CH

(Eng

lean

dN

g19

93)

Sub-

peri

ods:

Jan7

1–D

ec80

Jan8

1–D

ec90

Jan9

1–O

ct97

Con

tinu

ed

185


Dat

aD

ata

Fore

cast

ing

Fore

cast

ing

Eva

luat

ion

Aut

hor(

s)A

sset

(s)

peri

odfr

eque

ncy

met

hods

and

rank

hori

zon

and

R-s

quar

edC

omm

ents

67.

Mar

tens

and

Zei

n(2

004)

S&P5

00fu

ture

s,¥/

US$

futu

res,

Cru

de oilf

utur

es

Jan9

4–D

ec20

00Ja

n96–

Dec

2000

Jun9

3–D

ec20

00

Tic

kIm

plie

d BA

WV

XO

styl

eL

og-A

RFI

MA

GA

RC

H(r

anke

d,se

eco

mm

enta

lso)

Non

-ove

rlap

ping

1,5,

10,2

0,30

and

40da

ysah

ead.

500

daily

obse

rvat

ions

inin

-sam

ple

whi

chex

pand

son

each

itera

tion

Het

eros

ceda

stic

ityad

just

edR

MSE

.R

2ra

nges

25–5

2%(i

mpl

ied)

,15

–48%

(LM

)ac

ross

asse

tsan

dho

rizo

ns.B

oth

mod

els

prov

ide

incr

emen

tali

nfo.

toen

com

pass

ing

regr

.

Scal

eddo

wn

one

larg

eoi

lpri

ce.

Log

-AR

FIM

Atr

unca

ted

atla

g10

0.B

ased

onR

2,I

mpl

ied

outp

erfo

rms

GA

RC

Hin

ever

yca

se,a

ndbe

ats

Log

-AR

FIM

Ain

¥/U

S$an

dcr

ude

oil.

Impl

ied

has

larg

erH

RM

SEth

anL

og-A

RFI

MA

inm

ost

case

s.D

ifficu

ltto

com

men

ton

impl

ied’

sbi

ased

ness

from

info

rmat

ion

pres

ente

d

68.

McK

enzi

e(1

999)

21A

$bi

late

ral

exch

ange

rate

sV

ario

usle

ngth

from

1/1/

86or

4/11

/92

to31

/10/

95

DSq

uare

vs.p

ower

tran

sfor

mat

ion

(AR

CH

mod

els

with

vari

ous

lags

.Se

eco

mm

entf

orra

nk)

1da

yah

ead

abso

lute

retu

rns

RM

S,M

E,M

AE

.R

egre

ssio

nssu

gges

tall

AR

CH

fore

cast

sar

ebi

ased

.No

R2

was

repo

rted

The

optim

alpo

wer

iscl

oser

to1

sugg

estin

gsq

uare

dre

turn

isno

tth

ebe

stsp

ecifi

catio

nin

AR

CH

type

mod

elfo

rfo

reca

stin

gpu

rpos

e

186


69.M

cMill

an,

Spei

ghta

ndG

wily

m(2

000)

FTSE

100

FTA

llSh

are

Jan8

4–Ju

l96

Jan6

9–Ju

l96

D,

W,

MR

W,M

A,E

S,E

WM

AG

AR

CH

,T

GA

RC

H,

EG

AR

CH

,C

GA

RC

HH

IS,r

egre

ssio

n,(r

anke

d)

j=

1da

y,1

wee

kan

d1

mon

thah

ead

base

don

the

thre

eda

tafr

eque

ncie

s.U

sej

peri

odsq

uare

dre

turn

sto

prox

yac

tual

vola

tility

ME

,MA

E,

RM

SEfo

rsy

mm

etry

loss

func

tion.

MM

E(U

)an

dM

ME

(O),

mea

nm

ixed

erro

rth

atpe

naliz

eun

der/

over

pred

ictio

ns

CG

AR

CH

isth

eco

mpo

nent

GA

RC

Hm

odel

.Act

ual

vola

tility

ispr

oxie

dby

mea

nad

just

edsq

uare

dre

turn

s,w

hich

islik

ely

tobe

extr

emel

yno

isy.

Eva

luat

ion

cond

ucte

don

vari

ance

,hen

cefo

reca

ster

ror

stat

istic

sar

eve

rycl

ose

for

mos

tm

odel

s.R

W,M

A,E

Sdo

min

ate

atlo

wfr

eque

ncy

and

whe

ncr

ash

isin

clud

ed.

Perf

orm

ance

sof

GA

RC

Hm

odel

sar

esi

mila

rth

ough

nota

sgo

od

Out

:19

96–1

996

for

both

seri

es.

Con

tinu

ed

187


Dat

aD

ata

Fore

cast

ing

Fore

cast

ing

Eva

luat

ion

Aut

hor(

s)A

sset

(s)

peri

odfr

eque

ncy

met

hods

and

rank

hori

zon

and

R-s

quar

edC

omm

ents

70.

Noh

,Eng

lean

dK

ane

(199

4)

S&P5

00in

dex

optio

nsO

ct85

–Feb

92D

GA

RC

Had

j.fo

rw

eeke

ndan

dho

lsIm

plie

d BS

wei

ghte

dby

trad

ing

volu

me

(ran

ked,

pred

ict

optio

npr

ice

not‘

actu

alvo

l.’)

Opt

ion

mat

urity

.B

ased

on10

00da

ysro

lling

peri

odes

timat

ion

Equ

ate

fore

cast

abili

tyw

ithpr

ofita

bilit

yun

der

the

assu

mpt

ion

ofan

inef

ficie

ntop

tion

mar

ket

Reg

ress

ion

with

call+

puti

mpl

ieds

,da

ilydu

mm

ies

and

prev

ious

day

retu

rns

topr

edic

tnex

tday

impl

ied

and

optio

npr

ices

.Str

addl

est

rate

gyis

notv

ega

neut

rale

ven

thou

ghit

mig

htbe

delta

neut

ral

assu

min

gm

arke

tis

com

plet

e.It

ispo

ssib

leth

atpr

ofiti

sdu

eto

now

wel

l-do

cum

ente

dpo

st87

’scr

ash

high

erop

tion

prem

ium

71.

Paga

nan

dSc

hw-

ert

(199

0)

US

stoc

km

arke

t18

34–1

937

Out

:190

0–19

25(l

owvo

latil

ity),

1926

–193

7(h

igh

vola

tility

)

ME

GA

RC

H(1

,2)

GA

RC

H(1

,2)

2-st

epco

nditi

onal

vari

ance

RS-

AR

(m)

Ker

nel(

1la

g)Fo

urie

r(1

or2

lags

)(r

anke

d)

1m

onth

ahea

d.U

sesq

uare

dre

sidu

alm

onth

lyre

turn

sto

prox

yac

tual

vola

tility

R2

is7–

11%

for

1900

–25

and

8%fo

r19

26–3

7.C

ompa

red

with

R2

for

vari

ance

,R

2fo

rlo

gva

rian

ceis

smal

ler

in19

00–2

5an

dla

rger

in19

26–3

7

The

nonp

aram

etri

cm

odel

sfa

red

wor

seth

anth

epa

ram

etri

cm

odel

s.E

GA

RC

Hca

me

outb

estb

ecau

seof

the

abili

tyto

capt

ure

vola

tility

asym

met

ry.

Som

epr

edic

tion

bias

was

docu

men

ted

188


72.P

ong,

Shac

klet

on,

Tayl

oran

dX

u(2

004)

US$

/£

In:J

ul87

–D

ec93

Out

:Jan

94–

Dec

98

5-,3

0-m

inIm

plie

d AT

M,O

TC

quot

e(b

ias

adj.

usin

gro

lling

regr

.on

last

5ye

ars

mon

thly

data

)L

og-A

RM

A(2

,1)

Log

-AR

FIM

A(1

,d,1

)G

AR

CH

(1,1

)(r

anke

d)

1m

onth

and

3m

onth

sah

ead

at1-

mon

thin

terv

al

ME

,MSE

,re

gres

sion

.R

2

rang

esbe

twee

n22

and

39%

(1-m

onth

)an

d6

and

21%

(3-m

onth

)

Impl

ied,

AR

MA

and

AR

FIM

Aha

vesi

mila

rpe

rfor

man

ce.

GA

RC

H(1

,1)

clea

rly

infe

rior

.Bes

tco

mbi

natio

nis

Impl

ied

+A

RM

A(2

,1).

Log

-AR

(FI)

MA

fore

cast

sad

just

edfo

rJe

nsen

ineq

ualit

y.D

ifficu

ltto

com

men

ton

impl

ied’

sbi

ased

ness

from

info

rmat

ion

pres

ente

d

73.P

otes

hman

(200

0)S&

P500

(SPX

)&

futu

res

1Jun

88–

29A

ug97

DIm

plie

d Hes

ton

Opt

ion

mat

urity

(abo

ut3.

5to

4w

eeks

,non

-ov

erla

ppin

g).U

se5-

min

futu

res

infe

rred

inde

xre

turn

topr

oxy

‘act

ualv

ol.’

BS

R2

isov

er50

%.H

esto

nim

plie

dpr

oduc

edsi

mila

rR

2bu

tve

rycl

ose

tobe

ing

unbi

ased

Fte

stfo

rH

0:α

BS

=0,

βB

S=

1ar

ere

ject

edth

ough

t-te

stsu

ppor

tsH

0on

indi

vidu

alco

effic

ient

s.Sh

owbi

ased

ness

isno

tca

used

bybi

d–as

ksp

read

.Usi

ngin

σ,

high

-fre

quen

cyre

aliz

edvo

l.,an

dH

esto

nm

odel

,all

help

tore

duce

impl

ied

bias

edne

ss

Hes

ton

estim

atio

n:1J

un93

–29

Aug

97

futu

res

Tic

k

Impl

ied B

S(b

oth

impl

ieds

are

from

WL

Sof

all

optio

ns<

7m

onth

sbu

t>6

cale

ndar

days

)

S&P5

007J

un62

–May

93M

HIS

1,2,

3,6

mon

ths

(ran

ked)

Con

tinu

ed

189


Dat

aD

ata

Fore

cast

ing

Fore

cast

ing

Eva

luat

ion

Aut

hor(

s)A

sset

(s)

peri

odfr

eque

ncy

met

hods

and

rank

hori

zon

and

R-s

quar

edC

omm

ents

74.R

ando

lph

and

Naj

and

(199

1)

S&P5

002/

1/19

86–

31/1

2/88

Dai

lyM

RM

AT

MH

ISN

on-o

verl

appi

ng20

ME

,RM

SE,M

AE

,M

ean

reve

rsio

nm

odel

(MR

M)

sets

drif

trat

eof

vola

tility

tofo

llow

am

ean

reve

rtin

gpr

oces

sta

king

impl

ied A

TM

(or

HIS

)as

the

prev

ious

day

vol.

Arg

ueth

atG

AR

CH

did

notw

ork

asw

ell

beca

use

itte

nds

topr

ovid

ea

pers

iste

ntfo

reca

st,w

hich

isva

lidon

lyin

peri

odw

hen

chan

ges

invo

l.ar

esm

all

futu

res

optio

ns(c

rash

incl

uded

)op

enin

gT

ick

MR

MA

TM

impl

ied

days

ahea

d,re

-M

APE

GA

RC

H(1

,1)

estim

ated

usin

g

AT

Mca

llson

lyIn

:Fir

st80

HIS

20da

yIm

plie

d Bla

ck

expa

ndin

gsa

mpl

e.

obse

rvat

ions

(ran

ked

thou

ghth

eer

ror

stat

istic

sar

ecl

ose)

75.S

chm

alen

see

and

Tri

ppi

(197

8)

6C

BO

Est

ock

optio

ns29

/4/7

4–23

/5/7

5W

Impl

ied B

Sca

ll(s

impl

eav

erag

eof

alls

trik

esan

dal

lmat

uriti

es)

1w

eek

ahea

d.‘A

ctua

l’pr

oxie

dby

wee

kly

rang

ean

dav

erag

epr

ice

devi

atio

n

Stat

istic

alte

sts

reje

ctth

ehy

poth

esis

that

IVre

spon

dspo

sitiv

ely

tocu

rren

tvol

atili

ty

Find

impl

ied

rise

sw

hen

stoc

kpr

ice

falls

,neg

ativ

ese

rial

corr

elat

ion

inch

ange

sof

IVan

da

tend

ency

for

IVof

diff

eren

tsto

cks

tom

ove

toge

ther

.Arg

ueth

atIV

mig

htco

rres

pond

bette

rw

ithfu

ture

vola

tility

56w

eekl

yob

serv

atio

ns

(For

ecas

tim

plie

dno

tact

ual

vola

tility

)

190


76.S

cott

and

Tuc

ker

(198

9)

DM

/$,

£/$,

C$/

$,¥/

$&

SrFr

/$

Am

eric

anop

tions

onPH

LX

14/3

/83/

–13

/3/8

7(p

re-c

rash

)

Dai

lycl

osin

gtic

k

Impl

ied G

k(v

ega,

Infe

rred

AT

M,

NT

M)

Impl

ied C

EV

(sim

ilar

rank

)

Non

-ove

rlap

ping

optio

nm

atur

ity:3

,6an

d9

mon

ths.

Use

sam

ple

SDof

daily

retu

rns

topr

oxy

‘act

ualv

ol.’

MSE

,R

2ra

nges

from

42to

49%

.In

allc

ases

,α

>0,

β<

1.H

ISha

sno

incr

emen

tali

nfo.

cont

ent

Sim

ple

B-S

fore

cast

sju

stas

wel

las

soph

istic

ated

CE

Vm

odel

.Cla

imed

omis

sion

ofea

rly

exer

cise

isno

tim

port

ant.

Wei

ghtin

gsc

hem

edo

esno

tmat

ter.

Fore

cast

sfo

rdi

ffer

ent

curr

enci

esw

ere

mix

edto

geth

er

77.S

ill(1

993)

S&P5

0019

59–1

992

MH

ISw

ithex

ova

riab

les

HIS

(see

com

men

t)

1m

onth

ahea

dR

2in

crea

sefr

om1%

to10

%w

hen

addi

tiona

lva

riab

les

wer

ead

ded

Vol

atili

tyis

high

erin

rece

ssio

nsth

anin

expa

nsio

ns,a

ndth

esp

read

betw

een

com

mer

cial

-pap

eran

dT-

Bill

rate

spr

edic

tst

ock

mar

ketv

olat

ility

78.S

zakm

ary,

Ors

,Kim

and

Dav

idso

n(2

002)

Futu

res

optio

nson

S&P5

00,9

inte

rest

rate

s,5

curr

ency

,4en

ergy

,3m

etal

s,10

agri

cultu

re,3

lives

tock

Var

ious

date

sbe

twee

nJa

n83

and

May

2001

DIm

plie

d Bk,

NT

M

2Cal

ls+

2Put

seq

alw

eigh

tH

IS30

,G

AR

CH

(ran

ked)

Ove

rlap

ping

optio

nm

atur

ity,s

hort

estb

ut>

10da

ys.U

sesa

mpl

eSD

ofda

ilyre

turn

sov

erfo

reca

stho

rizo

nto

prox

y‘a

ctua

lvol

.’

R2

smal

ler

for

finan

cial

(23–

28%

),hi

gher

for

met

alan

dag

ricu

lt.(3

0–37

%),

high

est

for

lives

tock

and

ener

gy(4

7–58

%)

HIS

30an

dG

AR

CH

have

little

orno

incr

emen

tal

info

rmat

ion

cont

ent.

αim

plie

d>

0fo

r24

case

s(o

r69

%),

all3

5ca

sesβ

impl

ied

<1

with

robu

stSE

Con

tinu

ed

191


Dat

aD

ata

Fore

cast

ing

Fore

cast

ing

Eva

luat

ion

Aut

hor(

s)A

sset

(s)

peri

odfr

eque

ncy

met

hods

and

rank

hori

zon

and

R-s

quar

edC

omm

ents

79.T

aylo

rJW

(200

4)D

AX

,S&

P500

,H

ang

Seng

,FT

SE10

0,A

mst

erda

mE

OE

,Nik

kei,

Sing

apor

eA

llSh

are

6/1/

88–3

0/8/

95(e

qual

lysp

litbe

twee

nin

-an

dou

t-)

WST

ES

(E,A

E,

EA

E)

GJR (+

Smoo

thed

vari

atio

ns)

GA

RC

HM

A20

wee

ks,

Ris

kmet

rics

(ran

ked)

1w

eek

ahea

dus

ing

am

ovin

gw

indo

wof

200

wee

kly

retu

rns.

Use

daily

squa

red

resi

dual

retu

rns

toco

nstr

uct

wee

kly

‘act

ual’

vola

tility

ME

,MA

E,

RM

SE,

R2

(abo

ut30

%fo

rH

Kan

dJa

pan

and

6%fo

rU

S)

Mod

els

estim

ated

base

don

min

imiz

ing

in-s

ampl

efo

reca

ster

rors

inst

ead

ofM

L.S

TE

S-E

AE

(sm

ooth

tran

sitio

nex

pone

ntia

lsm

ooth

ing

with

retu

rnan

dab

solu

tere

turn

astr

ansi

tion

vari

able

s)pr

oduc

edco

nsis

tent

lybe

tter

perf

orm

ance

for

1-st

ep-a

head

fore

cast

s

80.T

aylo

rSJ

(198

6)15

US

stoc

ksFT

306

met

al£/

$

Jan6

6–D

ec76

Jul7

5–A

ug82

Var

ious

leng

thN

ov74

–Sep

82V

ario

usle

ngth

Var

ious

leng

th

DE

WM

AL

og-A

R(1

)A

RM

AC

H-A

bsA

RM

AC

H-S

qH

IS(r

anke

d)

1an

d10

days

ahea

dab

solu

tere

turn

s.2/

3of

sam

ple

used

ines

timat

ion.

Use

daily

abso

lute

retu

rns

devi

atio

nas

‘act

ualv

ol.’

Rel

ativ

eM

SER

epre

sent

one

ofth

eea

rlie

stst

udie

sin

AR

CH

clas

sfo

reca

sts.

The

issu

eof

vola

tility

stat

iona

rity

isno

tim

port

antw

hen

fore

cast

over

shor

tho

rizo

n.N

onst

atio

nary

seri

es(e

.g.E

WM

A)

has

the

adva

ntag

eof

havi

ngfe

wer

para

met

eres

timat

esan

dfo

reca

sts

resp

ond

tova

rian

cech

ange

fair

lyqu

ickl

y

5ag

ricu

ltura

lfu

ture

s4

inte

rest

rate

futu

res

AR

MA

CH

-Sq

issi

mila

rto

GA

RC

H

192


81.T

aylo

rSJ

(198

7)D

M/$

futu

res

1977

–83

DH

igh,

low

and

clos

ing

pric

es1,

5,10

and

20da

ysah

ead.

Est

imat

ion

peri

od,5

year

s

RM

SEB

estm

odel

isa

wei

ghte

dav

erag

eof

pres

enta

ndpa

sthi

gh,l

owan

dcl

osin

gpr

ices

with

adju

stm

ents

for

wee

kend

and

holid

ayef

fect

s

(see

com

men

t)

82.T

aylo

rSJ

and

Xu

(199

7)

DM

/$1/

10/9

2–30

/9/9

3In

:9m

onth

sO

ut:3

mon

ths

Quo

teIm

plie

d+

AR

CH

com

bine

dIm

plie

d,A

RC

HH

IS9

mon

ths

HIS

last

hour

real

ised

vol

(ran

ked)

1ho

urah

ead

estim

ated

from

9m

onth

sin

-sam

ple

peri

od.U

se5-

min

retu

rns

topr

oxy

‘act

ualv

ol.’

MA

Ean

dM

SEon

std

devi

atio

nan

dva

rian

ce

5-m

inre

turn

has

info

rmat

ion

incr

emen

tal

toda

ilyim

plie

dw

hen

fore

cast

ing

hour

lyvo

latil

ityD

M/$

optio

nson

PHL

XD

Frid

aym

acro

new

sse

ason

alfa

ctor

sha

veno

impa

cton

fore

cast

accu

racy

AR

CH

mod

elin

clud

esw

ithho

urly

and

5-m

inre

turn

sin

the

last

hour

plus

120

hour

/day

/wee

kse

ason

alfa

ctor

s.Im

plie

dde

rive

dfr

omN

TM

shor

test

mat

urity

(>9

cale

ndar

days

)C

all+

Put

usin

gB

AW

See

com

men

tfor

deta

ilson

impl

ied

and

AR

CH

Con

tinu

ed

193


Dat

aD

ata

Fore

cast

ing

Fore

cast

ing

Eva

luat

ion

Aut

hor(

s)A

sset

(s)

peri

odfr

eque

ncy

met

hods

and

rank

hori

zon

and

R-s

quar

edC

omm

ents

83.T

se(1

991)

Topi

xN

ikke

iSt

ock

Ave

rage

In:1

986–

1987

Out

:88–

89D

EW

MA

HIS

AR

CH

,GA

RC

H(r

anke

d)

25da

ysah

ead

estim

ated

from

rolli

ng30

0ob

serv

atio

ns

ME

,RM

SE,M

AE

,M

APE

ofva

rian

ceof

21no

n-ov

erla

ppin

g25

-day

peri

ods

Use

dum

mie

sin

mea

neq

uatio

nto

cont

rolf

or19

87cr

ash.

Non

norm

ality

prov

ides

abe

tter

fitbu

tapo

orer

fore

cast

.AR

CH

/GA

RC

Hm

odel

sar

esl

owto

reac

tto

abru

ptch

ange

invo

latil

ity.E

WM

Aad

just

toch

ange

sve

ryqu

ickl

y

84.T

sean

dT

ung

(199

2)Si

ngap

ore,

5V

Wm

arke

t&in

dust

ryin

dice

s

19/3

/75

to25

/10/

88D

EW

MA

HIS

GA

RC

H(r

anke

d)

25da

ysah

ead

estim

ated

from

rolli

ng42

5ob

serv

atio

ns

RM

SE,M

AE

EW

MA

issu

peri

or,

GA

RC

Hw

orst

.A

bsol

ute

retu

rns>

7%ar

etr

unca

ted.

Sign

ofno

nsta

tiona

rity

.Som

eG

AR

CH

nonc

onve

rgen

ce

194


85.V

asile

llis

and

Mea

de(1

996)

Stoc

kop

tions

12U

Kst

ocks

(LIF

FE)

In:

28/3

/86–

27/6

/86

WC

ombi

ne(I

mpl

ied

+G

AR

CH

)Im

plie

d(v

ario

us,s

eeco

mm

ent)

GA

RC

HE

WM

AH

IS3

mon

ths

(ran

ked,

resu

ltsno

tsen

sitiv

eto

basi

sus

eto

com

bine

)

3m

onth

sah

ead.

Use

sam

ple

SDof

daily

retu

rns

topr

oxy

‘act

ualv

ol.’

RM

SEIm

plie

d:5-

day

aver

age

dom

inat

es1-

day

impl

ied

vol.

Wei

ghtin

gsc

hem

e:m

axve

ga>

vega

wei

ghte

d>

elas

ticity

wei

ghte

d>

max

elas

ticity

with

‘>’

indi

cate

sbe

tter

fore

cast

ing

perf

orm

ance

.A

djus

tmen

tfor

div.

and

earl

yex

erci

se:

Rub

inst

ein

>R

oll>

cons

tant

yiel

d.C

rash

peri

odm

ight

have

disa

dvan

tage

dtim

ese

ries

met

hods

In2

(for

com

bine

dfo

reca

st):

28/6

/86−

25/3

/88

Out

:6/7

/88−

21/9

/91

86.V

ilasu

so(2

002)

C$/

$,FF

r/$,

DM

/$,¥

/$,

£/$

In:1

3/3/

79–

31/1

2/97

DFI

GA

RC

HG

AR

CH

,IG

AR

CH

(ran

ked,

GA

RC

Hm

argi

nally

bette

rth

anIG

AR

CH

)

1,5

and

10da

ysah

ead.

Use

dda

ilysq

uare

dre

turn

sto

prox

yac

tual

vola

tility

MSE

,MA

E,a

ndD

iebo

ld-

Mar

iano

’ste

stfo

rsi

g.di

ffer

ence

Sign

ifica

ntly

bette

rfo

reca

stin

gpe

rfor

man

cefr

omFI

GA

RC

H.B

uilt

FIA

RM

A(w

itha

cons

tant

term

)on

cond

ition

alva

rian

cew

ithou

ttak

ing

log.

Tru

ncat

edat

lag

250

Out

:1/1

/98–

31/1

2/99

Con

tinu

ed

195


Dat

aD

ata

Fore

cast

ing

Fore

cast

ing

Eva

luat

ion

Aut

hor(

s)A

sset

(s)

peri

odfr

eque

ncy

met

hods

and

rank

hori

zon

and

R-s

quar

edC

omm

ents

87.W

alsh

and

Tso

u(1

998)

Aus

tral

ian

indi

ces:

VW

20,V

W50

&V

W30

0

1Jan

93–

31D

ec95

5-m

into

form

H,D

and

Wre

turn

s

EW

MA

GA

RC

H(n

otfo

rw

eekl

yre

turn

s)H

IS,I

EV

(im

prov

edex

trem

e-va

lue

met

hod)

(ran

ked)

1ho

ur,1

day

and

1w

eek

ahea

des

timat

edfr

oma

1-ye

arro

lling

sam

ple.

Use

squa

reof

pric

ech

ange

s(n

on-c

umul

ativ

e)as

‘act

ualv

ol.’

MSE

,RM

SE,

MA

E,M

APE

Inde

xw

ithla

rger

num

ber

ofst

ock

isea

sier

tofo

reca

stdu

eto

dive

rsifi

catio

n,bu

tget

sha

rder

assa

mpl

ing

inte

rval

beco

mes

shor

ter

due

topr

oble

mof

nons

ynch

rono

ustr

adin

g.N

one

ofth

eG

AR

CH

estim

atio

nsco

nver

ged

for

the

wee

kly

seri

es,p

roba

bly

too

few

obse

rvat

ions

In:1

year

Out

:2ye

ars

88.W

eian

dFr

anke

l(1

991)

SrFr

/$,

DM

/$,

¥/$,

£/$

optio

ns(P

HL

X)

2/83

–1/9

0M

Impl

ied

GK

AT

Mca

ll(s

hort

est

mat

urity

)

Non

-ove

rlap

ping

1m

onth

ahea

d.U

sesa

mpl

eSD

ofda

ilyex

chan

gera

tere

turn

topr

oxy

‘act

ualv

ol.’

R2

30%

(£),

17%

(DM

),3%

(SrF

r),

0%(¥

).α

>0,

β<

1(e

xcep

ttha

tfo

r£/

$,α

>0,

β=

1)w

ithhe

tero

sced

.co

nsis

tent

SE

Use

Eur

opea

nfo

rmul

afo

rA

mer

ican

styl

eop

tion.

Als

osu

ffer

sfr

omno

nsyn

chro

nici

typr

oble

m.O

ther

test

sre

veal

that

impl

ied

tend

sto

over

pred

ict

high

vol.

and

unde

rpre

dict

low

vol.

Fore

cast

/impl

ied

coul

dbe

mad

em

ore

accu

rate

bypl

acin

gm

ore

wei

ght

onlo

ng-r

unav

erag

e

Spot

rate

sD

196


89.W

esta

ndC

ho(1

995)

C$/

$,FF

r/$,

DM

/$,¥

$/$,

£/$

14/3

/73–

20/9

/89

WG

AR

CH

(1,1

)IG

AR

CH

(1,1

)A

R(1

2)in

abso

lute

AR

(12)

insq

uare

sH

omos

ceda

stic

Gau

ssia

nke

rnel

(no

clea

rra

nk)

j=

1,12

,24

wee

kses

timat

edfr

omro

lling

432

wee

ks.

Use

jpe

riod

squa

red

retu

rns

topr

oxy

actu

alvo

latil

ity

RM

SEan

dre

gres

sion

test

onva

rian

ce,

R2

vari

esfr

om0.

1%to

4.5%

Som

eG

AR

CH

fore

cast

sm

ean

reve

rtto

unco

nditi

onal

vari

ance

in12

to24

wee

ks.I

tis

diffi

cult

toch

oose

betw

een

mod

els.

Non

para

met

ric

met

hod

cam

eou

twor

stth

ough

stat

istic

alte

sts

for

dono

tre

ject

null

ofno

sign

ifica

nce

diff

eren

cein

mos

tcas

es

In:1

4/3/

73–

17/6

/81

Out

:24/

6/81

–12

/4/8

9

90.W

iggi

ns(1

992)

S&P5

00fu

ture

s4/

82–1

2/89

DA

RM

Am

odel

with

2ty

pes

ofes

timat

ors:

1w

eek

ahea

dan

d1

mon

thah

ead.

Com

pute

actu

alvo

latil

ityfr

omda

ilyob

serv

atio

ns

Bia

ste

st,e

ffici

ency

test

,reg

ress

ion

Mod

ified

Park

inso

nap

proa

chis

leas

tbia

sed.

C-t

-Ces

timat

oris

thre

etim

esle

ssef

ficie

ntth

anE

Ves

timat

ors.

Park

inso

nes

timat

oris

also

bette

rth

anC

-t-C

atfo

reca

stin

g.87

’scr

ash

peri

odex

clud

edfr

oman

alys

is

1.Pa

rkin

son/

Gar

men

-K

lass

extr

eme

valu

ees

timat

ors

2.C

lose

-to-

clos

ees

timat

or(r

anke

d)

Con

tinu

ed

197


Dat

aD

ata

Fore

cast

ing

Fore

cast

ing

Eva

luat

ion

Aut

hor(

s)A

sset

(s)

peri

odfr

eque

ncy

met

hods

and

rank

hori

zon

and

R-s

quar

edC

omm

ents

91.X

uan

dTa

ylor

(199

5)

£/$,

DM

/$,¥

/$&

SrFr

/$PH

LX

optio

ns

In:J

an85

–Oct

89D

Impl

ied B

AW

NT

MT

Sor

shor

tG

AR

CH

Nor

mal

orG

ED

HIS

4w

eeks

(ran

ked)

Non

-ove

rlap

ping

4w

eeks

ahea

d,es

timat

edfr

oma

rolli

ngsa

mpl

eof

250

wee

ksda

ilyda

ta.U

secu

mul

ativ

eda

ilysq

uare

dre

turn

sto

prox

y‘a

ctua

lvol

.’

ME

,MA

E,

RM

SE,W

hen

αim

plie

dis

set

equa

lto

0,β

impl

ied

=1

cann

otbe

reje

cted

Impl

ied

wor

ksbe

stan

dis

unbi

ased

.Oth

erfo

reca

sts

have

noin

crem

enta

lin

form

atio

n.G

AR

CH

fore

cast

perf

orm

ance

not

sens

itive

todi

stri

butio

nal

assu

mpt

ion

abou

tre

turn

s.T

hech

oice

ofim

plie

dpr

edic

tor

(ter

mst

ruct

ure,

TS,

orsh

ort

mat

urity

)do

esno

taff

ect

resu

lts

Cor

resp

ondi

ngfu

ture

sra

tes

Out

:18

Oct

89–4

Feb9

2

92.Y

u(2

002)

NZ

SE40

Jan8

0–D

ec98

DSV

(of

log

vari

ance

)1

mon

thah

ead

estim

ated

from

prev

ious

180

to22

8m

onth

sof

daily

data

.Use

aggr

egat

eof

daily

squa

red

retu

rns

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JWBK021-IND JWBK021-Poon March 14, 2005 23:38 Char Count= 0

Index

absolute returns volatility, 10–11actual volatility, 101ADC (asymmetric dynamic covariance)

model, 65–6ARCH (autoregressive conditional

heteroscedasticity) models, 7, 10,17, 31, 37–44, 121, 122, 126

Engle (1982), 37–8forecasting performance, 43–4LM-ARCH, 51, 53see also GARCH

autoregressive conditionalheteroscedasticity modelssee ARCH models

ARFIMA model, 34, 51, 53, 127ARIMA model, 34ARMA model, 7, 34Asian financial crisis, 9asymmetric dynamic covariance (ADC)

model, 65–6at the money (ATM), 76

implied volatility, 101, 106, 107,116–17

options, 87, 91, 106, 117, 119autocorrelation, 19

of absolute returns, 8asset returns, 7lack of, 7

autocorrelation function, decay rate, 7autoregressive model, 126

backtest, 132–3three-zone approach to, 133–5

backward induction, 86

Bank for International Settlements (BIS),129

Barone-Adesi and Whaley quadraticapproximation, 89–90, 92–5

Basel Accords, 129–31Basel Committee, 129–31BEKK model, 66biased forecast, 29bid–ask bounce, 16, 90, 91, 117Black model, 127Black-Scholes model, 49–50, 71–95

binomial method, 80–6matching volatility with u and d,

83–4two-step binomial tree and

American-style options, 85–6dividend and early exercise premium,

88–90dividend yield method, 88–9Heston price and, 99–102implied volatility smile, 74–7, 97investor risk preference, 91–2known and finite dividends, 88measurement errors and bias, 90–2no-arbitrage pricing, 77–80

stock price dynamics, 77partial differential equation, 77–8solving the partial differential

equation, 79–80option price, 97for pricing American-style options,

72, 85–6for pricing European call and put

options, 71–7

215


216 Index

Black-Scholes model (continued )risk, 91testing option pricing model, 86–7

Black-Scholes implied volatility (BSIV),49, 50, 73–4, 91, 100

break model, 46Brownian motion, 18

Capital Accord, 130CGARCH (component GARCH), 55Chicago Board of Exchange (CBOE), 2close-to-open squared return, 14component GARCH (CGARCH), 55conditional variance, 22conditional volatility, 10constant correlation model, 66

deep-in-the-money call option, 76deep-out-of-the-money call option, 76delta, 89, 90Diebold and Mariano asymptotic test, 25,

26–7Diebold and Mariano sign test, 25, 27Diebold and Mariano Wilcoxon sign-rank

test, 27discrete price observations, 15

EGARCH, 34, 41, 43, 121, 122equal accuracy in forecasting models,

tests for, 25error statistics, 25EVT-GARCH method, 135EWMA (exponentially weighted moving

average), 31, 33, 40, 44explained variability, proportion of,

29–30exponential GARCH (EGARCH), 34, 41,

43, 121, 122exponential smoothing method, 33exponentially weighted moving average

(EWMA), 31, 33, 40, 44extreme value method see high-low

method

factor ARCH (FARCH) model, 66FARCH (factor ARCH) model, 66fat tails 7FIEGARCH, 50–1, 52–4FIGARCH, 50, 51–2, 53, 122financial market stylized facts, 3–9forecast biasedness, 117–19forecast error, 24

forecasting volatility, 16–17fractionally integrated model, 46, 50–4

positive drift in fractional integratedseries, 52–3

forecasting performance, 53–4see also FIGARCH; FIEGARCH

fractionally integrated series, 54

GARCH, 18, 34, 38–9, 121, 122, 126component GARCH (CGARCH), 55EVT-GARCH, 135exponential (EGARCH), 34, 41, 43,

121, 122GARCH(1,1) model, 12, 17, 38,

40, 46GARCH(1,1) with occasional break

model, 55GARCH-t , 135GJR-GARCH, 41–2, 43, 44, 122integrated (IGARCH), 39–40, 45quadratic (QGARCH), 42regime switching (RSGARCH), 57–8,

122smooth transition (STGARCH), 58TGARCH, 42

generalized ARCH see GARCHgeneralized method of moments (GMM),

60generalized Pareto distribution, 137Gibbs sampling, 61GJR-GARCH, 41–2, 43, 44, 46, 122Gram–Charlier class of distributions, 12

Heath-Jarrow-Morton model, 127Heston formula, 99, 102Heston stochastic volatility option

pricing model, 98–9assessment, 102–4

zero correlation, 103nonzero correlation, 103–4

Black-Scholes implied and, 99–102market price of volatility risk, 107–13

case of stochastic volatility, 107–8constructing the risk-free strategy,

108–10correlated processes, 110–11Ito’s lemma for stochastic variables,

107market price of risk, 111–13

volatility forecast using, 105–7heteroscedasticity-adjusted Mean Square

Error (HMSE), 23–4


Index 217

high-low method (H-L), 12–14, 17, 21historical average method, 33historical volatility models, 31–5

forecasting performance, 35modelling issues, 31–2regime switching, 34–5single-state, 32–4transition exponential smoothing,

34–5types, 32–5

HISVOL, 121, 122

implied standard deviation (ISD), 116,121, 122

implied volatility method, 30, 91, 101at the money, 101, 106, 107Black-Scholes (BSIV), 49, 50, 73–4,

91, 100implied volatility smile, 74–7, 97importance sampling, 60independent and identically distributed

returns (iid), 16, 136in-sample forecasts, 30integrated cumulative sums of squares

(ICSS), 55integrated GARCH (IGARCH), 39–40,

45integrated volatility, 14, 39, 40interest rate, level effect of, 58in-the-money (ITM) option, 87intraday periodic volatility patterns,

15–16inverted chi-squared distribution, 62IOSCO (International Organization of

Securities Commissions), 130Ito’s lemma, 77, 107, 108, 110

Jensen inequality, 11

kurtosis, 4, 17

large numbers, treatment of, 17–19Legal & General, 4level effect in interest rates, 58leverage effect (volatility asymmetry), 8,

37, 56, 148likelihood ratio statistic, 139LINEX, 24liquidity-weighted assets, 130LM-ARCH model, 51, 53log-ARFIMA, 51, 53, 127lognormal distribution, 2

long memory effect of volatility, 7long memory SV models, 60

market risk, 130Markov chain, 60maximum likelihood method, 11MCMC (Monte Carlo Markov chain), 59MDH (mixture of distribution

hypothesis), 148mean 2Mean Absolute Error (MAE), 23, 44Mean Absolute Percent Error (MAPE),

23, 44Mean Error (ME), 23Mean Logarithm of Absolute Errors

(MLAE), 24Mean Square Error (MSE), 23MedSE (median standard error), 44mixture of distribution hypothesis

(MDH), 148Monte Carlo Markov chain (MCMC), 59,

60–3parameter w, 62–3volatility vector H , 61–2

moving average method, 33multivariate volatility, 65–9

applications, 68–9bivariate example, 67

nearest-to-the-money (NTM), 116–17near-the-money (NTM), 76, 106negative risk premium, 102nonsynchronous trading, 15normal distribution, 2

operational risk, 131option forecasting power, 115–19option implied standard deviation,

115–16option pricing errors, 30stochastic volatility (SV) option pricing

model, 97–113Ornstein-Uhlenbeck (OU) process,

108orthogonality test, 28–30outliers, removal of, volatility persistence

and, 18–19out-of-sample forecasts, 30out-of-the-money (OTM) put option, 76,

87, 92, 119out-of-the money (OTM) implied

volatility, 101


218 Index

quadratic GARCH (QGARCH), 42quadratic variation, 14–15quasi-maximum likelihood estimation

(QMLE), 60, 136

random walk (RW) model, 32–3, 44, 126range-based method, 12, 17realized bipower variation, 15realized volatility, 14–16recursive scheme, 25regime-switching GARCH (RSGARCH),

57–8, 122regime-switching model, 19, 46regression-based forecast efficiency,

28–30risk 1risk management, volatility models in,

129–41risk-neutral long-run level of volatility,

98risk-neutral price distribution, 101risk-neutral probability measure, 83,

85risk-neutral reverting parameter, 98risk-neutrality, 49risk-weighted assets, 130rolling scheme, 25Root Mean Square Error (RMSE), 23, 44RSGARCH, 57–8, 122

serial correlation, 28Sharpe ratio, 1–2sign test, 25simple regression method, 33–4skewness, 4smile effect, 75smirk, 75smooth transition GARCH (STGARCH),

58smooth transition exponential smoothing,

34–5squared percentage error, 24squared returns models, 11Standard and Poor market index

(S&P100), 4standard deviation as volatility measure,

1, 2STGARCH 58stochastic volatility (SV), 31, 59–63, 121

forecasting performance, 63innovation, 59–60

MCMC approach, 60–3parameter w, 62–3volatility vector H , 61–2

option pricing model, 97–113stock market crash, October, 1987 18strict white noise, 16strike price effect, 101Student-t distribution, 17switching regime model, 54

tail event, 135, 136TAR model, 34, 43, 58Taylor effect, 8, 46TGARCH, 42Theil-U statistic, 24threshold autoregressive model (TAR),

34, 43, 58time series, 4–7trading volume weighted, 117trimming rule, 18

UK Index for Small Capitalisation Stocks(Small Cap Index), 4

unbiased forecast, 29unconditional volatility, 10

value-at-risk (VaR), 129, 131–510-day VaR, 137–8evaluation of, 139–41extreme value theory and, 135–9model, 136–7multivariate analysis, 138–9

variance as volatility measure, 1variance–covariance matrix, 65VAR-RV model, 51, 53VDAX, 126VECH model, 66vega, 89, 90vega weighted, 117VIX (volatility index), 126

new definition, 143–4old version (VXO), 144–5reason for change, 146

volatility asymmetry see leverage effectvolatility breaks model, 54–5volatility clustering, 7, 147volatility component model, 46, 54volatility dynamic, 132volatility estimation, 9–17

realized volatility, quadratic variationand jumps, 14–16


Index 219

scaling and actual volatility,16–17

using high–low measure, 12–14using squared returns, 11–12

volatility forecasting records, 121–8getting the right conditional variance

and forecast with the ‘wrong’models, 123–4

predictability across different assets,124–8

exchange rate, 126–7individual stocks, 124–5other assets, 127–8stock market index, 125–6

which model?, 121–3volatility forecasts, 21–30

comparing forecast errors of differentmodels, 24–8

error statistics and form of εt , 23–4form of Xt , 21–2see also volatility forecasting

recordsvolatility historical average, 126

Index compiled by Annette Musker

volatility index see, VIXvolatility long memory

break process, 54–5competing models for, 54–8components model, 55–7definition, 45–6evidence and impact, 46–50forecasting performance, 58regime-switching model

(RS-GARCH), 57–8volatility persistence, 17–19, 45, 136volatility risk premium, 119volatility skew, 76, 116volatility smile, 101, 116volatility spillover relationships, 65volume-volatility, 147–8

weighted implied, 116–17weighted least squares, 117Wilcoxon single-rank test, 25wildcard option, 91

yen–sterling exchange rate, 4

a practical guide to forecasting financial market...

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