NOTE TO USERS

This reproduction is the best copy available.

®

UMI

THREE ESSAYS ON VOLATILITY

by

Stefano Mazzotta Department of Finance, McGill University, Montreal

A Thesis Submitted to

MCGILL UNIVERSITY

In Partial Fulfillment of the Requirements for the Degree of

DOCTOR OF PHILOSOPHY

Finance

April 2005

© Stefano Mazzotta 2005

1+1 Library and Archives Canada

Bibliothèque et Archives Canada

Published Heritage Branch

Direction du Patrimoine de l'édition

395 Wellington Street Ottawa ON K1A ON4 Canada

395, rue Wellington Ottawa ON K1A ON4 Canada

NOTICE: The author has granted a non­exclusive license allowing Library and Archives Canada to reproduce, publish, archive, preserve, conserve, communicate to the public by telecommunication or on the Internet, loan, distribute and sell th es es worldwide, for commercial or non­commercial purposes, in microform, paper, electronic and/or any other formats.

The author retains copyright ownership and moral rights in this thesis. Neither the thesis nor substantial extracts from it may be printed or otherwise reproduced without the author's permission.

ln compliance with the Canadian Privacy Act some supporting forms may have been removed from this thesis.

While these forms may be included in the document page count, their removal does not represent any loss of content from the thesis.

• •• Canada

AVIS:

Your file Votre référence ISBN: 0-494-12906-9 Our file Notre référence ISBN: 0-494-12906-9

L'auteur a accordé une licence non exclusive permettant à la Bibliothèque et Archives Canada de reproduire, publier, archiver, sauvegarder, conserver, transmettre au public par télécommunication ou par l'Internet, prêter, distribuer et vendre des thèses partout dans le monde, à des fins commerciales ou autres, sur support microforme, papier, électronique et/ou autres formats.

L'auteur conserve la propriété du droit d'auteur et des droits moraux qui protège cette thèse. Ni la thèse ni des extraits substantiels de celle-ci ne doivent être imprimés ou autrement reproduits sans son autorisation.

Conformément à la loi canadienne sur la protection de la vie privée, quelques formulaires secondaires ont été enlevés de cette thèse.

Bien que ces formulaires aient inclus dans la pagination, il n'y aura aucun contenu manquant.

2

TABLE OF CONTENTS

ABSTRACT 4

ABSTRACT 5

ACKNOWLEDGMENTS 7

INTRODUCTION . . . 10

CHAPTER 1. LEVERAGE AND FEEDBACK IN INTERNATIONAL ASSETS 12 1.1. The Theory . . . . . . . . . . . . . . . . . 12

1.1.1. Theories of Asymmetric Volatility . . . . . . . . . 12 1.1.2. International Asset Pricing Theory . . . . . . . . 21 1.1.3. Foreign Exchange Risk and Market Imperfections 22

1.2. Empirical Applications and Evidence . . . . . . . . . 27 1.2.1. Econometric Models of Asymmetric Volatility . . 27 1.2.2. Volatility Feedback Effect ..... . . . . . . . . 30 1.2.3. Volatility in the International Finance Literature 34

1.3. Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . 36

CHAPTER 2. How IMPORTANT IS VOLATILITY ASYMMETRY FOR THE

RISK PREMIUM OF INTERNATIONAL ASSETS?

2.0.1. Introduction. 2.0.2. Related Work . . . . 2.0.3. The Data . . . . . .

2.1. International Asset Returns 2.1.1. The Asset Pricing Model. 2.1.2. IAPM Implementation. . .

2.2. Second Moment Asymmetry ... 2.2.1. M-NGARCH: a Simple Illustration. 2.2.2. M-NGARCH Estimation.. . . . . .

2.3. Encompassing Test for the Risk Premium. 2.3.1. CME test results . . . . . . . . . .

38 39 41 43 45 45 48 52 55 57 62 63

2.3.2. Risk Premium and Asymmetry . . 66 2.3.3. Robustness at the Weekly and Monthly Frequencies 70

2.4. Conclusion. . . . . . . . . . . . . . . . . . 71 2.A. Tables and Figures . . . . . . . . . . . . . 73 2.B. The Conditional Mean Encompassing Test 92

TABLE OF CONTENTS-Continued

CHAPTER 3. ASSESSING THE QUALITY OF VOLATILITY, INTERVAL,

AND DENSITY FORECASTS . . ...

3.1. Introduction and Background . . . . . . . 3.2. Volatility Forecast Evaluation ...... .

3.2.1. The Forecasting Object of Interest 3.2.2. Volatility Forecasts ........ . 3.2.3. Predictability Regressions .. . . . 3.2.4. Volatility Forecast Evaluation Results . 3.2.5. Bias and Efficiency . . . .....

3.3. Interval Forecast Evaluation . . . . . . . 3.3.1. Interval Evaluation Methodology 3.3.2. Interval Evaluation Results

3.4. Density Forecast Evaluation . . . . . . . 3.4.1. Graphical Density Forecast Evaluation 3.4.2. Tests of the Unconditional Distribution. 3.4.3. Tests of the Conditional Distribution 3.4.4. Density Slices Tests .

3.5. Conclusion. . . . . 3.A. Tables and Figures . . . . .

CHAPTER 4. FOREIGN EXCHANGE OPTION AND RETURNS BASED

3

95 96 99 99 99

101 103 107 108 110 112 114 115 116 118 120 122 125

CORRELATION FORECASTS . . . . . 154 4.1. Introduction. . . . . . . . . . . 155 4.2. Correlation Forecast Evaluation 159

4.2.1. Data issues ....... 159 4.2.2. The Forecasting Object of Interest 160 4.2.3. The Measures of Correlation . . . . 162

4.3. Correlation Forecast Evaluation Methodology and Results 166 4.3.1. Efficiency and Bias . . . . . . . . . . . . . . . . . . 169

4.4. Two Applications of Correlation Forecasts . . . . . . . . . 170 4.4.1. Scenario Analysis for the Euro Nominal Effective Ex­

change Rate Index . . . . . . . . . . . . . . . . . . .. 171 4.4.2. Exchange Rate Intervention and Correlation Among Cross-

Rates 175 4.5. Concluding Remarks 178 4.A. Tables and Figures . 180

CHAPTER 5. CONCLUSIONS AND FUTURE RESEARCH 196

REFERENCES 198

4

ABSTRACT

This dissertation is in the form of one survey paper and three essays on the topic of volatility. The unifying feature that permeates the entire thesis is the focus on the measurement and use of conditional second moment of equities and currencies as a measure of risk for asset pricing and policy purposes in the context of international markets.

The survey examines selected papers from the international finance liter­ature and from the volatility literature with a focus on the theoretical and empirical relationship between first and second unconditional and conditional moments of domestic and international asset returns. It then specifically pro­poses several areas for investigation related to international finance topics.

The first essay investigates the importance of asymmetric volatility when computing the risk premium of international assets. The results indicate that condition al second moment asymmetry is significant and time-varying. They also show that, if the priee of risk is time-varying, the world market and foreign exchange risk premia estimated without allowing for time-varying asymmetry are less consistent with the data. Furthermore, they imply that asymmetry is more pronounced when the business condition is such that investors require higher compensation to bear risk.

In the second essay we start from the consideration that financial deci­sion makers often consider the information in currency option valuations when making assessments about future exchange rates. The purpose of this essay is then to systematically assess the quality of option based volatility, interval and density forecasts. We use a unique dataset consisting of over 10 years of daily data on over-the-counter currency option prices. We find that the implied volatilities explain a large share of the variation in realized volatility. FinaIly, we find that wide-range interval and density forecasts are often misspecified whereas narrow-range interval forecasts are weIl specified.

In the third essay we examine whether the information contained in vari­ous measures of correlation among exchange rates can be used to assess future currency co-movement. We compare option-implied correlation forecasts from a dataset consisting of over 10 years of daily data on over-the-counter currency option prices to a set of return-based correlation measures and assess the rel­ative quality of the correlation forecasts. We find that while the predictive power of implied correlation is not always superior to that of returns based correlations measures, it tends to provide the most consistent results across currencies. Predictions that use both implied and returns-based correlations generate the highest adjusted R2 's, explaining up to 42 per cent of the realized correlations.

5

ABSTRACT

Résumé: Cette dissertation se comporte d'un chapitre de revue de littéra­ture et de trois essais qui se rapporte sur la volatilité. Le point commun qui imprègne la thèse entière se porte sur la mesure et l'utilisation des seconds moments conditionnels des actions ordinaires et des devises comme mesure de risque pour l'évaluation des capitaux et leurs implications dans le contexte des marchés internationaux.

La revue de littérature résume les majeurs contributions en modélisation des actifs financiers dans un contexte international ainsi que le rapport théorique et empirique entre le premier et le second moments inconditionnels et condi­tionnels des rendements de capitaux domestiques et internationaux. Aussi la revue de littérature propose différents domaines de recherches en finance internationale.

Le premier essai étudie l'importance de la volatilité asymétrique lors de l'évaluation de la prime de risque des capitaux internationaux. Les résultats indiquent que l'asymétrie en seconds moments conditionnels est significative et varie avec le temps. Les résultats montrent également que si le prix du risque varie avec le temps, les primes de risque du marché global et des taux de change estimés sans tenir compte de la variabilité de l'asymétrie sont moins consistent avec les données. En outre, l'asymétrie est plus prononcée quand la conjoncture économique est telle que les investisseurs exigent une compensation plus élevée pour endurer le risque.

Dans le deuxième essai nous supposons que les décideurs financiers consid­èrent l'information contenue dans les prix des options en devises pour estimer l'évolution des taux de change dans le futur. Le but de cet essai est alors d'estimer la performance des prévisions de la volatilité, de l'intervalle et de la fonction de densité basée sur les prix des options en devises. Nous employons une base de données unique qui comporte dix ans de données journalières sur des prix over-the-counter d'option en devises. Nous trouvons que les volatilités implicites expliquent une grande part de la variation de la volatilité réalisée. En conclusion, nous trouvons que les prévisions de la densité et des intervalles larges sont misspécifiés tandis que les prévisions des intervalles étroits sont précises.

Dans le troisième essai nous examinons si l'information contenue dans di­verses mesures de corrélation parmi des taux de change peut être employée pour évaluer le co-mouvement des devises dans le futur. Nous comparons les prévisions des corrélations extraites d'une base de données se composant de dix ans de données quotidiennes de prix over-the-counter d'options en devises aux corrélations basées sur les rendements et évaluons la qualité relative des prévisions de corrélation. Nous trouvons que bien que la puissance prédictive

6

de la corrélation «implicite» n'est pas toujours supérieure à celle des mesures de corrélations des rendements, elle tend à fournir des résultats conformes à travers les devises. Les prévisions qui emploient simultanément les corrélations implicites et les corrélations des rendements produisent le plus haut R2 ajusté, expliquant jusqu'à 42 % des corrélations réalisées.

7

ACKNOWLEDGMENTS

l sincerely thank the co-chair of my thesis committee Prof. Vihang Errunza,

and the members of my thesis committee, Prof. Lawrence Kryzanowski, and

Prof. Jan Ericsson for guidance on my research. l am particularly grateful

to the co-chair of my thesis committee, Prof. Peter Christoffersen for his

unwavering encouragement and support. Peter has had a deep influence on my

formation. He is not only a refined scholar, but also a man of great wisdom

and compassion.

l am in debt with all the members of the McGill faculty. In particular, l

thank Kris Jacobs, Benjamin Croitoru, Dietmar Leisen, and Adolfo De Motta.

Without their talent, dedication, and patience the Finance Ph.D. program

would have not existed.

l want to thank my fellow finance Ph.D. students, Ines Chaieb, Rodolfo

Oviedo, Basma Majerbi, Xiaofei Li, and Marcelo Braga Dos Santos, who un­

fortunately passed away just before receiving his Ph.D.

Ines, Rodolfo, and l spent countless hours together studying, discussing

and challenging different ideas. How much l learned from them can be hardly

described.

With regard to the first essay, l also thank Francesca Carrieri, Louis Eder­

ington, Wayne Ferson, John Galbraith, Bruno Gérard, Sergei Sarkissian, and

Bas Werker for the insightful discussions.

For the second and third essays, we have benefited from several visits to

the External Division of the European Central Bank whose hospitality is grate­

fullyacknowledged. Very useful comments were provided by Torben Andersen,

Lorenzo Cappiello, Bruce Lehmann, Filippo di Mauro, Stelios Makrydakis,

Nour Meddahi and Neil Shephard. My special thanks go to Filippo di Mauro,

for encouraging and supporting our collaboration with the ECB.

8

Many other people have contributed to my Ph.D. studies at McGill. 1 would

like to thank especially the direct or of the Ph.D. program, Jan Jorgensen,

Prof. Susan Christoffersen, Prof. Mo Chaudhury, the administrator, Stella

Scalia, the secretaries of the finance area, Susan Lovasik, and Che Doran, and

the computer support team, Marc Belisle, Sani Sulu, Edem Dzirasah, and

especially Pierre Cambron, and Joe Caruso. A thought of gratitude also go es

to Prof. Leda Matteuzzi Mazzoni at my Alma mater, Bologna University.

1 also want to deeply thank the founder of Soka University, Japan, Daisaku

Ikeda, who guided and inspired me during the last 18 years of my life.

1 gratefully acknowledge the financial support by the Institut de Finance

Mathématique de Montréal (IFM2), and the Centre Interuniversitaire de Recherche

en Économie Quantitative (CIREQ). 1 also thank Inquire Europe and Inquire

UK for the award given to the first essay at the EFA 2004 conference in Maas­

tricht.

1 finally want to thank my wife Mizuki and my daughter Rosalba for their

love and support. It is to them that 1 dedicate this thesis, with my deepest

love and gratitude.

9

Contributions of the Authors

1 am solely responsible for the first essay. Nonetheless, 1 would like to thank

for inspiration, support and guidance Prof. Errunza, Prof. Kryzanowski, and

especially Prof. Christoffersen.

The second essay, "Assessing the Quality of Volatility, Interval, and Density

Forecasts from OTC Currency Options" is joint work with the co-chair of my

thesis committee, Peter Christoffersen. Prof. Christoffersen and 1 have made

equally substantial contributions to this essay.

The third essay, "Foreign Exchange Option and Returns Based Correlation

Forecasts: Evaluation and Two Applications" is joint work with OUi Castrén,

at the European Central Bank. Dr. Castrén and 1 have made equally substan­

tial contributions to this essay.

The responsibility of any remaining error is shared accordingly.

The OTC volatilities used in the second and third essays were provided by

Citibank N.A.

10

INTRODUCTION

This dissertation is in the form of one survey and three essays on the topic of

volatility. The unifying feature that permeates the entire work is the focus on

the conditional second moment of equities and currencies returns as a measure

of risk for asset pricing and policy purposes in the context of international

markets.

The survey examines selected papers from the international finance liter­

ature and from the volatility literature with a focus on the theoretical and

empirical relationship between first and second unconditional and conditional

moments of domestic and international asset returns.

The first essay investigates the importance of asymmetric volatility when

computing the risk premium of international assets. The results indicate that

condition al second moment asymmetry is significant and time-varying. They

also show that, if the price of risk is time-varying, the world market and foreign

exchange risk premia estimated without allowing for time-varying asymmetry

are misspecified. Furthermore, they imply that asymmetry is more pronounced

when the business condition is such that investors require higher compensation

to bear risk.

The second essay is joint work with Peter Christoffersen. Here we start from

the consideration that financial decision makers often consider the information

in currency option valuations when making assessments about future exchange

rates. The purpose of this essay is then to systematically assess the quality

of option based volatility, interval and density forecasts. We use a unique

dataset consisting of over 10 years of daily data on over-the-counter currency

option prices. We find that the implied volatilities explain a large share of the

variation in realized volatility. Finally, we find that wide-range interval and

11

density forecasts are often misspecified whereas narrow-range interval forecasts

are well specified.

The third essay is joint work with Olli Castrén (European Central Bank).

In this essay we examine whether the information contained in various mea­

sures of correlation among exchange rates can be used to assess future currency

co-movement. We compare option-implied correlation forecasts from a dataset

consisting of over 10 years of daily data on over-the-counter currency option

priees to a set of return-based correlation measures and assess the relative

quality of the correlation forecasts. We find that while the predictive power of

implied correlation is not always superior to that of returns based correlations

measures, it tends to provide the most consistent results across currencies.

Predictions that use both implied and returns-based correlations generate the

highest adjusted R2s, explaining up to 42 per cent of the realized correla­

tions. We then apply the correlation forecasts to two policy-relevant topics,

to pro duce scenario analyses for the euro effective exchange rate index, and

to analyze the impact on cross-currency co-movement of interventions on the

JPY /USD exchange rate.

The last chapter summarizes the main findings of the three essays and

concludes with the presentation of sorne topics that are left for future research.

Chapter 1

LEVERAGE AND FEEDBACK EFFECT IN

INTERNATIONAL ASSETS: A SURVEY

Introd uction

12

This survey examines selected papers from the international finance literature

and from the volatility literature with a focus on first and second uncondi­

tional and conditional moments of international asset returns. The survey is

organized as follows: Section 1 introduces sorne theories of asymmetry and im­

portant international asset pricing models. Section 2 examines the empirical

application and the evidence with particular focus on leverage, volatility feed­

back, and volatility modeling in international asset pricing models. Section 3

concludes and introduces the first essay.

1.1 The Theory

1.1.1 Theories of Asymmetric Volatility

This section presents theories related to asymmetric volatility. It also intro­

duces the three common explanations of asymmetric voiatility in a more formaI

way: the stricto sensu lever age proposed by Black (1976), the volatility feed­

back, and balance sheet and growth option effects.

A full-ftedged intertemporal rational expectation equilibrium modei of as­

set prices that endogenously generate volatility asymmetry and time varying

expected returns is provided by Veronesi (1999). The key assumption of the

13

model is that dividends are generated by realizations of a Gaussian diffusion

pro cess whose drift rate shifts between high and low state at random times.

Identical investors cannot observe the drift rate of the dividend process; they

must infer it from the observation of past dividends' realizations. Investors'

uncertainty is maximum when they assign .5 probability to each state.

The main result is that the equilibrium priee of the assets is an increasing

and canvex functian of investors' posterior probability of the high state.

To informally illustrate this result, let 1T(t) denote investors' posterior prob­

ability that the state is high at t. Suppose investors believe times are good

so that 1T(t) is dose to 1. A bad pieee of news decreases 1T(t) and therefore

decreases future expected dividends and increases investors' uneertainty about

the true drift rate of the dividend proeess. This would push 1T(t) doser to .5.

Risk-averse investors want to be compensated for bearing more risk; henee they

will require an additional discount on the price of the asset. The important

consequenee is that the priee drops by more than it would in a present-value

model.

Suppose instead that investors believe times are bad and henee 1T(t) is doser

to O. A good piece of news increases their expectation of future dividends but

also raises their uneertainty. Henee the equilibrium priee of the asset increases,

but not as much as it would in a present-value model. Formally, the priee

function is increasing but convex in 1T(t).

This is the rational presented for the feedback effect. Its theoretical jus­

tification is in the investors' willingness to "hedge" against changes in their

level of uneertainty. The form of the priee function signifies that investar tend

ta aver react ta bad news in gaad times and under react ta gaad news in bad

times, making the priee of the asset more sensitive to news in good times than

in bad times.

The model has important implications for expected returns and volatility.

14

The volatility can be decomposed into two components: the first is an "uneer­

tainty component" that describes the sensitivity of priees to news of the risk

neutral investor and is related only to investors. This component is a symmet­

ric function of 7r(t), maximized at 7r(t) = .5. The second additional component

is a "risk-aversion component" which stems mainly from investors' degree of

risk aversion. An important feature is that the risk aversion component is

positive when 7r(t) is high and negative when 7r(t) is low. This characteristic

of risk aversion yields an asymmetric effect on the priee sensitivity to news.

The properties of the equilibrium priee function implies that if investors are

uncertain about whether a shift in regime has occurred, return volatility should

be high. Moreover, even if during sorne recessions uneert ai nt y is not very high,

it is still the case that volatility should be higher than in booms. Sinee the

pricing function is increasing and convex in 7r(t) pereentage volatility decreases

quickly when 7r(t) approaches 1 so that this is the point of minimum return

volatility. These results also imply persistenee in return volatility changes,

because investors' beliefs need several good or bad realizations to change.

The model gives theoretical support to the assumption that expected re­

turns should be proportional to (expected) stock volatility, as postulated by

Merton (1980).

Also related to a business cycle explanation is Whitelaw (2000). The study

poses two questions: (1) Are empirical results of weak or negative relation

between conditional expected returns and condition al volatility consistent both

with general equilibrium models and with the time series properties of variables

such as consumption growth which drive equity returns in these models? (2)

What features are neeessary to generate this counter intuitive behavior of

expected returns and volatility? The model is framed in the context of a

representative agent, exchange economy.

15

By no arbitrage arguments, the Euler equationi is

(1.1 )

where mHI = (3(cHdCt)-a, (3 is a discount factor that captures impatience,

C is consumption, Ct is the coefficient of relative risk aversion of an investor

endowed with a power utility function, and r m,t+1 is the return on the market

at time t + 1. Equation (1.1) can be rewritten as

Identifying dividend with consumption as is common in the literature, and

using the fact that the market return can be decomposed as

_ (CHI) (SHI/CHI + 1) rm,t+1 - --

Ct St/Ct (1.3)

where sand C are respectively the price of the market and the dividend­

consumption, (1.2) becomes

Et[rm,HI - r{] = -r{VOltlrm,HI]Volt[mHI]

COTT, [/3 ( ~:' ) -Π, ( ";:,) (S'+1~~/: + 1) 1 (1.4)

If, for the sake of illustration, the st! Ct ratio is kept constant, then

(1.5)

is negative as long as Ct is positive. For a negative relation between the re­

turn's conditional moments to hold, it must be the case that when the volatility

1 Following the widespread convention, Et indicates the conditional expectation E[ -lId. The same is true for conditional moments where the subscript t reads "conditional on the

information at time t"

16

is high the correlation is low. However, this is impossible with a fixed price

dividend ratio. To duplicate the features of the data the model is devised in a

way that allows the price dividend ratio variation to partially offset the vari­

ation in the dividend growth, at least in sorne state of the world. The paper

models consumption growth as an autoregressive process, with two regimes in

which the parameters differ. The probability of a regime shift is modeled as a

function of the level of consumption growth, yielding time-varying transition

probabilities.

For regimes that are sufficiently far apart in terms of the time-series behav­

ior of consumption growth, the regime switching probability will control the

condition al volatility. In particular, states with a high probability of switching

to a new regime will have high volatility. At the same time, increasing the

probability of a regime switch may decrease the correlation between equity

returns and the pricing kernel, thus reducing the risk premium. This second

effect will occur because the price dividend ratios, which depend on expected

future consumption growth, will be related to the regime and not to short-run

consumption growth.

The two-regime specification is able to identify the expansionary and con­

tractionary phases of the business cycle consistent with the NB ER business

cycle dating. The model generates results that are roughly consistent with the

empirical evidence of a negative or weak relation between first and second con­

ditional moments. In substance, expected returns and conditional volatility

exhibit a complex, nonlinear relation. They are negatively related in the long

run and this relation varies widely over time.

The key features of the specification are regime parameters that imply

different means of consumption growth across the regimes and state-dependent

regime switching probabilities. In contrast, a single-regime model calibrated

to the same data generates a strong positive, and essentially linear, relation

17

between expected returns and volatility.2

Note that despite both Veronesi's and Whitelaw's theoretical approaches

introduce business cycle fluctuation to match empirical findings, results point

in opposite directions.

The Leverage Effect. According to the Modigliani and Miller (1958) theorem,

the fundamental asset of a firm is its entire value: the different way in which

ownership is split is not relevant. It follows that when a firm is not close to

bankruptcy, the volatility of a stock's return should come entirely from the

fluctuations in the total firm value. In a firm that has both equity and debt

in its capital structure, the debt holders' claim on firm value is limited to the

face value of the bonds, so nearly aIl variations in total firm value will be

transmitted to the equity, except when the firm is close to insolvency.

Let PM,t denote the market index, Ti,t the return of asset i and It the

information set at time t. The return of an asset can be expressed as Ti,t+l =

Eh,t+llltl+ ci,t+l· Similarly for the market, TM,t+l = E[TM,t+llltl+ CM,t+l·

Define condition al variances and covariances O";,t+l = vaT [Ti,t+l lIt], O"~,t+l =

vaT[TM,t+lIItl and O";M,t+l = COVh,t+l,TM,t+lIItl·

Definition: A Teturn Ti,t displays asymmetric volatility if

(1.6)

In words, the variance of the return Ti,t+l conditional on the information

set available at time t and on the innovation Ci,t being negative is different from

the variance of Ti,t+l conditional on the same information set and on Ci,t being

positive. The inequality sign is in practice thought indicate "larger than" .

Assume debt is riskless. Let Di,t-l and Ei,t-l be the debt and the equity.

Let also rLl be the return on debt fixed at time t-1, set to be constant during

2In the empirical part, the parameters are estimated by maximum likelihood using

monthly consumption data over the period 1959 -1996.

18

time t - 1 and t. Let also r\t be the total return of the firm's asset and ri,t be

the return on Ei,t-l. Then, by definition

E· 1 E· 1 f 2,t- + (1 2,t-) _ -ri,t - rt- 1 - ri,t Ei t-l + Di t-l Ei t-l + Di t-l

" "

which implies:

f _( Di,t-l)(_ f) ri,t - r t - 1 - 1 + ~ ri,t - r t - 1 i,t-l

L tt ' L Di t-1 h e mg i,t-l = ~ we ave

var[ri,t - rLIllt-ll = var[(l + Li,t-l) (r'i,t - rLl)IIt- 1l

var[ri,tllt-ll = (1 + Li,t-l)2var[1\tllt-ll

(1.7)

(1.8)

(1.9)

(1.10)

The last expression shows that if Li,t-l changes, the volatility of equity

will change even when the volatility of firms' return var[r\tllt-ll is constant.

In particular, both changes in Ei,t-l and Di,t-l can affect equity's volatility.

This is the lever age effect in the strict sense and the only use made of this

expression in this thesis.

Volatility Elasticity. Ignoring the dependence on asset i, expression (1.10) can

be rewritten in terms of volatilities as (Jf = (1 + DEt -1 kY, where (Jf and (Ji t-1

indicate respectively the volatility of the stock and volatility of the total value

of the firm. It is insightful to consider the elasticities of equity volatility ÇE

with respect to the equity assuming constant firm volatility

ÇE = 8(Jf Et-l = _ Dt-l 8Et-l (Jf Et-l + Dt-l

(1.11)

Similarly for the debt,

è _ 8(Jf Dt- 1 Dt- 1 <"D - S

8Dt-l (Jt Et-l + Dt-l (1.12)

19

Both elasticities are bounded: -1 :::; ÇE :::; 0, and ° :::; ÇD :::; 1. Moreover,

the elasticity of the stock ÇL volatility with respect to Lt is equal to one.3

Covariance Asymmetry. Under the same specifications

Cov[(ri,t - rLl), (rM,t - rLl)IIt- 1] (1.13)

= (1 + Li,t-l)(1 + LM,t-l)COV[(l"i,t - rLl), (rM,t - rLl)IIt- 1]

From the previous discussion it is also apparent that the same forces that

affect variance also affect covariance. This effect may also be important as the

volatility feedback effect would be st ronger if the response of the covariances

to market shocks is also asymmetric. Formula (1.13) shows that, even if the

firm's covariance with the market is constant, the covariance of stock returns

will change if Li,t-l or LM,t-1Change.

For the conditional betas the effect of lever age is not univocal as

(1 + Li,t-l) -!\t-l = (1 + LM,t-l) f3i,t-l (1.14)

The Feedback Effect. The feedback effect can be illustrated by assuming that

the conditional CAPM holds and, consistently with empirical evidence, that

the volatility is persistent. Making those assumptions explicit

(1.15)

âVar[ri,t+1I It-l] ---=----=---'--'----=- > ° is large, but less than 1

ârr,t+l (1.16)

At the market level, bad news has two effects. First, it increases current

volatility in the market. Second, since volatility is persistent, it will induce

3These theoretical bounds are of particular interest for empirical investigation as in

Figlewski and Wang (2000)

20

investors to revise conditional volatility. In the mean-variance world underlying

the CAPM, increased condition al volatility at the market level commands a

higher required return, leading to an immediate decline in the current value of

the market. This can be seen from the fact that a fundamental relation of the

condition al CAPM iS4

E[rM"IM,-d - rL ~ (t, :J -1 M'-lvar [rM"I Mt-l1 (1.17)

h ei - E[U"i,t(WDIMt-l]' t t l . d"d l . k . M d W i w ere t = - E[u:,t(wDIMt- 1] lS 0 a m lVl ua ns averslOn, t an tare

respectively the total and individual wealth at t, and rM,t = ::~l . Decline in the price will "feed back" into the conditional volatility starting

over the cycle, hence the name feedback. If the leverage effect is at work, it

will reinforce the feedback effect. Good news determines higher current period

volatility and an upward revis ion of the conditional volatility. Increases in

price due to the updating of the information set when the good news arrives

should be to sorne extent offset by the increased conditional volatility. Hence,

the net impact on stock return volatility - also through the lever age effect - is

not clear.

At the firm level, if the shock is idiosyncratic, the covariance between the

market return and firm return do es not change, and no change in the required

risk premium is needed. A necessary condition for volatility feedback to be

observed is that the covariance of the firm's return increases in response to

market shocks. It follows that idiosyncratic shocks should generate volatility

asymmetry only through lever age.

Volatility feedback at the firm level occurs when market shocks increase the

covariance of the firm's return with the market and the conditional variance

such that firm betas remain constant and the change in the price of risk can

4Cfr. Huang and Litzenberger (1988).

21

be reflected in the expected return.

Growth Option Approach ta Leverage. Duffee (2002) provides yet another ex­

planation of asymmetry that links the changes in the balance sheet of the firms

to volatility. Firm's riskier assets, and particularly growth options, could play

an important role in defining the sign of the correlation between return and

volatility. This sign cannot be theoretically determined; it depends on the

relative weight of the different classes of assets. To illustrate the idea, suppose

that the value of an entirely equity financed firm is vt = Et + Ct, where Ct is

the value of a "growth" call option.

The volatility of the stock is af = (1 + 8) i af, where 8 = :~: is the delta

of the option. To see how the volatility af changes when vt changes compute

the elasticity of the stock volatility with respect to the total value of the firm

(1.18)

The sign of this elasticity is not determined. When the option is deep out

of the money, it is positive. When the call is deep in the money, the 8 will

tend to 1, 88/8Et tend to zero, and thus the sign of (1.18) is negative.5

Despite the uncertainty with regard to the sign, from this formulation it is

clear that if growth options constitute an important economic element of the

balance sheet of a firm it may affect the sign of the relationship.

1.1.2 International Asset Pricing Theory

If the financial markets are perfect with no barriers to international invest­

ment and the consumption opportunity sets are the same across countries,

an investor can achieve the same expected lifetime utility given his wealth

independently from his location.

5This is so since the value of the caU cannot exceed that of the underlying asset.

22

In an economy where the law of one price holds for the only consumption

good, returns are normal multivariate and there is a risk free asset earning

r, the CA PM holds.6 In perfectly segmented markets, the pricing error of

using the domestic CAPM instead of the international CAPM (ICAPM), ai,

should be zero for aH assets because the CAPM must hold for aH the domestic

countries. In this world, aH assets are domestic since only domestic investors

can hold them.

If instead investors have access to foreign markets the relevant portfolio

is the world market. By using the domestic CAPM when the global CAPM

should be used instead a pricing error would occur.

If in addition, the differences across countries in the composition of national

consumption baskets, relative prices of goods, the evolution of relative prices

over time, capital controls, taxation, access to information affect how asset

are priced in different countries, developing a coherent theory of international

asset pricing becomes a chaHenging task. The international finance literature

has grown into two different, and largely autonomous, streams: models that

consider exchange risk, and models that examine other market imperfections.

The combinat ion of both approaches into a single tractable model is still in its

infancy. The foHowing subsection introduces sorne important ideas related to

exchange rate risk and market imperfections.

1.1.3 Foreign Exchange Risk and Market Imperfections

A model that remains central to understand the role of exchange rate risk in

international markets is Solnik (1974), extended by Sercu (1980).

The main results of the paper can be subsumed in two important separation

theorems. The first theorem states:

6This brief introduction to International Asset pricing models follows Karolyi and Stulz

(2002).

23

Theorem 1 (First Separation Theorem). Investors are indifferent between

holding the original assets or the stock market portfolio (hedged against ex­

change risk) and the n bonds.

Furthermore, the demands for stocks and bonds are shown to be separable

and hence the proportion of the risky assets in the fund can be computed. It

follows that for each country a two fund separation theorem holds: investors

will hold a country-specifie risky fund and the riskless asset.

Using a no-triangular arbitrage argument and a particular structure of the

covariance matrix structure of asset returns, Solnik derives a more general fund

separation theorem which is the central finding of the paper. The theorem

states:

Theorem 2 (Second Separation Theorem). All investors will be indifferent

between holding a portfolio of the original securities and a combination of three

funds. The three funds are: 1) The world market portfolio (hedged against the

exchange risk); 2) A portfolio of bonds bearing the exchange risk; 3) The home

country risk free asset. The funds are independent of investor preferences; they

refiect instead assets characteristics.

Notice that, since the two risky funds are held in the same proportion,

independently of the investor, this is in reality a two fund separation theorem

with a risky fund made of two components, namely the hedged world market

portfolio and a portfolio of bonds bearing the exchange risk. In practice how­

ever, a n + 1 fund separation results, as the domestic bond is different for each

country.

The particular structure of the covariance matrix used in Solnik (1974) is

not essential to the result as it is shown in the generalization by Sercu (1980).

The latter paper derives a pricing formula that states that the risk premium of

a stock consists of two parts: The expected cost of hedging the stock against

24

exchange risk and a CAPM like risk premium for the risk not associated with

exchange rates. Similar pricing equations are still the workhorse of numerous

studies7 in the most widely known formulation in Adler and Dumas (1983).

One important criticism of Solnik (1974) should be mentioned. This model

assumes that asset prices are independent from exchange rates. As noted by

Dumas, however, the only case in which this assumption is reasonable is when

countries have perfectly isolated economies. Integrated capital markets do not

seem compatible with perfectly isolated economies.

Foreign Exchange Risk and PPP. Adler and Dumas (1983) extends Solnik

(1974) by more realistically assuming that inflation is stochastic in each coun­

try, and allowing for differences in national consumption preferences. In this

setting, domestic bond investing is not anymore safe in real terms and cannot

be a perfect hedge against inflation. The assumption of stochastic inflation

also determines that correlations between risky asset returns and changes in

inflation will affect the variance of real returns.

ln an international finance context, Purchasing Power Parity (PPP) is rel­

evant as it measures the similarity of consumption opportunities in different

countries. Difference in PPP is what distinguishes a nation from another: in­

vestors with different PPP have different real returns and hence they generally

hold different portfolios.

PPP is a relation between weighted average price levels, not individual

commodity prices. Consumer Price Parity (CPP), which refers to an arbitrage

condition between two identical goods traded in two locations in absence of

frictions, is sufficient for PPP to hold, but it is not necessary.

Sufficient conditions for PPP to hold exactly are:

(1) The Consumer Price Indices (CPI) used as the base for computation

7The the essay in the following chapter makes use of this model.

25

is representative of consumers' possibilities and preferences. (2) Consumer

preferences are homothetic, i.e. the proportion of goods consumed does not

change with the level of wealth.

The literature suggests that ppp is violated both instantaneously, and at

any forecasting horizon. Consequently, Adler and Dumas (1983) argue that

heterogeneity of national consumption should be at the foundation of any

sensible international asset pricing model.

In addition, since ppp is not forecastable, it can be modeled as a martin­

gale. However, the random fluctuation of prices is small compared with that

of exchange rates. This suggests that ex change rates can be, and in practice

often are, used in substitution of PPP.

Using a set up similar to Solnik-Sercu, Adler and Dumas (1983) derive

optimal portfolio weights W

w = Ct + (1- Ct) [

~-1 (E(r) - rf) 1 [ ~-lW 1 1 - ~-1 (E(r) - rf) 1 - ~-lW

(1.19)

where Ct is the risk tolerance, ~ is the N x N covariance matrix of asset

returns and w is the N x 1 vector of covariances of the N assets with the

investor's rate of inflation.

For investor l the notation can be simplified to

(1.20)

As the Ct is equal to 1 for the case of an investor with logarithmic prefer­

ences, Wlog corresponds to the portfolio of an investor with logarithmic pref­

erences.8 The component Wh is the portfolio of an investor with zero risk

C 8log P = logC - logP implies that priees drop out of the objective function of the

investor.

26

tolerance. Hence, it is the global minimum variance in real terms, and it is

independent of expected returns. This result mirrors the second two fund sep­

aration theorem of Solnik (1984). The difference is that the second element of

the equation is the vector of portfolio weights whose nominal return is most

highly correlated with inflation. This portfolio is investor specifie and it is

identical to the home currency T-Bill if there is no inflation, reducing to the

Solnik (1974)'s case.

Assuming that demand for assets originates from investors who ho Id the

optimal portfolio (1.19), and that supply is given, it is possible to aggregate

across investors. The result is a CAPM with the number of country L, covari­

ances term with the inflation rate 'Tr, the covariance with the market, and the

intercept

L

ri = rI + ÀmCov(ri' rm) + L XICov(ri, 'Trj)

j=l

(1.21)

This model is testable by replacing inflation rates with ex change rates. It

is also the starting point of the first essay.

Imperfections in International Markets. The issue of whether and to what

extent international markets are integrated constitutes an important topic in

the international finance literature. One way to model segmentation is to

assume that is costly for domestic investors to hold foreign assets. Using a tax

on both, long and short positions, Stulz (1981b) shows that aIl foreign assets

with a {3 larger than sorne beta {3* plot on either one of two security market

lines. The presence of such a tax determines that sorne foreign assets with a

{3* smaller than {3* are not held by domestic investors even if their expected

return is increased slightly.

Errunza and Losq (1985) theoretically and empirically investigate the pric­

ing and portfolio implications of investment barriers. In a two country set up

27

two kind of investors and two types of securities are considered: respectively,

restricted and unrestricted, and eligible and ineligible. Restricted investors

cannot hold ineligible securities. This structure, named mildly segmented,

leads to the existence of "super" risk premiums for ineligible securities and

to a breakdown of the standard separation result. The empirical study uses

an extended database including LDC markets and provides sorne support for

the mild segmentation hypothesis.

A recent attempt to jointly consider exchange risk and imperfections is in

Bayraktar (1999). The paper derives international equity pricing relations by

taking into account exchange rate risk under various forms of market struc­

tures in a mean-variance, two-country, two-period, two-goods framework. Seg­

mented, mildly segmented and integrated market structures are analyzed. The

result is that, as long as investors are consuming the imported goods, the

stochastic exchange rate is one of the important determinants of real equity

prices even when markets are integrated. This is because the exchange rate

through terms of trade affects the purchasing power of the investors. Despite

the theoretical interest of this model, the formulae derived are rather cumber­

sorne and no testable implication is provided.

1.2 Empirical Applications and Evidence

1.2.1 Econometrie Models of Asymmetric Volatility

Glosten, Jagannathan and Runkle (1993) propose an econometric GARCR-M

specification with monthly data that allows for asymmetric volatility and find

support for a negative relation between conditional expected monthly returns

and conditional variances of monthly returns.9 The GARCR model is modified

by allowing (1) seasonal patterns in volatility, (2) returns positive and nega-

9For a more general survey of volatility modeling see Gysels, Harvey and Renaults (1996)

28

tive innovations to have different impacts on conditional volatility, and (3)

nominal interest rates to predict conditional variance. The study finds that

monthly conditional volatility may not be as persistent as is commonly be­

lieved. Moreover, positive unanticipated returns result in a downward revis ion

of the conditional volatility whereas negative unanticipated returns determine

an upward revision of conditional volatility. This is in contrast to Nelson (1991)

and Engle and Ng (1993), which use daily data on the stock index to document

that both large positive and negative returns lead to an upward revision of the

conditional volatility, although negative returns lead to a larger revision.

Nelson (1991) proposes an exponential model of volatility. The specification

for the conditional variance for a EGARCH(l, 1) is

(1.22)

where Zt is an innovation from a Generalized Error Distribution (GED).

Note that the left-hand side is the log of the conditional variance. This implies

that the lever age effect is exponential, and hence the forecasts of the conditional

variance are guaranteed to be nonnegative. The presence of lever age effects

can be tested by the hypothesis that ae < O. The model can also be estimated

assuming normally distributed errors.

The model's desirable feature of accommodating the negative correlation

between current returns and future returns volatility is somehow counterbal­

anced by the difficulty of forecasting multiple period ahead.

Babsiri and Zakoïan (2001) notes that although asymmetric GARCH mod­

els allow positive and negative changes to have different impacts on future

volatilities, the two components of the innovation have, up to a constant, the

same volatilities. This is not appropriate if past negative innovations have

been typically of higher magnitudes than positive ones, as it is often found in

29

the market data.

The model, proposed and applied to the French CAC 40 index, develops

a structure that allows for time varying skewness and kurtosis and for two

kind of asymmetries (1) different volatility pro cesses for up and down moves

in equity markets (contemporaneous asymmetry,lO which relates to the shape

of the conditional distribution at any point in time); (2) asymmetric reactions

of these volatilities to past positive and negative changes (which the paper

calls dynamic asymmetry or "leverage" effect, but corresponds to the notion

of asymmetry used in this survey).

The empirical results confirm the existence of both types of asymmetries,

but suggest that contemporaneous asymmetry may be more important and

that the asymmetries found in the literature may need reconsideration.

A general specifications of the volatility dynamic that nests most existing

work has been suggested by Rentschel (1995).

Under the usual assumptions, the volatility dynamic a; can be specified as

follows

a; = 130 + f3 1aLl + f32aLd(Zt-l)

Where Zt V\ N(O; 1).

(1.23)

Different GARCR models are characterized by differences in the innovation

functions !. Most relevant to the cases are

Leverage: !(Zt-l) = (Zt-l - ())2

Power: !(Zt-l) = (Zt-l _())2'Y, which nests Leverage and their Box-Cox

transformation

lOThis can be thought of as conditional skewness.

30

(1.24)

Which converges to Nelson EGARCR when 'Y goes to zero.

Kroner and Ng (1998) argue that most time-varying covariance models usu­

ally impose the strong restrictions of symmetry on how past shocks affect the

forecasted covariance matrix. The paper shows that the choice of a misspecified

multivariate volatility model can lead to substantially misguided conclusions

in any application that involves forecasting dynamic covariance matrices such

as estimating the optimal hedge ratio or deriving the risk minimizing port­

folio. The paper uses a general model that nests early models of correlation

including BEKK and a Constant Correlation models and their asymmetric ex­

tensions. The study focuses on the dynamic relation between large and small

firm returns. It finds that symmetric models are generally misspecified, es­

pecially in the dynamics of the covariance. Large-firm returns are found to

affect the volatility of small-firm returns, but small-firm returns do not have

much effect on large-firm volatility. Moreover, asymmetric effects in both the

variances and covariances are detected: bad news about large firms can cause

volatility in both small-firm returns and large-firm returns. Furthermore, the

conditional covariance between large-firm returns and small-firm returns tends

to be higher following bad news about large firms than good news.

1.2.2 Volatility Feedback Effect

At the aggregate level, D.S. stock returns are negatively correlated with both

contemporaneous volatility and future volatility. Christie (1982) examines

the relation between the variance of equity returns and sever al explanatory

variables to find that equity variance has a strong positive association with

31

financialleverage. The negative elasticity of volatility with respect to value of

equity is found to be attribut able to financialleverage to a substantial degree.

Brown, Harlow, and Tinic (1988), develop and test the uneertain informa­

tion hypothesis as a means of explaining the response of rational, risk-averse

investors to the arrivaI of unanticipated information. The uncertain informa­

tion hypothesis predicts that following news of a dramatic financial event, both

the risk and expected return of the affected companies increase systematically,

and that priees react more strongly to bad news than good. The empirical in­

vestigation of over 9000 marketwide and firm-specific events pro duces results

consistent with these predictions. The paper show that stock price reactions to

unfavorable news events tend to be larger than reactions to favorable events,

which is attributed to volatility feedback.

On the opposite side, Poterba and Summers (1986) argue that volatility

feedback could not be important because changes in volatility are too short­

lived to have a major effect on stock priees.

Campbell and Hentschel (1992) present the first formalized empirical model

of volatility feedback that investigates how changes in volatility affect required

stock returns and thus the level of stock priees. The explanation hinges on

volatility persistence and varying risk premia. Large pieees of good news have

positive impact on stock priee sinee they imply larger future dividends. How­

ever, they also increase conditional volatility. The greater future volatility

increases the required rate of return on stock, which offsets the stock priee

increase, at least partially. When a large piece of bad news hits the market,

the direct negative impact on the dividends is amplified by the higher required

rate of return on stock indueed by the higher volatility.

This mechanism could explain why negative stock returns are more common

than large positive ones. Large negative returns imply also excess kurtosis. In

contrast, a small pieee of news would lower conditional volatility and increase

32

the stock price. Volatility feedback therefore implies that stock priee move­

ments are correlated with future volatility. Moreover, the volatility feedback

mechanism can explain skewness, exeess kurtosis of returns even if the under­

lying shocks to the market are conditionally normally distributed. While the

study tries to provide sorne economic justification, the model is not based on

any equilibrium notion.

Bekaert and Wu (2000) use the market portfolio and portfolios with dif­

ferent leverage constructed from Nikkei 225 stocks. It rejects the hypothesis

of leverage and it finds support for the volatility feedback hypothesis. The

sample period is from January 1, 1985 to June 20, 1994 of daily observations.

One coneern for this kind of studies is that debt data is the last available

book value. II Indeed the conclusion that lever age variables explain litt le of

the volatility behavior of the Japanese stock returns could have been driven

by the fact that lever age is poorly measured. It is also unclear whether the

assumption that debt is riskless is innocuous.

Wu (2001) expands on Campbell and Hentschel (1992). The main differ­

enee is that Wu (2001) do es not use a Q-GARCH and instead models the

volatility as a stochastic proeess directly. Then the Efficient Method of Mo­

ments (EMM) is used to estimate the parameters of the (under identified)

model. The model allows both the lever age effect and the volatility feedback

effect. It finds that both the lever age effect and volatility feedback are im­

portant determinants of asymmetric volatility, and that volatility feedback is

significant both statistically and economically. One critique to this model is

that it assumes that aggregate risk premia are determined only by the level of

aggregate volatility. Since risk premia are positive on average, the model in

l1The practice of using book value of the debt is standard in the literature.

practice forces the risk premia to positively vary with volatility.12

Consider the following regression also used in Christie (1982)

33

(1.25)

The lever age hypothesis entails that for firms with large D / E ratio, Ào

should be more negative than that for firm with lower D / E. The usual inter­

pretation is that a negative Ào implies that to a decrease in rt corresponds an

increase in 0"t+1.13 This does not need to be the case. The OLS14 estimation

procedure of the two regressions

yield that Ào = À2 - À}.

(1.26)

(1.27)

For a large sample of domestic firms, Duffee (1995) finds that the negative

sign of Ào is largely due to a large and positive À1' which entails a positive

contemporaneous relation between firm stock returns and firm stock return

volatility.15 This positive relation is strongest for both small firms and firms

with litt le financial leverage. At the aggregate level, the sign of this contem-

poraneous relation is reversed.

12This critique is due to Duffee (2002).

13In early studies such as Black and Christie's the volatility is the sample standard de-

viation of returns over available sub intervals. Black uses daily returns for the monthly

volatility, and Christie similarly constructs quarterly estimates. 14This can be more easily seen by writing equation (1.25) in matrix and vector form:

(loga+1 -loga) = rÀ + ê. The OLS estimate X= (r'r)-lr' (loga+1 -log a) , which is also

the difference of two parameters vectors in the other two equations.

15Potential non stationarities are dealt with by subtracting log at-l from the left hand

side in both equations. Moreover, changes in volatility are the focus of interest, not levels.

34

Figlewski and Wang (2000) use both returns and directly measured leverage

to examine the effect of financialleverage as it applies to the individual stocks

in the S&P100 (OEX) index, and to the index itself. They find a strong

asymmetry associated with falling stock priees, but also numerous anomalies

that call into question leverage changes as a viable explanation. The papers

concludes that the "lever age effect" is rather a "down market effect" that may

have little direct connection to firm capital structure.

1.2.3 Volatility in the International Finance Literature

Using a methodology close to that proposed by Glosten, Jagannathan and Run­

kle (1993), Bekaert and Harvey (1997) model asymmetry in the international

finance context. Using monthly data from 20 emerging countries, they primar­

ily investigate the relation between market liberalizations and local market

volatility.

The model proposed is a GARCH-M with asymmetry of the form

ri,t = f-li,t-l + Ei,t (1.28)

(1.29)

2 2 f3 2 S 2 (J. t = Ci + a(J· t-l + ·e· t-l + 'Y. i e· t-l t, t, t t, 1, t, (1.30)

(1.31)

Where f-li,t-l is the conditional mean return for country i, and Ei,t the un­

expected return, which is driven by Ew,t, a world shock and ei,t a purely id­

iosyncratic shock. The dependence of local shock on world shock is determined

35

by Vi,t-l. The skedastic function follows a GAReH pro cess with each element

having the usual meaning plus the addition of the last term Si, which is a

dummy variable that take value 1 when the idiosyncratic shock is negative.

The motivation for introducing the dummy variable is the possible presence of

leverage.

Similar to Bekaert and Harvey (1995), the conditional mean is modeled

allowing for time varying influences of local and world factors, i.e.

(1.32)

where X t is a set of world variables including the world market dividend

yields, in excess of the 30-day Eurodollar rate, the Moody's Baa minus the Aaa

bond yield and the change in the 30-day Eurodollar rate. Xi,t represents local

information and includes a constant, the equity return, the exchange rate, the

dividend yield, the ratio of market capitalization to GDP and the ratio of trade

to GDP, all of which are lagged. As for the coefficient of the world volatility

+ ' X*' Vi,t = qi,O qi,l i,t (1.33)

Xt~ includes market capitalization to GDP, the size of the trade sector

(export plus import divided by GDP) which may proxy for the degree of inte­

gration. A non-linear model is also used for robustness.

The inclusion of the asymmetry parameter improves the fit for the world

market: a likelihood ratio test rejects the hypothesis of no asymmetry. How­

ever, the estimated coefficient "( results not significant in 10 out of 20 countries

and has the positive sign in 3 of the significant cases. The result could be

interpreted as a weak sign that lever age is present, but it is far from being

conclusive and does not shed any light on weekly and daily frequencies. This

36

lack of power could also be determined by inefficiencies inherent to the non

full-information likelihood estimation procedure.

Building on the methodology developed in De Santis and Gerard (1997),

De Santis and Gerard (1998) ask two questions: (1) Is currency risk priced?

(2) If it is, what is the compensation that an investor can expect for bearing

exchange risk?

Dumas and Solnik (1995) had already asked similar questions using the

methodology proposed by Harvey (1991), but without taking into account con­

ditional second moments. Thus, even if both studies find the price of exchange

rate risk is significantly different from zero, no statement can be made about

its magnitude relative to the market premium. De Santis and Gerard (1998)

implement a fully parametric approach that allows the simultaneous analy­

sis of international equity market and currency deposit and the estimation of

time-varying conditional prices and measures of risk. The paper determines

the relative magnitude of market risk premium and currency risk premium by

using the time series of the first and second conditional moments. The main re­

suIt, in accordance with the theoretical model, shows that both, currency and

market risk are priced factors. It follows that IAPM that only uses the world

market portfolio to measure risk and explain conditional expected returns is

misspecified. Moreover, time-varying second moments are not sufficient to de­

tect either market or currency risk as relevant pricing factors. The assumption

that the prices of aIl sources of risk vary through time is also needed. These

findings are consistent with those of Dumas and Solnik.

1.3 Conclusion

This survey presented a selection from the theoretical and empiricalliterature

related to volatility asymmetries and international finance asset pricing. It

37

emerges that while the study of asymmetries for domestic market have pro­

gressed substantially, theoretical and empirical investigation of similar features

in international assets have been lagging behind.

The following three essays make contributions with respect to the mea­

surement and use of conditional second moment of equities and currencies as

a measure of risk for asset pricing and policy purposes in the context of in­

ternational markets. The first essay investigate the importance of covariance

asymmetry for international asset pricing. The second and third essays fo­

cus on the measurement and forecast of volatility and correlation of foreign

exchange rates, with an emphasis on policy.

38

Chapter 2

How IMPORTANT 18 A8YMMETRIC VOLATILITY FOR

THE RI8K PREMIUM OF INTERNATIONAL A88ET8?

Stefano 11azzotta

Abstract. This essay investigates the importance of asymmetric volatil­ity when computing the risk premium of international assets. The results indicate that conditional second moment asymmetry is significant and time­varying. They also show that, if the price of risk is time-varying, the world market and foreign exchange risk premia estimated without allowing for time­varying asymmetry are misspecified. Furthermore, they imply that asymmetry is more pronounced when the business condition is such that investors require higher compensation to bear risk.

JEL Classification: GlO; G12; G15; C52,. Keywords: Time-varying covariance asymmetry; International asset pric­

ing; Risk premia estimation

39

2.0.1 Introduction

The question of how risk affects returns is a fundamental one in finance. The

notion of risk used in the literature is generally based on the second moment

of returns, often condition al on some information set. This essay focuses on

one particular aspect of the trade off between risk and return. It investigates

the importance of asymmetric response of the conditional volatility to return

innovations with regard to the estimation of the risk premium of international

assets.

Asymmetric volatility refers to the negative correlation between current

returns and future volatility. In other words, positive returns tend to be fol­

lowed by lower volatility than negative returns of the same magnitude. The

occurrence of this phenomenon in domestic individual stocks and indices, of­

ten referred to as the leverage effect, l has received substantial empirical and

theoretical attention. Black (1976) first conjectured that asymmetry could

be determined by changes in the capital structure of firms. Another credible

hypothesis is that the asymmetric volatility response to returns shock could

be due to time-varying risk premia. This is often referred to as the volatility

feedback effect. Such an asymmetry is a well-established empirical fact in V.S.

assets. With regard to international asset returns, the evidence2 with respect

to the existence of asymmetry of the second moment of international returns

is still sparse.

It is conceivable that the asymmetric response is not only present in in­

ternational assets, but it also varies over time. Indeed, both the "lever age"

hypothesis and the "volatility feedback" conjecture are compatible with time

1 It is not uncommon that the leverage effect is identified with volatility asymmetry. This

identification is not in general correct since leverage effect suggests a causality relation that

is far from being established. 2See e.g. Cappiello, Engle, and Sheppard (2003).

40

variation in the asymmetric response. In fact, if the asymmetric response was

largely due to the change in the firms' capital structure, as maintained by the

leverage effect hypothesis, then the changes in the relative weights of equity

and debt in the firms' balance sheet would determine time variation in the

asymmetric response at the firm level. If volatility feedback was the main rea­

son for asymmetry, it is possible to conjecture that investors' reaction to news,

particularly bad news, may be more pronounced depending on the phase of

the business cycle.

This paper asks three interrelated questions from the asset pricing per­

spective of a D.S. investor: 1) Do conditional second moments of returns of

developed countries' assets respond asymmetrically to returns innovations? 2)

If present, does asymmetry vary over time? 3) If investors are compensated for

market and exchange rate risk, does it matter whether they take into account

asymmetries? Indeed, the question of whether asymmetry affects pricing is

possibly more important than the presence of asymmetry itself.

The original results of this essay can be summarized as follows. Firstly, it is

found that asymmetry is not only present, but also significantly time-varying

for the assets of developed countries.

Secondly, not allowing for time variation in the asymmetry yields mislead­

ing estimates of the world market risk premium and of the foreign exchange

risk premia. This result is potentially of interest for portfolio management and

asset allocation.

Finally, this essay proposes a novel econometric empirical model that con­

veniently parameterizes the alternative hypothesis of constant or time-varying

asymmetry. The proposed methodology generalizes the International Asset

Pricing Model of Adler and Dumas (1983) (IAPM) with the multivariate

GARCH specification implemented by De Santis and Gérard (1998), hence­

forth DG. The DG parameterization of the IAPM ideally lends itself to the

41

investigation of the economic importance of second moments asymmetry as it

allows a quantification of the market risk premium and of the exchange risk

premia.

The rest of the essay is organized as follows: After reviewing a selection

of the pertinent literature, Section l presents the international asset pricing

model. In Section II the multivariate G ARC H specification, allowing for the

constant and time-varying asymmetry in the conditional second moments, is

introduced along with estimation results. In Section III statistical tests high­

lighting the importance of second moments asymmetry for the estimation of

the risk premium are illustrated. The economic implications of the results are

also discussed in this section. Section IV concludes.

2.0.2 Related Work

Christie (1982) was among the first to examine the relation between the vari­

ance of equity returns and several explanatory variables. What he found was

that equity variance has a strong positive association with financial lever age

and the negative elasticity of volatility with respect to value of equity should

be ascribed to financialleverage to a substantial degree. Brown, Harlow, and

Tinic (1988) develop and test the uncertain information hypothesis as a means

of explaining the response of rational, risk-averse investors to the arrivaI of

unanticipated information. The paper shows that stock price reactions to un­

favorable news events tend to be larger than reactions to favorable events,

which is attributed to volatility feedback.

Campbell and Hentschel (1992) present the first formalized empirical modei

of volatility feedback that investigates how changes in volatility affect required

stock returns and thus the level of stock prices. The explanation hinges on

voiatility persistence and varying risk premia. Large pieces of good news have

a positive impact on stock price since they imply larger future dividends. How-

42

ever, they also increase conditional volatility. The greater future volatility

increases the required rate of return on stock, which offsets the stock price

increase, at least partially. When a large piece of bad news hits the market,

the direct negative impact on dividends is amplified by the higher required

rate of return on stock induced by the higher volatility.

Bekaert and Wu (2000) use the market portfolio and portfolios with differ­

ent lever age constructed from Nikkei 225 stocks. The paper rejects the lever age

hypothesis and finds support for the volatility feedback hypothesis.

The seminal works mentioned above focus on asymmetric volatility. For the

international markets, Bekaert and Harvey (1997) use monthly data from 20

emerging countries to model asymmetry in order to primarily investigate the

relation between market liberalization and local market volatility. The model

proposed is a GARCH-M with asymmetry. The skedastic function follows a

GARCH pro cess with the addition of term Si, a dummy variable that takes

the value 1 when the idiosyncratic shock is negative. The motivation for intro­

ducing the dummy variable, is the possible presence of covariance asymmetry.

The conditional mean is modeled allowing for time-varying influences of local

and world factors. The inclusion of the asymmetry improves the fit for the

world market while a likelihood ratio test rejects the hypothesis of no asym­

metry. However, the estimated coefficient for the asymmetry is not significant

in 10 out of 20 countries and has the positive sign in 3 of the significant cases.

The result could be interpreted as evidence, however slight, that asymmetry is

present, but is far from being conclusive and do es not shed any light on weekly

and daily frequencies.

Dependence of asset returns conditional on sign and magnitude has also

gained attention in the last few years. Longin and Solnik (2001) find that

returns correlation increase in down market and conclude that the asymmet­

ric correlation pattern should become a key property of any multivariate eq-

43

uity return model to match. Indeed, if international returns display second

moment asymmetry any model that does not allow for asymmetry would be

misspecified. The conclusion is reinforced by Ang and Bekaert (2002), who

have developed a model with time-varying investment opportunities set in a

regime switching pro cess approach that is capable of replicating the asymmetry

displayed by the data.

Harvey and Siddique (2000) document significant time-variation in condi­

tional skewness measures for both the U.S. stock market and a broader world

market portfolio. The study also finds that allowing for skewness helps to

explain many of the episodes of negative ex ante market risk premia.

Cappiello, Engle, and Sheppard (2003) investigate the presence of asym­

metric condition al second moments in international equity and bond returns

through an asymmetric version of the two-stage procedure (Dynamic Condi­

tional Correlation) of Engle and Sheppard (2002). Whilst equity index returns

have been found to show strong asymmetries in conditional volatility, in con­

trast bond index returns do not exhibit this behavior. However, both bonds

and equities exhibit asymmetry in conditional correlation.

2.0.3 The Data

The analysis was carried out usmg a sample of daily observation starting

January 1990 and ending December 2002. As robustness check, weekly and

monthly data starting January 1980 and ending December 2002 were also an­

alyzed. Data availability dictated the longest sample period. The samples

include 3398 daily, 1200 weekly, 276 monthly observations. The results pre­

sented in detail here refer to the daily estimation. A summary of the findings

for the weekly and monthly frequencies is also provided. AlI models are esti­

mated in US dollars.

44

The data are indices from Germany (DAX 30), Japan (Nikkei 225), the

U.K. (FTSE 100) and the US (S&P 500), in US dollars, i.e. the G4 countries.

The world return is the DS-Market total return index. The conditional risk free

rates are the Euro-Mark 1-Month lending, the Euro-Yen 1-Month lending, the

UK T-Bill1-Month and the EURO-Dollar 1-Month, all middle rates. The data

series used to compute the instruments are the US Corporate bond Moody's

ND BAA and ND AAA for the default premium (USDP), the Euro-Dollar 1-

Month deposit and the US Theasury constant maturities 1 O-year , for the term

premium (USTP). AlI the data are from Datastream.

The default premium (USDP) and the term premium (USTP) have been

widely used in international asset pricing literature, as well as in the studies

cited ab ove , and are now considered to sorne extent as standard instruments.

Avramov (2002) studies 14 predictors that have been widely used in the lit­

erature. He finds that the term premium is robust both in sample and out

of sam pIe. He also notices that the USTP's good performance is due to its

ability to capture exposure related to shifts in interest rates and economic con­

ditions that affect the likelihood of default. The fact that the term premium

may proxy for time-varying risk aversion makes it a particularly suit able vari­

able for explaining second moment asymmetry, as the degree of asymmetric

response could be infiuenced by investors' attitudes towards risk.

The USDP, known also under the name of "junk spread" , is also intuitively

appealing to explain asymmetry as it could capture sorne aggregate measure

of firms lever age.

The World Market Dividend Yield was also preliminarily considered as a

potential instrument, but it was found to have less explanatory power and it

was dropped for the sake of parsimony. In addition, the robustness of dividend

yield as return predictor has been recently put into question. Goyal and Welch

(2003) have found that the dividend yield forecast returns only over horizons

45

longer than 5-10 years.

The choice of instruments is also justified by the fact the USDP and USTP

are the less correlated pair of instruments with a correlation coefficient of 0.24

versus a correlation of 0.28 for the USDP and DY and 0.55 for the USTP and

DY, all of which are highly significant.

2.1 International Asset Returns

In the following section the presence of time-varying asymmetries is investi­

gated in an International Asset Pricing Model context for the G4 countries3 :

Germany, Japan, UK, and US. Due to the large share of market capitalization

relative to the world market, this set of countries is the one that is often the

object of study. The question of the possible time-varying asymmetry in the

second cross moments will be addressed along with the possible relevance of

model misspecifications with regard to the estimation of the conditional first

moments, and in particular the risk premia estimates.

2.1.1 The Asset Pricing Model.

The IAPM of Adler and Dumas (1983) suggests that exposure to foreign ex­

change should be priced when purchasing power parity does not hold.

The following are the main assumptions of the model. The world economy

has L + 1 countries with M risky securities of which Lare currency deposits.

All returns ri,t are expressed in the reference currency, which in this essay is

the USD. ppp is violated by assumption, and thus investors from different

3Preliminary univariate econometric analysis of stock returns in a country by country

setting performed by the author provided evidence that asymmetry is present and signifi­

cantly time-varying for Germany, the U.K. and the U.S. For Japan, asymmetry is significant

but constant. Results are not presented in this paper.

46

countries have a different degree of appreciation for real returns, i.e. the same

nominal return is worth differently according to the investor's country.

This set of assumptions implies that the optimal portfolios differ across

countries and that the expected return must include market and currency

premmm.

The fundamental pricing equation is

L+l

Et-1(ri,t) = Ol,t-1COVt-l(ri,t,rm,t) + LOe,t-1COVt-l(ri,t,7fe,t) (2.1) e=2

i = 1, ... , M with M equal to the number of assets

where

Oe,t-l = 'l/Jt-l(Jc -1) ~;~~1 is the priee offoreign exchange risk for currency,

and

Ol,t-l = 'l/Jt-l = 2:L+1 k-.L is the world priee of market risk. c=1 W t _1 .pc

The coefficient of relative risk aversion for investors from country c is 'l/Je,

while 'l/Jt-l is an average of the risk aversion of each group weighted by the

correspondent relative wealth Wwc,t-l, 7f ct is the inflation of country c measured t-1 '

in the referenee currency, r m,t is the excess return on the world portfolio of

an traded stock. From the definition of 'l/Jt-l it is apparent that the priee of

market risk can be negative only if the risk aversion 'l/Je is negative.

For the purpose of testing, a further assumption is that inflation is non­

stochastic, which can be justified by the fact that in developed markets the

varianee of inflation is negligible with respect to the variance of the exchange

rate. Nonstochasticity of inflation implies that the random component that

needs to be modeled in 7f e,t is the relative change in the ex change rate between

the currency of country c and the referenee currency. This assumption makes

47

it possible to use the return on country c' s currency deposit as a proxy for the

exchange rate risk.

Under this set of assumptions DG write a system of equations

for the q equity portfolios,

for the L currency deposits, and

for the world portfolio.

DG find that the exchange rate is prieed extending the results from Dumas

and Solnik (1995) and Harvey (1991), and Ferson and Harvey (1993) and

provide a measure of its magnitude. To do so they implement a fully parametric

approach that allows the simultaneous analysis of international equity market

and currency deposits and the estimation of time-varying conditional priees

and measures of risk. Notice that this model provides a precise origin for a

time-varying priee of risk. If the risk aversion coefficients for each country's

investor are constant the variation of the priee of risk is determined by a time­

varying wealth share.4

4In principle, since the market eapitalization is an arguably good proxy for wealth it

would be possible to estimate the risk aversion parameter 'l/Jc, instead of estimating the priee

of risk directly. This avenue is not pursued here for eonsisteney with the previous literature.

48

2.1.2 IAPM Implementation.

For the purpose of this essay the model is implemented as follows. The mean

equation is:

L+l

rt = 61,t-lhl,t + L 6c,t-lh n+c,t + Ct Ctl~t-l 1./\ N(O, Ht ) (2.2) c=2

where rt is a 8 x 1 vector of returns, ~t-l is the set of information available

at time t - 1, Ht is the conditional covariance matrix of asset returns and hn+c,t

are columns of H t . The resulting structure of the system of 8 equations is as

follows: the first four equations in the system are for the equity indices, the

following three are for the Eurocurrency deposits and the last one is for the

world market portfolio.

The model assumes that the conditional second moments follow a diago­

nal GARCH pro cess in which the second moments in Ht depend only on past

squared residuals and an autoregressive component. This assumption has been

usually found satisfactory for monthly and weekly data for developed countries.

For the sample at hand, dependence is generally not found significant beyond

the first lag. 5 Moreover, in this sample non contemporaneous correlations are

generally insignificant. This implies that there should not be any concern re­

garding the difference in trading hours across international markets and the

possible non contemporaneous spillovers for daily observations and gives sup­

port to the proposed specification.

The advantage of assuming a specifie form for the covariance pro cess is

that it allows estimating the magnitude of the conditional prices of risk for the

5It is of course entirely possible that the true data generating pro cess is not exactly a

multivariate GARCH(l, 1). This assumption is however found to be satisfactory in a large

body of the literature and also the diagnostics for this sample suggest that this is a suitable

approximation.

49

market and the foreign exchange factors.

The conditional covariance matrix H t can be written as:

Ht = CC' + aa'ét-lé~_l + bb'Ht - 1 (2.3)

where a and b are vectors. Notice that this specification is a restriction

of the more general form: Ht = C'C + A'ét-lé~_lA + B'Ht-1B imposing the

BEK K diagonality restriction on A and B introduced by Engle and Kroner

(1995).

The direct parameterization of the constant matrix CC' in the conditional

covariance requires the estimation of C and is the general alternative to covari­

ance targeting. It, however, does not require the iterative procedure necessary

to estimate the unconditional variance Ho. DG assume that the system is

covariance stationary and estimate Ho iteratively. They find that this assump­

tion does not substantially affect the dynamics of the conditional covariance.

Here the constant matrix CC' is estimated along with the other parameters

by Quasi Maximum Likelihood (QML). The disadvantage of estimating CC'

is a loss of parsimony, which can however be afforded here thanks to the large

sample size.

The theory developed by Adler and Dumas (19S3) dictates that the market

price of risk should be positive. This positivity restriction of the price of risk

should hold at any point in time, as any violation would imply a negative

risk aversion. Adler and Dumas also suggest that one of the hypotheses to be

tested is that the price of market risk is positive.

In the asset pricing literature, to constrain the price of risk to be positive

as implied by theory, the price of risk 6t is often modeled as an exponential

function of the instrument (see e.g. DG (1997) , and DG (199S)). There are,

however, many authors who in the absence of theoretical indications about

50

the functional form relating priee of risk and the instruments prefer to use a

linear specification. Ferson (1989) and Ferson, Forerster and Keim (1993), use

for instanee a linear specification, and DG do also recognize the plausibility of

this choiee.

Arguably, a good model should be relatively robust to parameterization

about which the theory do es not provide any indication. The positivity of the

priee of risk of such a model should stem from estimation despite a specification

that allows the priee of risk to be estimated negative; any positivity constraint

used for computational convenienee should not be binding. Clearly, a model

that imposes positivity on priee of risk cannot indicate whether or not the sign

of the estimate priee of risk conforms to theory prediction. Here, the time­

varying priees of risk were modeled as linear functions of the instruments, thus

leaving unconstrained the sign of the priee of market risk.

The potentially time-varying risk premium is defined as

(2.4)

c 1, .. .4

Remarkably, in this study the priee of market risk is estimated positive

despite the linear specification. When taken as a constant, the priee of market

risk turns out to be either positive and significant or negative but insignificant.

For the model with time-varying priee of risk the priee of risk is also implied

to be positive by the coefficients estimated significantly.6 Figure 2.8 shows the

estimated priee of risk for two models with time vaying price of risk, which are

detailed inthe following sections. These results increase the confidenee in the

suitability of this type of model to explain international returns.

60nly for the purpose of plotting, time-varying priee of risk and asymmetry parameters

that are less significant than 10% are ignored.

51

As aH returns are measured in USD, there are three sources of foreign

exchange risk: DEM, JPY and GBP. The empirical model is then

4

ri,t O'l,t-l COVt-l (ri,t, r m,t) + L O'c,t-l COVt-l (ri,t, r4+c,t) + Ei,t (2.5) c=2

1, ... 8 (2.6)

where O'1,t-l is the world price of market risk, O'2,t, O'3,t, and O'4,t are respec­

tively the world price of foreign exchange risk of the DEM, of the JPY and of

the GBP.

Given that international markets are assumed to be integrated, the price

of each source of risk is common across aIl investors; it follows that there is no

subindex i on the O"s.

In the condition al implementation of the Adler and Dumas model it lS

possible that intertemporal hedging may play a role in the sense of Merton

(1973).

Dumas and Solnik (1995) point out that if the model is conditional it

should also be intertemporal since investors anticipate the future variations of

the instrumental variables and hedge them over their lifetime. To assess the

relative importance of intertemporal and exchange risk they run a "horse race"

between a model that allows for intertemporal risk and a model that allows

for exchange rate risk. They cannot conclude in favor of either and speculate

that under certain conditions exchange rate risk premia may equivalent to

intertemporal risk premia.7

More recently, Ng (2004) and Chang et al. (2005) develop and test an

international model that allow for both exchange risk and intertemporal risk,

7Note that the conditional formulation of the Adler and Dumas model is equivalent to

that of an ICAPM model where the state variables of the investment opportunity set are

the foreign exchange rate.

52

in addition to market risk. They both find that market risk is the most impor­

tant component of the risk premium. Their results show that for the foreign

exchange assets, the intertemporal hedging demand components are rather

unimportant. DG (1998) indirectly test for the relevance of the intertemporal

component. They cannot reject the null hypothesis that the intertemporal

component is not priced. For these reasons the following section will use equa­

tion (2.5) leaving the investigation of the importance of asymmetry for models

that allow for intertemporal and exchange risk for future research.

2.2 Second Moment Asymmetry

The definition of asymmetric conditional covariance is as follows:

Definition 1. Return ri,t and rj,t display asymmetric conditional covariance

if for two innovations for asset i Ci,t and -Ci,t of a given magnitude but opposite

sign and two innovations for asset j C j,t and -c j,t

This inequality is also often expressed as "lower than" in such a way as to

convey the idea that joint negative innovations tend to increase the conditional

covarIance.

The model specifies the errars as CtlJt-l ~ N(O, Ht ). Ignoring the condi­

tioning set for notational simplicity, by the assumption that Ct = ZtHt1/2 the

standardized residuals Zt are defined as

(2.8)

The diagonal multivariate GARCH(l, 1) specification is as follows

53

(2.9)

Focusing on the ARCH component of the conditional covariance pro cess

and using (2.8), it is possible to rewrite the model as follows

The model becomes

H CC' AH1/

2" H 1

/2 A' BH B' t = + t-l Zt-l Zt-l t-l + t-l (2.11)

In a univariate setting, an asymmetric G ARC H parameterization known

as N-GARCH appeared first in Engle and Ng (1993). This essay proposes a

multivariate generalization of the N -G ARC H as an alternative model which

allows very naturally for second moment asymmetry. This model will be called

BEKK multivariate NGARCH, or M-NGARCH for brevity. The asymmet-

ric conditional covariance is specified as:

(2.12)

This specification clearly nests the symmetric one and straightforwardly

lends itself to higher order GARCH or ARCH specifications.8 Moreover, the

proof of proposition 2.5 in Engle and Kroner (1995) shows that the BEK K

model yields a positive definite covariance matrix for aIl values of Ct-l.

8This model is also similar to the Quadratic ARCH model developed in Sentana (1995).

The main difference is that here asyrnmetry is applied to standardized innovations and it is

potentially time-varying.

54

The conditions under which BEKK's model yield a positive definite covari­

ance matrix are also sufficient for the above model (2.12) to yield a positive

definite covariance matrix. This follows immediately by redefining S;-l =

(Zt-l - e)Htl~î·

In the case of time-varying asymmetry, smoothness may be an impor­

tant advantage of this parameterization over variants of that first proposed

by Glosten, Jagannathan and Runkle (1993) which makes use of indicator

variables.

The most parsimonious way to include asymmetry is to restrict e to be con­

stant and equal for aIl the assets. This specification may indeed be appropriate

when modeling assets of the same class.

The sign of e in the typical case in which positive returns tend to be fol­

lowed by lower conditional second variance than negative returns of the same

magnitude should be positive. In addition, if e is higher than Zt for aIl t

the relationship between innovation and conditional second moment will be

monotonic.

Alternatively, it is possible to allow for a different ei for each asset in the

model, or the specification can be extended to allow for time-varying asymme­

try including a set of explanatory variables

(2.13)

where et = !(IVt) is taken to be a function of a set of instrumental vari­

ables known at time t. The main difference is that asymmetry is applied to

standardized innovations and it is potentially time-varying.

The function considered in this implementation is identical to that of the

priee of risk: it is simply a linear combination of the instruments and a con­

stant term. The default premium depends on the expected default loss of

55

risky bonds and a time-varying risk premium while the term premium may

proxy for time-varying risk aversion. These two explanatory variable can be

related to second moment asymmetry, as the degree of asymmetric response is

likely to be infiueneed by investors' attitude towards risk. Moreover, the use of

the same set of instruments in the same functional form as used for the price

of risk should provide enough evidenee that any time variation found in the

asymmetry parameter is not the spurious product of an ad hoc choiee of instru­

ments or functional forms. The specification for the time-varying asymmetry

is therefore

f(I1;t)

~ 1, ... 8

(2.14)

(2.15)

The model outlined above allows testing for the significance and dynamics

over time of the priee of risk. In particular, in the models with time-varying

prices of risk, if 61 is significant, but neither 62, nor 63 are significant, we

conclude that 6t is different from zero and constant. The significance and

dynamics over time of (Ji,t can be explored in a similar fashion.

2.2.1 M-NGARCH: a Simple Illustration.

For the sake of illustration, consider the case of a two asset model with constant

asymmetry parameter (J = [(JI (J2]. Let

Ct-l = [CI,t-1 C2,t-I]' Zt-l = [Zl,t-l Z2,t-I]' and (J = [(JI (J2]

H t = [~l'l,t ~1'2,t], 2,1,t 2,2,t

56

SI 2 t-1 ] . , ,

S2,2,t-1

Then the ARCH component of equation (2.12) is

(2.16)

which can be expanded as

[ a1,1 0] [Sl,1,t-1 Sl,2,t-1]' [ Zl,t-l _ 0

1 Z2,t-l - O

2]'

o a2,2 S2,1,t-1 S2,2,t-1

[

SIl t-1 SI 2 t-1] [aIl , , , , ,

S~l~-l S2~~-1 0 (2.17)

Expressing the model in vectorized form and omitting G ARC H and the

constant terms gives a better idea of the way in which asymmetry is captured .

h12t = , , ... + a11 a 2 2 (SIl t-1 (Zl t-1 - el) + S21 t-1 (Z2 t-1 - e 2 )) , , , " , , ,

(Sl,2,t-1 (Zl,t-1 - el) + S2,2,t-1 (Z2,t-1 - e 2 )) + ... (2.18)

h2,2,t = ... + a~,2 (Sl,2,t-1 (Zl,t-1 - el) + S2,2,t-1 (Z2,t-1 - e 2 )) + ...

Notice that even in the expression for the variances the standardized resid­

uals and elements of St-1 play a role. Their role is in fact that of reconstituting

the dependence structure originally in the Ct which the standardization proce­

dure removes. Under the alternative that asymmetry is present, the reintro­

duction of the dependence structure operates on the "symmetrized" residuals.

57

A plot of the response function of the Ct-l is shown in Figure 2.1 for values

of BI and B2 similar to those estimated and equal to 4, with parameter al,l

[ 1 0.5] and a2,2 equal to 0.12 and a covariance equal to . The asymmetric

0.5 1 response is apparent from the plot. In this particular illustration the zero point

for the conditional covariance innovation is C = [ 4.899 4.899]. Whenever

both innovations are lower than 4.899, covariance increases. In particular,

wh en both innovations are negative the increase in the covariance is larger

than when they are positive and of the same magnitude. This is the precise

meaning of covariance asymmetric response as used in this essay.

2.2.2 M-NGARCH Estimation.

Presented in this section are estimation and standard statistical testing results

of individual parameters, Wald tests for joint hypotheses, and LR tests for

model specification. More insight into the economic implications of asymmetry

and more specifie tests appear in the subsequent section.

AlI the models estimated belong to the multivariate G ARC H (1, 1) family.

Presented here is a subset of six different specifications. The dimensions of

interest for comparison across models taken into consideration are: the spec­

ification of the price of covariance risk 6, which is treated as a constant or

time-varying parameter using the US default spread and US term spread; the

specification of the asymmetry which is alternatively constrained to be zero,

or allowed to be constant and different for each asset, or allowed to be time­

varying and different for each asset using the same information set as the price

of risk. Table A summarizes the asset pricing models considered in this section.

The QML estimation was performed using the BFGS Quasi-Newton method

using an array of starting values. The Bollersiev and Wooldridge (1992) robust

standard errors were computed using numerical Hessian and scores.

58

Tables 2.5-2.10 in the end of chapter Appendix show the parameter esti­

mates for all the parameters excluding the constant: in parentheses are the

p-values from a robust t-test of significanee. The matrix of constants contains

several parameters that are not significantly estimated.9 Although care should

be used when analyzing results that depend on the covariance level, typically

the constant matrix estimates affect an models in a similar manner and should

not affect any of the results that depend on second moment dynamics.

Table A: IAPM models

Model Priee of Risk 6 Asymmetry ()

1 Constant 6 0

2 Constant 6 Constant ()

3 Constant 6 Time-varying ()t

4 Time-varying 6t 0

5 Time-varying 6t Constant ()

6 Time-varying 6t Time-varying ()t

G ARC H and ARCH parameters are highly significant in all models with

magnitudes comparable to previous studies.10

The models with constant 6' sare estimated as an interesting starting point

despite the widespread view, supported by the results in this essay, that the

priee of risk is more likely to change over time. In the first model, with constant

9DG finds that estimating the constant matrÏx or using the iterative procedure yield

covariance estimates that do not differ much in the dynamics, although the latter method

equalizes the sample covariance with the estimated unconditional covariance. The constant

matrix is usually of little interest as it contains no information with regard to the dynamics,

hence it is not shown.

lOIn interpreting the coefficients notice that returns are in daily percent while the instru-

ments are expressed in yearly percent.

59

priee of risk and no asymmetry, two out of four priees of risk are found to be

positive and significant at any conventionallevel: 81 , equal to 0.058 is the priee

of covariance with the global market, and 84 , equal to 0.013 is the priee of

GBP risk. Positiveness of the priee of market risk is one of the tests suggested

by Adler and Dumas (1983).11

The remaining two priees of exchange risk are found to be negative and

insignificant.

In model 2 the priee of risk 8 is kept constant and one asymmetry parameter

(Ji for each asset i is introdueed in the covarianee. AH estimated 8's are found

to be insignificant. The asymmetry parameters, (J3 for the UK and (J4 for the

US are found to be positive at 1.376 and 1.631 respectively, and significant.

Model 3 aHows for time variation in (J i,t. Interestingly, a time-varying asym­

metry restores the significanee for the estimates of 82 and 83 , The comparison

of the goodness of competing second moments specifications to explain condi­

tional first moments is formalized in the foHowing section where the conditional

mean encompassing tests are introdueed. One intuitive way to think about why

a different specification of the conditional covarianee may affect priee of risk

estimation is to consider that conditional second moments estimated under

different specification function in a fashion similar to regressors in a linear re­

gression framework. In the same way that "better" regressors pro duce "better"

estimates of regression coefficients, conditional second moments that have ex­

planatory power with respect to returns provide estimates of the priee of risk

that are significant.

Small p-values for the several coefficients of the instruments show strong

support for time-varying asymmetry for aH assets, exeept the German index.

AH the interest rates on the currency deposits have highly significant time­

Il In the robustness section the same parameter is found to be positive and significant at

10 per cent level at weekly frequency, and insignificant at monthly frequency.

60

varying asymmetry parameters, albeit one order of magnitude smaller for the

JPY and the GBP.

Model 4 features time-varying priees of market and currency risks 6c,t, with

c = 1...4, and no asymmetry; the significanee of the estimates of 6c,t conforms

to DG's (1998) results: the evidenee of variation over time of the priee of risk

is quite strong for all the currency risks and only slightly less for the world

market with a p-value of 0.061 for the USTP's coefficient.

Model 5 confirms the finding of the previous model 4 as far as the prices of

risk are coneerned. The constant O~s are estimated to be positive and highly

significant for all the assets.

Model6 features both time-varying priee of risk 6c,t and time-varying asym­

metry Oi,t. With regard to the priee of risk, the variation over time is proved ro­

bust to model specification for aU factors within the integrated markets frame­

work: aU 6c,t are again found to be significantly time-varying.

The market priee of risk 61 and 61,t require particular attention sinee, ac­

cording to the theoretical model, they should always be estimated positive

despite the fact that no restriction is imposed by the linear parameterization

used here. The absence of any restriction do es not affect the estimated sign of

market priee of risk estimates: 61 is always positive and significant in models 1

and 2 or negative and insignificant in model 3. The time-varying priee of mar­

ket risk 61,t is also implied positive by the parameters estimated significantly

at 10% level or less. Figure 2.8 plots 61,t for the model without asymmetry and

for the model with time-varying asymmetry. They are both positive for the

duration of the sam pIe period. However, the price of risk estimated without

aUowing for time-varying asymmetry is approximately double the magnitude

of the price of risk estimated aUowing for no asymmetry. The positiveness of

the priees of market risk estimates is reassuring as it is consistent with the

theoretical model and increases confidence in these asset pricing methodolo-

61

gles. A discussion of this results in relation with asymmetry is provided in the

following section.

Overall, asymmetry is found to be highly significant for all the assets. When

such a high number of parameters is estimated, however, tests for individu al

parameters may not prove to be completely reliable. For this reason, further

testing was performed using a Wald test for the joint significanee of parameters

of the instruments. To test the null hypothesis that either Oc,t for c = 1...4,

or (Ji,t are constant, it is possible to jointly consider all the parameters in the

respective linear combinat ion of instruments, excluding the constant term. If

the null is true, all of these parameters should be zero. The formaI testing of

this hypothesis can be performed through a Wald test. The results of these

tests which are presented in the table 2.12 show overwhelming support for both

Oc,t and ei,t time-varying for any model discussed in this essay.

The likelihood ratio test for nested models presented in Table 2.11 shows

a pair-wise comparison of comparable models. The tests consistently reject

models without asymmetry when compared either with models with constant

(Ji, or with models with time-varying (Ji,t. The LR tests also reject constant

asymmetry with respect to time-varying asymmetry specifications.

Perhaps not surprisingly, when nested specifications allowing the priee of

risk to be either constant or time-varying are compared, the tests are incon­

clusive. This could be due to the intrinsic difficulty in estimating the me ans

of asset returns or to the lack of power of the test.

In substance, these results show a strong rejection of the null hypothesis

according to which asymmetry is not present or that, if present, it is constant.

62

2.3 Encompassing Test for the Risk Premium

The central question of interest is whether the presence of asymmetry in the

conditional second moments does affect the estimates of the conditional first

moments. If the asymmetry of the type studied in this essay was in fact only

a statistical property of the international assets returns with no influence on

the risk premia estimates, its interest would be rather limited. However, if

the estimates of the risk premia in the mean equation differ significantly from

one another under competing specifications that allows for asymmetry - either

constant or time-varying - and the model that allows for asymmetry fits the

data better according to sorne measure, then using a model that does not allow

for asymmetry would yield misguided estimates of the total risk premium and

of its components.

As it is made clear in a more formaI way in the following section, asymmetry

does substantially affect the estimation of the risk premia.

Wooldridge (1990) proposes a general class of tests designed to detect con­

ditional mean misspecifications based on the encompassing principle. The

Conditional Mean Encompassing (CME) test exploits the correlation between

the residual under the null and the gradient of the alternative. Roughly speak­

ing, the encompassing principle states that a model that is weIl specified under

the null should be able to explain the characteristics of a rival model, i.e. it

should encompass it.

The CME test do es not appear prominently in the asset pricing literature.

Asset pricing models of the type used in this essay fall short of prescribing how

the conditional covariance should be parameterized. In implementing these

models, the choice of the one particular covariance model is often based on

considerations that leave aside the theoretical foundation of the model itself.

Yet, the choice of one covariance model may substantially affect the risk

63

premia estimates precisely because of the difficulty which estimating the condi­

tional mean of returns entails. In other word, different covariance models may

weIl yield estimates that look similar on the surface. However, the important

question is: will the risk premia of asset pricing models that use competing co­

variance as covariance risk estimates look similar? In substance, the fact that

different covariance specifications may yield risk premia that substantially di­

verge from one another raises the question as to which model the analyst should

select.

Clearly, while neither t-tests of individual parameter significance nor Wald

or LR tests can answer this question, the CME test cano

Of interest to practitioners is the monetary quantification of the impor­

tance of time-varying asymmetry for portfolio management. In particular, it

is plausible that for practical reasons one may be interested in comparing the

pricing error, either absolute or in percentage terms, between a model that al­

lows for time-varying asymmetry and a model that does not. In such instances,

the loss function for the parameter estimation should be designed accordingly.

Then, using for example Nonlinear Least Squares (NLS), it would be possible

to estimate the parameters and compare the absolute or relative pricing errors

or perform the CME tests.

An illustration of the test and how it is used in this essay is presented in

the technical Appendix at the end of the chapter.

2.3.1 CME test results

The results of the most important CME tests are shown in table 2.13. Empha­

sis is on the models that allow for time-varying price of risk. In this section,

an alternative model is considered better than a model weIl specified under the

null if, according to the CME test, a departure from the null in direction of

the alternative is detected. On the contrary, an alternative model is not better

64

than the model under the null if there is no information left in the model errors

that the alternative model is capable of explaining.

The CME test between the benchmark model with time-varying 6c,t but no

asymmetry under the null and the model with time-varying 6c,t and constant

()i do es not reject the null with a p-value of 0.447. This result shows that the

alternative model does not fit the conditional mean any better than the model

under the null.

This result indicates that adding a constant asymmetry parameter to the

covariance specification when the prices of risk are time-varying does not pro­

duce better estimates of the risk premium. This finding is interesting in light

of the increasing attention paid to asymmetry.

When the benchmark model is tested against the model with time-varying

6c,t and time-varying ()i,t, remarkably the joint CME test for all the components

of the conditional mean strongly rejects the model un der the null with a p­

value of 0.034. This result indicates that, for asset pricing in this sample, it is

important to allow for time-varying asymmetry. The components of the risk

premium related to 63 ,t (the JPY deposit) 64 ,t are rejected with p-values of

0.036 and 0.055. The market price of risk 61,t and 62,t (the DEM deposit) are

not rejected.

If the model with time-varying !Sc,t and constant ()i is considered under the

null against the alternative model specified with 6c,t and time-varying ()i,t, the

rejection is even st ronger with a p-value of 0.002. The individual significance of

the factors is generally higher compared to that of the previous test. This result

confirms that including a constant asymmetry () is not sufficient to improve the

estimates of the conditional mean, and hence to estimate better risk premia.

Interestingly, the CME test of the model with constant 6i under the null

against the model with constant !Sc and constant ()i rejects the null with a p­

value of 0.024. It follows that if one decides not to use instruments (such as the

65

USDP or the USTP), it would still be better to include constant asymmetry

parameters in the covariance specification in or der to get better estimates of

the risk premium.

Similarly rejected is the same model when compared to the one with time­

varying (Ji,t under the alternative with a p-value of 0.014. The comparison

between the models with (Ji and the model with (Ji,t is inconclusive having a

p-value of 0.329.12

Diagnostic tests on the standardized residuals of aIl models were performed.

Not surprisingly, normality tests show that for aIl the models the normality

assumption is not appropriate. The Bera-J arque test rejects the null hypothesis

of normality for aIl residuals and for aIl models. This gives support to the use

of robust standard errors for inference.

The difference between models is small when testing for skewness and kur­

tosis separately. In general the asymptotic test statistics of residuals' kurtosis

under the null of normality are 1 or 2 orders of magnitude larger than those

for the similar tests for skewness. 13

The conclusion from this battery of tests is that for the purpose of estimat­

ing risk premia, when the price of risk is time-varying, asymmetry should also

be time-varying. A constant asymmetry does not provide any improvement.

Indeed, when the price of risk is allowed to be time-varying, the condition al

mean estimated by a model that allows for time-varying asymmetry is more

consistent with the data. However, if the price of risk is modeled as a con­

stant, then better estimates of the risk premia can still be obtained including

a constant asymmetry parameter in the covariance specification.

These are important findings from an economic perspective, as the estima-

12The last two tests are not reported in the table to save space.

13 A somewhat related approaeh to the one presented here would be to use a model that

directly priees skewness. There is no clear mapping between these two methodologies, but

their aecuracy for asset pricing could be studied using a CME test.

66

tion of risk premia is the fundamental objective of any asset pricing model.

2.3.2 Risk Premium and Asymmetry

The findings in this essay corroborate the previous results with regard to the

time-varying nature of the price of risk. In addition, they extend and link the

factors driving the price of risk variability to the variability of the covariance

risk. It is remarkable that the same set of instrumental variables is capable

of explaining the variability of the price of risk and one particular aspect of

covariance risk, i.e. covariance asymmetry. If one believes that these instru­

ments are suit able indicators for the changing business climate and that the

family of IAPM models considered here are a reasonable approximation for

the way assets are priced in developed international markets, then the results

are consistent with the idea that the same economic factors affect both the

investor's willingness to bear risk, by mean of the price, and the amount of

risk itself, through the conditional covariance.

Table 2.10 shows that all the significant coefficients for both the prices

of risk and the asymmetry parameters et have a positive sign. This is an

indication that the factors that drive the price of risk affect the magnitude

and direction of the asymmetric response in a similar way. Notice also that

for et the default premium factor is significantly positive for all but German

and the US indexes, for which is insignificant. One economic interpretation of

this finding, which is also appealing to intuition, is that when the economic

condition is bad not only do investors require a higher price for bearing risk,

but they also react more asymmetrically to news.

Notice also that the term premium, which is often used as a proxy for time

varying risk aversion, is significantly positive for 5 out of the 8 assets. This

evidence suggests that the variation over time of investors' aggregated risk

aversion affects both the price of risk, and the asymmetric response in second

67

moments in the same direction. An increase in the slope of the yield curve

results in a higher price of risk and more asymmetric response in covariance.

Total and factors risk premia for comparable models are shown in the plot

of the annualized risk premia for the multivariate models. These plots are

drawn by setting all the insignificant coefficients of the instrumental variables

to zero and taking only that part of the linear combination having significant

parameters for each asset. This is done to eliminate the undue influence of

insignificant components of the price of risk estimates which obfuscate the

effect of significant factors by exaggerating the contribution of insignificant

ones.

Interesting insights emerge from the plot of the total risk premium. With

sorne exceptions, allowing for time-varying asymmetry generally yields lower

estimates of the total risk premium for equities and for the world market

index and higher estimates for the foreign currency deposits. One exception

is the second half of the sample for the Japanese Yen denominated deposit.

The plot implies that an investor who takes into account the time variation

of asymmetry from early 1997 to 2001 would content herself with a much

lower risk premium compared to an investor who gives no consideration to

asymmetry or who considers it constant.

In figures 2.4-2.6 sorne considerable differences among models emerge from

the inspection of the foreign currency risk premia especially in the first half of

the sample. The estimates of the components of the risk premia due to the

DEM risk are sizably higher both for German and U.K. indices and deposits.

For Japan and the World Market, the differences are not as considerable, while

for the US index they are negligible.

The evolution over time of these risk premia seems to be related to the

various stages of the introduction of the Euro currency. The peaks around

1992 and 1993 coincide with the European Monetary System crises. This

68

finding is interesting, as it indicates that investors who took into account time­

varying asymmetry would have required higher compensation to bear DEM

risk during those times of turmoil. The uncertainty started to dissipate in

January 1994 with the start of the second stage of EMU. In December 1995 the

European Council agreed to name the European currency unit to be introduced

at the start of Stage Three, the "euro", and confirmed that Stage Three of

EMU would start on 1 January 1999. The successful introduction of the Euro

currency coincides with an excursion in the negative territ ory for the DEM

risk premium.

Figure 2.5 shows that there are several similarities between the risk premia

determined by the GBP risk and those determined by the DEM in the first half

of the sample. One example of how they differ is that for the former the premia

for investing in DEM deposit and GBP deposit remain positive and steadier

throughout the second half of the sample. This joint evidence suggests that

events affecting the Euro area did not affect in the same way the risk premia

determined the GBP risk in the second half of the 90's.

Figure 2.6 shows that for the risk premia resulting from the JPY risk, taking

into account the time-varying asymmetry yields estimates of the risk premia

that are generally lower than those that were obtained without any consider­

ation given to time-varying asymmetry. For both, the Japanese equities and

the Yen deposits, from 1997 to 2000 the divergence is striking. Models with no

asymmetry or constant asymmetry show peaks of risk premia above 40% per

year. However, once time-varying asymmetry is taken into consideration the

contribution to the total premium of the risk premia required to invest either

in Japanese equities or in Japanese deposits shrinks to a much smaller level.

It seems that the changing risk aversion captured by the instruments plays an

important role in determining the compensation required by the investors. In

other words, the empirical link between price of risk and the asymmetric re-

69

sponse in the covariance risk indicates that sorne fundamental factor captured

by the instruments has an effect on both, the price of risk and the amount of

it as expressed by the covariances shown in the pricing equation.

This conjecture seems to apply to all the stock indices, as it can be inferred

by the plots of conditional covariances for the time-varying asymmetric models

as well as the symmetric ones. 14 While the estimated conditional covariances

are hardly distinguishable between models, the corresponding market prices

of risk are rather different. It seems that small differences in the estimated

conditional covariances lead to sizable differences in the estimated risk premia,

which can be due to the fact that it is objectively difficult to estimate expected

returns.

Figure 2.7 plots the parameter that is the main focus of interest in this

essay: the time-varying asymmetry. This plot is drawn by setting all the

insignificant coefficients of the instrumental variables to zero and taking only

that part of the linear combinat ion with significant parameters for each asset.

The pi ct ures show a wide variation in the range of the varying asymmetry.

However, there is a general pattern which is common to all assets,15 namelyan

increase from the beginning of the sample period until a decline around 1999.

This in turn is followed by increase and a j ump16 around the end of 2001 for

Japan, UK, DEM deposit, JPY deposit and the World market, that is for all

assets for which varying asymmetry is significantly explained by the default

premmm.

It is worth noting that the pattern of the asymmetry parameter for Ger­

many, the U.S. and GBP is similar to the world market price of risk. This

implies that, ceteris pari bus, when investors require higher risk premia for in­

vesting in the world market the asymmetric response in the covariance of these

14Plots of the conditional covariances are not shown to conserve space.

15This is not surprising as these are linear combinations of only two factors.

16The V.S. default premium jumps from .87 to 1.28 on December 7, 2001.

70

assets tends to be more pronounced. Historically, the peaks occur around 1991,

coinciding with the first Gulf War, and late 1998, which coincided with the

LTCM crisis. The asymmetry parameter sharply increased after the year 2000

to the end of the sample, which saw a period of great geopolitical uncertainty

and unfavorable business conditions. The world market asymmetry parameter

is driven not only by the term premium, but also by the default premium, and

hence it appears to be slightly different.

In order to conserve space, the analogous set of plots for those models

having a constant price of risk is not included. It is, however, the case that the

risk premium estimated by the model that allows for time-varying asymmetry

is much flatter and smaller than that estimated by the symmetric model.

2.3.3 Robustness at the Weekly and Monthly Frequencies

The daily analysis presented so far shows crisp results in terms of the overall

importance of time varying asymmetry along with the significance of individ­

ual parameters. For asset pricing and portfolio management however lower

frequency are often of interest. Robustness check were performed using weekly

and monthly data over a longer period. 17

The weekly analysis corroborates the results obtained at the daily level.

Although the level of significance of individual parameters decreases, the evi­

dence regarding the time variability of asymmetry and its importance for asset

pricing is unchanged. The statistical evidence provided by the Wald, LR and

CME tests remain as strong as at the daily frequency. This is remarkable as

the number of weekly observations is substantially smaller than that of daily.

The monthly results, despite the much smaller number of observations, still

show strong statistical evidence of time variation of both the price of risk and

the asymmetry parameter. The Wald test that an the time-varying coefficients

17The results are not reported in detail to conserve space.

71

of the asymmetry parameters shows a p-value of 0 up to the third decimal.

Similarly, the LR tests reproduce a strong rejection pattern similar to that

obtained at daily and weekly frequencies. Models featuring no asymmetry are

rejected by models with constant asymmetry, which in turn are rejected by

models with time-varying asymmetry. At monthly level however, no power is

left to compare models from an asset pricing perspective using the CME test.

This is possibly due to the small sample size.

2.4 Conclusion

This essay provides three main contributions. Firstly, allowing for time-varying

asymmetry is important for the estimation of the market risk premium and

foreign exchange risk premia. The conditional mean tests show that when

the price of risk is time varying, models that do not allow for time-varying

asymmetry yield conditional mean estimates, i.e. risk premia that are mis­

specified with respect to those provided by models that allow for time-varying

asymmetry. This finding has potentially important consequences for portfolio

management and asset allocation. However, if the price of risk is modeled as

a constant, it is of benefit to include a constant asymmetry parameter in the

covariance specification as it yields better estimates of the risk premia.

As an intermediate step towards reaching the first conclusion, this essay

also shows that, the returns from developed countries display time-varying

asymmetry which is significantly driven by the same economic factors that

drive the price of risk. For integrated markets, when according to the business

climate investors require a higher price for bearing risk, they also react more

asymmetrically to news.

Thirdly, this essay also proposes and implements an econometric empirical

model that takes into account second moment asymmetry in a multivariate

72

GAReR context. The technique is general and can be easily extended to take

into account more articulated forms of asymmetry as described in Rentschel

(1995).

73

2.A Tables and Figures

Germany Japan UK US DEM dep JPY dep GBP dep World Market Mean

Std Skewness Kurtosis JBTest p-val LB(21) p-val

-0.002 -0.052 0.001 0.014 0.000 0.001 0.007 0.003 1.460 1.692 1.090 1.034 0.677 0.732 0.594 0.800 -0.271 0.332 -0.092 -0.115 0.069 0.918 -0.127 -0.163 7.662 6.468 5.368 7.043 4.892 11.600 6.703 6.476 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.048 0.011 0.000 0.013 0.510 0.058 0.118 0.000

TABLE 2.1. Descriptive Statistics.

Germany Japan UK US DEM dep JPY dep GBP dep World Market Germany 1 0.225 0.6080.375 0.232 0.094 0.152 0.647

Japan 1 0.2220.093 0.140 0.379 0.114 0.561 UK 1 0.348 0.124 0.051 0.264 0.624 US 1 -0.149 -0.048 -0.095 0.748 DEM dep 1 0.415 0.682 0.073 JPY dep 1 0.324 0.176 GBP dep 1 0.091

TABLE 2.2. Sample Correlation.

74

Lag Cermany Japan UK US DEM dep JPY dep CBP dep World Market 1 0.001 -0.008 0.045 0.008 -0.019 -0.010 -0.003 0.213 2 -0.035 -0.041 -0.056 -0.020 0.003 0.006 0.017 -0.025 3 -0.023 -0.007 -0.063 -0.046 -0.022 -0.017 -0.010 -0.017 4 0.019 -0.002 -0.011 -0.013 0.020 -0.011 0.029 0.014 5 -0.021 -0.012 -0.033 -0.012 0.001 0.019 0.030 -0.032 6 -0.051 -0.030 -0.035 -0.032 -0.038 -0.022 -0.056 -0.043 7 0.015 -0.013 -0.010 -0.030 0.027 0.016 0.011 -0.021 8 0.010 0.043 0.028 -0.003 -0.016 0.012 0.003 0.015 9 -0.005 0.036 0.020 0.012 0.006 -0.005 0.016 0.035 10 -0.019 0.021 -0.008 0.028 0.024 0.057 0.028 0.021

TABLE 2.3. Autocorrelation function for Ti t LCL -0.0343 VCL 0.0343. ,

Lag Cermany Japan UK US DEM dep JPY dep CBP dep World Market 1 0.138 0.090 0.187 0.204 0.081 0.109 0.081 0.154 2 0.192 0.113 0.246 0.189 0.078 0.065 0.160 0.216 3 0.194 0.125 0.200 0.181 0.063 0.029 0.108 0.142 4 0.159 0.133 0.168 0.121 0.047 0.039 0.102 0.131 5 0.132 0.098 0.185 0.146 0.034 0.024 0.087 0.134 6 0.123 0.085 0.171 0.163 0.082 0.064 0.145 0.100 7 0.128 0.109 0.140 0.139 0.036 0.015 0.065 0.118 8 0.145 0.071 0.197 0.138 0.047 0.014 0.036 0.133 9 0.138 0.063 0.127 0.136 0.051 0.033 0.153 0.112 10 0.100 0.064 0.165 0.141 0.088 0.014 0.101 0.106

TABLE 2.4. Autocorrelation function for Ti,t LCL -0.0343 VeL 0.0343.

75

The empirical model 1 is

4

ri,t = 61 COVt-l (ri,t, r m,t) + L 6cCOVt-l (ri,t, r4+c,t) + Ei,t i = 1, ... 8 c=2

EtHt-1/2 - Zt VOl N(O, 1)

61, is the constant world price of market risk, 62, 63, and 64 are respectively the world price of foreign exchange risk of the DEM, of the JPY and of the GBP. AH returns are measured in USD, there are three sources of foreign exchange risk: DEM, JPY and GBP. The conditional covariance is specified as

Ht = CC' + AHtl~i'(Zt-d(Zt-l)H;~iA + BHt- 1B'

where A and B are diagonal matrix with typical element ai and f3i and C is lower triangular.

01 02 03 04 Const 0.058 -0.014 -0.016 0.013

( 0.000) ( 0.939) ( 0.991) ( 0.000)

ARCH 01 02 03 04 05 06 07 Os

0.138 0.161 0.130 0.162 0.114 0.115 0.138 0.149 ( 0.000) ( 0.000) ( 0.000) ( 0.000) ( 0.000) ( 0.000) ( 0.000) ( 0.000)

GARCH /31 /32 /33 /34 /35 /36 /37 /3s

0.991 0.988 0.992 0.988 0.994 0.994 0.991 0.990 ( 0.000) ( 0.000) ( 0.000) ( 0.000) ( 0.000) ( 0.000) ( 0.000) ( 0.000)

TABLE 2.5. Multivariate Model 1: Constant priee of risk 6. P-value in brack-ets.

76

The empirical model 2 is

4

ri,t = (hCOVt-l(ri,t,rm ,t) + L OcCOVt-l(ri,t,r4+C,t) + éi,t i = 1, ... 8 c=2

étHt-1/2 _ Zt V"\ N(O, 1)

01 is the constant world price of market risk, 02, 03, and 04 are respectively the world price of foreign exchange risk of the DEM, of the JPY and of the GBP. AlI returns are measured in USD, there are three sources of foreign exchange risk: DEM, JPY and GBP. The conditional covariance is specified as

Ht = CC' + AHt~i'(Zt-l - e)'(Zt-l - e)HJ~iA + BHt- 1B'

where A and B are diagonal matrices with typical element ai and (3i and C is lower triangular. e is an 8 x 1 vector containing the asymmetry parameters.

Multivariate Model 2: Constant price of risk 0 and constant asymmetry e. P-value in brackets

01 02 153 154 Const 0.ül8 -0.019 0.011 0.028

( 0.120) ( 0.772) ( 0.452) ( 0.418)

81 82 83 84 85 86 87 8s Const -0.180 3.480 1.376 1.631 0.570 -1.607 0.082 -0.212

( 0.649) ( 0.442) ( 0.036) ( 0.003) ( 0.420) ( 0.598) ( 0.387) ( 0.873)

ARCH al a2 a3 a4 a5 a6 a7 as 0.140 0.162 0.126 0.163 0.113 0.120 0.137 0.149

( 0.000) ( 0.000) ( 0.000) ( 0.000) ( 0.000) ( 0.000) ( 0.000) ( 0.000)

GARCH /31 /32 /33 /34 /35 /36 /37 /3s 0.991 0.988 0.993 0.987 0.994 0.993 0.991 0.989

( 0.000) ( 0.000) ( 0.000) ( 0.000) ( 0.000) ( 0.000) ( 0.000) ( 0.000)

TABLE 2.6. Multivariate Model 2: Constant price of risk o and constant asymmetry e. P-value in brackets.

77

The empirical model 3 is

4

ri,t = 51 COVt-1 (ri,t, r m,t) + L 5cCOVt-1 (ri,t, r4+c,t) + Ci,t i = 1, ... 8 c=2

Ct Ht-1/

2 Zt V"I N(O, I)

51 is the constant world price of market risk, 52, 53, and 54 are respectively the world price of foreign exchange risk of the DEM, of the JPY and of the GBP. An returns are measured in USD, there are three sources of foreign exchange risk: DEM, JPY and GBP. The conditional covariance is specified as

Ht = CC' + AHt1~i'(Zt_1 - et-d(Zt-1 - et-1)H:~iA + BHt_1B'

where A and B are diagonal matrices with typical element ai and f3i and C is lower triangular. The specification for the time-varying asymmetry is

where et is an 8 x 1 vector containing the time-varying asymmetry parameters.

&1 &2 &3 &4

Const -0.001 -0.005 0.013 0.012 ( 0.529) ( 0.909) ( 0.060) ( 0.028)

81 ,t 82,t 83,t 84 ,t 85 ,t 86,t 87 ,t Const 0.522 0.767 -1.817 0.708 -0.860 -0.114 -0.765

( 0.044) ( 0.017) ( 0.982) ( 0.087) ( 0.920) ( 0.955) ( 0.876) USDP -1.375 1.409 2.964 0.436 0.658 0.051 0.064

( 0.971) ( 0.035) ( 0.009) ( 0.053) ( 0.011) ( 0.000) ( 0.052) USTP -1.127 0.352 0.494 0.228 0.176 0.033 -0.261

( 0.997) ( 0.007) ( 0.017) ( 0.082) ( 0.026) ( 0.118) ( 0.908)

ARCH G1 G2 G3 G4 G5 G6 G7

0.139 0.162 0.123 0.163 0.111 0.124 0.134 ( 0.000) ( 0.000) ( 0.000) ( 0.000) ( 0.000) ( 0.000) ( 0.000)

GARCH (31 (32 (33 (34 (35 (36 (37 0.991 0.988 0.993 0.987 0.994 0.992 0.992

( 0.000) ( 0.000) ( 0.000) ( 0.000) ( 0.000) ( 0.000) ( 0.000)

8s,t

0.046 ( 0.070)

0.117 ( 0.018) -0.062

( 0.944)

Gs

0.149 ( 0.000)

(3s

0.989 ( 0.000)

TABLE 2.7. Multivariate Model 3: Constant price of risk 5 and time varying asymmetry et. P-value in brackets.

78

The empirical model 4 is

4

ri,t = 61,t-l COVt-l (ri,t, r m,t) + L 6c,t-l COVt-l (ri,t, r4+c,t) + Ci,t i = 1, ... 8 c=2

ctHt-1/2 = Zt '-" N(O, 1)

61,t-l is the world priee of market risk, 62,t, 63,t, and 64,t are respectively the world priee of foreign exchange risk of the DEM, of the JPY and of the GBP. AH retums are measured in USD, there are three sources of foreign exchange risk: DEM, JPY and GBP. The potentially time-varying risk premium is defined as

The conditional covariance is specified as

Ht = CC' + AHtl~î'(Zt_l)'(Zt_l)Htl~îA + BHt-1B'

where A and B are diagonal matriees with typical element ai and f3i and C is lower triangular.

51 ,t 52 ,t 53,t 54,t

Const 0.152 -0.068 -0.097 0.019 ( 0.000) ( 0.975) ( 1.000) ( 0.093)

USDP -0.111 0.053 0.088 0.033 ( 1.000) ( 0.001) ( 0.000) ( 0.024)

USTP 0.009 0.010 omo -0.026 ( 0.061) ( 0.146) ( 0.000) ( 1.000)

ARCH al a2 a3 a4 a5 a6 a7 as

0.138 0.161 0.130 0.162 0.114 0.115 0.138 0.149 ( 0.000) ( 0.000) ( 0.000) ( 0.000) ( 0.000) ( 0.000) ( 0.000) ( 0.000)

GARCH (31 (32 (33 (34 (35 (36 (37 (3s 0.991 0.988 0.992 0.988 0.994 0.994 0.991 0.990

( 0.000) ( 0.000) ( 0.000) ( 0.000) ( 0.000) ( 0.000) ( 0.000) ( 0.000)

TABLE 2.8. Multivariate Model 4: Time-varying priee of risk 6t . P-value in brackets.

79

The empirical model 5 is

4

ri,t = 61,t-lCOVt-l(ri,t, rm,t) + L 6c,t-lCOVt-l(ri,t, r4+c,t) + éi,t i = 1, ... 8 c=2

étHt-1/2 = Zt "'" N(O, 1)

61,t-l is the world price of market risk, 62,t, 63,t, and 64,t are respectively the world priee of foreign exchange risk of the DEM, of the JPY and of the GBP. AlI returns are measured in USD, there are three sources of foreign exchange risk: DEM, JPY and GBP. The potentially time-varying risk premium is defined as

The conditional covariance is specified as

Ht = CC' + AHtl~î'(Zt_l - B)'(Zt-l - B)Hi~îA + BHt-1B'

where A and B are diagonal matrices with typical element ai and f3i and C is lower triangular. B is an 8 x 1 vector containing the asymmetry parameters.

81 ,t 82 ,t 83 ,t 84,t

Const 0.073 -0.083 -0.103 0.081 ( 0.000) ( 0.996) ( 0.999) ( 0.003)

USDP -0.059 0.060 0.127 -0.027 ( 1.000) ( 0.000) ( 0.000) ( 1.000)

USTP -0.002 0.015 0.014 -0.028 ( 0.774) ( 0.000) ( 0.000) ( 1.000)

81 82 83 84 85 86 87 8s Const 5.293 4.364 2.995 9.840 9.731 5.007 0.771 5.850

( 0.000) ( 0.000) ( 0.000) ( 0.000) ( 0.000) ( 0.000) ( 0.001) ( 0.000)

ARCH al a2 a3 a4 a5 a6 a7 as

0.139 0.162 0.125 0.163 0.113 0.122 0.137 0.149 ( 0.000) ( 0.000) ( 0.000) ( 0.000) ( 0.000) ( 0.000) ( 0.000) ( 0.000)

GARCH (31 (32 (33 (34 (35 (36 (37 (3s 0.991 0.988 0.993 0.987 0.994 0.993 0.991 0.989

( 0.000) ( 0.000) ( 0.000) ( 0.000) ( 0.000) ( 0.000) ( 0.000) ( 0.000)

TABLE 2.9. Multivariate Model 5: Time-varying price of risk 6t and constant asymmetry B. P-value in brackets.

80

The empirical model 6 is

4

ri,t = (h,t-1 COVt-1(Ti,t, rm,t) + L Oc,t-1 COVt-1(Ti,t, T4+c,t) + éi,t i = 1, ... 8 c=2

étHt-1/2 _ Zt V"\ N(O, I)

01,t-1 is the world priee of market risk, 02,t, 03,t, and 04,t are respectively the world priee of foreign exchange risk of the DEM, of the JPY and of the GBP. The time-varying risk premium and covariance are respectively

Ot = 01 + 02USDPt + 03USTPt

Ht = CC' + AHt1~i'(Zt_1 - et-d(Zt-1 - et-1)H:~iA + BHt- 1B'

where A and B are diagonal. The time-varying asymmetry is

where et is an 8 x 1 vector of the time varying asymmetry parameters.

Ol,t 02,t 03,t 04,t Const 0.080 -0.062 -0.018 -0.040

( 0.056) ( 1.000) ( 0.997) ( 0.986) USDP -0.109 0.000 0.006 0.150

( 0.992) ( 0.472) ( 0.050) ( 0.001) USTP 0.017 0.043 0.023 -0.059

( 0.000) ( 0.003) ( 0.026) ( 0.961)

Bl,t B2,t B3,t B4,t B5,t B6,t B7,t Bs,t Const -0.644 2.268 -8.969 4.191 -1.325 -4.454 -4.349 -4.977

( 0.860) ( 0.000) ( 0.999) ( 0.000) ( 0.999) ( 1.000) ( 1.000) ( 1.000) USDP -0.643 4.188 12.871 -6.226 4.998 6.279 0.759 6.862

( 0.934) ( 0.001) ( 0.000) ( 0.996) ( 0.037) ( 0.000) ( 0.008) ( 0.000) USTP 0.375 -3.011 0.755 0.104 -1.207 -0.076 2.274 0.328

( 0.000) ( 1.000) ( 0.004) ( 0.009) ( 0.993) ( 1.000) ( 0.007) ( 0.002)

ARCH al a2 a3 a4 a5 a6 a7 a8 0.140 0.162 0.121 0.163 0.111 0.125 0.132 0.149

( 0.000) ( 0.000) ( 0.000) ( 0.000) ( 0.000) ( 0.001) ( 0.000) ( 0.000)

GARCH /31 /32 /33 /34 /35 /36 /37 /38 0.991 0.988 0.993 0.987 0.994 0.992 0.992 0.989

( 0.000) ( 0.000) ( 0.000) ( 0.000) ( 0.000) ( 0.000) ( 0.000) ( 0.000)

TABLE 2.10. Multivariate Model 6: Time-varying price of risk Ot and time-varying asymmetry et. P-value in brackets.

81

Model1 Model2 Model3 Model4 Model5 Model2 70.753

( 0.000)

Model3 128.643 57.891 ( 0.000) ( 0.000)

Model4 4.940 ( 0.764)

Model5 76.021 5.269 71.081 ( 0.000) ( 0.728) ( 0.000)

Model6 137.257 8.614 132.317 61.236 ( 0.000) ( 0.376) ( 0.000) ( 0.000)

TABLE 2.11. Likelihood ratio test for nested models. The table shows a pair-wise comparison of nested models. The model taken as restricted bench­mark is indicated in the column heading. The rows indicate the unrestricted model to with respect to which the test was performed. P-val's in brackets. The tests consistently reject models without asymmetry when compared either with models with constant (Ji, or with models with time-varying (Ji,t. The LR tests also reject constant asymmetry with respect to time-varying asymmetry specifications. However, when nested specifications allowing the priee of risk to be either constant or time-varying are compared, the tests are inconclusive. These results show a strong rejection of the null hypothesis according to which asymmetry is not present or that, if present, it is constant.

82

Model3 Wald Statistics p-val Wald Test for an the IVs coefficients in e Ho: an the IVs coeffficients in e = 0

930.460 ( 0.000)

Model4 Wald Test for an the IVs coefficients in 5 Ho: an the IVs coeffficients in 5 = 0

406.851 ( 0.000)

Model5 Wald Test for an the IVs coefficients in 5 Ho: an the IVs coeffficients in 5 = 0

7070.337 ( 0.000)

Model6 Wald Test for an the IV s coefficients in 5 Ho: an the IVs coeffficients in 5 = 0

246.945 ( 0.000) Wald Test for an the lVs coefficients in e Ho: an the lVs coeffficients in e = 0

5496.153 ( 0.000)

TABLE 2.12. Wald test of significance for the joint nun hypothesys that an the coefficient of the time varying priee of risk 5 and or the asymmetry parameter e are equal to zero.

Ho: The model with time-varying (j is well specified Ha: The model with time-varying (j and constant e fits the conditional mean better

CME Statistic p-val

28.149 ( 0.254)

24.459 ( 0.436)

35.427 ( 0.062)

29.362 ( 0.207)

Ho: The model with time-varying (j is well specified

97.200 ( 0.447)

83

Ha: The model with time-varying (j and time-varying e fits the conditional mean better

CME Statistic p-val

30.749 ( 0.161)

31.910 ( 0.129)

37.864 ( 0.036)

35.982 ( 0.055)

Ho: The model with time-varying (j and constant e is well specified

122.842 ( 0.034)

Ha: The model with time-varying (j and time-varying e fits the conditional mean better

CME Statistic 45.767 p-val ( 0.005)

35.570 ( 0.060)

31. 722 ( 0.134)

Ho: The model with constant (j is well specified

34.221 ( 0.081)

139.741 ( 0.002)

Ha: The model with constant (j and constant e fits the conditional mean better

CME Statistic p-val

17.366 ( 0.027)

6.560 ( 0.585)

18.802 ( 0.016)

8.955 ( 0.346)

All (js 49.668 ( 0.024)

TABLE 2.13. Conditional Mean Encompassing test. P-val in brackets. This battery of tests shows that for the purpose of estimating risk premia, when the price of risk is time-varying, asymmetry should also be time-varying. A constant asymmetry do es not provide any improvement. In fact, when the price of risk is allowed to be time-varying, the risk premium estimated by a model that allows for time-varying asymmetry is more consistent with the data. However, if the price of risk is modeled as a constant, then better estimates of the risk premia can still be obtained including a constant asymmetry parameter in the covariance specification.

84

4 .... ...........

3

............ 2

FIGURE 2.1. Response function of the innovation vector Et_Ifor values of BI and B2 similar to those estimated and equal to 4, with parameter al,l and a2,2

equal to 0.12 and the variance equal to 1 and covariance equal to .5. The asymmetric response is apparent from the plot. The zero point for the con­ditional covariance innovation is E = [4.899 4.899] Whenever both innovations are lower than 4.899 covariance increases. In particular, when both innova­tions are negative, the increase in the covariance is larger than when they are positive and of the same magnitude.

85

Japan

1990 1992 1994 1996 1998 2000 2002 1990 1992 1994 1996 1998 2000 2002

United Kingdom United States

1990 1992 1994 1996 1998 2000 2002 1990 1992 1994 1996 1998 2000 2002

DEM deposit JPY deposit

100

1990 1992 1994 1996 1998 2000 2002 1990 1992 1994 1996 1998 2000 2002

GBP deposit World Market

FIGURE 2.2. Annualized total risk premia of models allowing for time varying price of risk. Dashed line: No asymmetry; Dotted line: constant asymmetry, Continous line: Time-varying asymmetry. With sorne exceptions, allowing for time-varying asymmetry generally yields lower estimates of the total risk premium for equities and for the world market index and higher estimates for the foreign currency deposits. One exception is the second half of the sample for the Japanese Yen denominated deposit. The plot implies that an investor who takes into account the time variation of asymmetry from early 1997 to 2001 would content herself with a much lower risk premium compared to an investor who gives no consideration to asymmetry or who considers it constant.

86

Germany Japan

Of---------------------j

1990 1992 1994 1996 1998 2000 2002 1990 1992 1994 1996 1998 2000 2002

United Kingdom United States

60 1 :\. ;111

1 1\ ~f 1 1 Il 1 1 \'~i

If l 'l, 1. Il, '\~d~,I~\ li ~ "i'i t .. ), , ' 'It \ \ ~ÛiI~ :I~ , ,'. ,:.J \ I~ï! . '.1 ÎIt~,~, ~, VIi

. t,/I"~'

40

20

o

1990 1992 1994 1996 1998 2000 2002 1990 1992 1994 1996 1998 2000 2002

DEM deposit JPY deposit

60 60

40 40 1

20 ,,~, I~\!,~ Fl'\ " \... '

o!k./ ~ .,~ , 1\ '';'41 lt: \if .... Q? W~1 r \If "

~ 20 ,i\, ,. ~", li ..

~~"\C\I>~l:.L.k." !'4~~ ft.!\ o 'V 'i'i' il If. 1990 1992 1994 1996 1998 2000 2002 1990 1992 1994 1996 1998 2000 2002

GBP deposit World Market

60

40

FIGURE 2.3. Annualized risk premia for the market factor. Dashed line: No asymmetry; Dotted line: constant asymmetry, Continous line: Time-varying asymmetry. Notice the similarites between US stock and the World market in these and in the previous plot.

87

Germany Japan

60

40

1990 1992 1994 1996 1998 2000 2002 1990 1992 1994 1996 1998 2000 2002

United Kingdom United States

60 60

40 40

20

O ctA", /" J~ ) \il W "i;?' • w ,,''9'' ; i;<1L'

1990 1992 1994 1996 1998 2000 2002 1990 1992 1994 1996 1998 2000 2002

DEM deposit JPY deposil

60 60

40 40

~1 W "-J 1990 1992 1994 1996 1998 2000 2002

GBP deposit World Market

60 60

40

FIGURE 2.4. Risk premia for the DEM risk. Dashed line: No asymmetry; Dotted line: constant asymmetry, Continous line: Time-varying asymmetry. The estimates of the components of the risk premia due to the DEM risk are sizably higher both for German and U.K. indices and deposits. For Japan and the World Market, the differences are not as considerable, while for the US index they are negligible. The evolution over time of these risk premia seems to be related to the various stages of the introduction of the Euro currency. The peaks around 1992 and 1993 coincide with the European Monetary System cnses.

88

Germany Japan

60 60

40

1990 1992 1994 1996 1998 2000 2002 1990 1992 1994 1996 1998 2000 2002

United Kingdom United States

60 60

40 40

20

1990 1992 1994 1996 1998 2000 2002 1990 1992 1994 1996 1998 2000 2002

DEM deposit JPV deposit

60 60

40 40

20

1990 1992 1994 1996 1998 2000 2002 1990 1992 1994 1996 1998 2000 2002

GBP deposit World Market

60 60

40 40

20 1 l',

o )~o.A. '.,

FIGURE 2.5. Bisk premia for the JPY risk. Dashed line: No asymmetry; Dotted line: constant asymmetry, Continous line: Time-varying asymmetry. Taking into account the time varying asymmetry yields estimates of the risk premia that are generally lower than those obtained without taking into ac­count time varying asymmetry. Notice that for the Japanese equities and for the Yen deposits from 1997 to 2000 the divergence is striking. Models with no asymmetry or constant asymmetry show peaks of risk premia above 40% per year for both assets. However, once time varying asymmetry is taken into con­sideration the contribution to the total premium of the risk premia required to invest either in Japanese equities or in Japanese deposits shrinks to much a smaller level.

89

Germany Japan

60

1990 1992 1994 1996 1998 2000 2002 1990 1992 1994 1996 1998 2000 2002

United Kingdom United States

60 60

40

20

1990 1992 1994 1996 1998 2000 2002 1990 1992 1994 1996 1998 2000 2002

DEM deposit JPY deposit

60 60

40

1990 1992 1994 1996 1998 2000 2002 1990 1992 1994 1996 1998 2000 2002

GBP deposit World Market

60 60

40

FIGURE 2.6. Risk premia for the GBP risk. Dashed line: No asymmetry; Dotted line: constant asymmetry, Continous line: Time-varying asymmetry. The risk premia determined by the British Pound risk show severai similarities with those due to the DEM in the first half of the sampie. One difference is that for the former the premia for investing in DEM deposit and GBP deposit remain steadier and positive throughout the second half of the sampie.

90

Germany Japan

-1 1990 1992 1994 1996 1998 2000 2002 1992 1994 1996 1998 2000 2002

United Kingdom United States 4.8 ....----~~--~--~--------__,

4

3.8 ~-~-~-~-~-~-~--' 1992 1994 1996 1998 2000 2002 1990 1992 1994 1996 1998 2000 2002

DEM deposit JPY depos~

2~-~-~-~-~-~-~--'

1992 1994 1996 1998 2000 2002 1990 1992 1994 1996 1998 2000 2002

GBP depos~ World Market

-5 ~--~--~--~--~--~~~ 2~--~--~--~--~--~~~

1990 1992 1994 1996 1998 2000 2002 1990 1992 1994 1996 1998 2000 2002

FIGURE 2.7. Time varying asymmetry parameter ()t.

91

0.2~----~------~------r-----~------'-----~---.

0.18

0.16

0.14

0.12

0.1

0.08

0.06

0.04~----~------~------~----~------~----~--~

1990 1992 1994 1996 1998 2000 2002

FIGURE 2.8. Price of world market risk as implied by significant parameters, i.e. parameters estimated with a p-value of at least 0.1. Dashed line: model 4, which has no asymmetry parameter e. Continous line: model 6, which features a time-varying parameter et. Despite the absence of any positivity restriction they are both positive during aH the period.

92

2.B The Conditional Mean Encompassing Test

Formally, in the conditional mean encompassing test the statement of the null

is of the form Ho : Model X is weIl specified; the statement of the alternative

is of the form Ha : Model Y fits the conditional mean better. Notice that

this test does not say anything about the difference of the same parameter in

two competing models, and as such, although it provides a guidance for model

selection, it does not provide a measure of mispricing.

The CME tests can be computed using any vIT-consistent estimators and

they are valid without any assumption on the variance of the dependent vari­

able. This feature is of particular interest in this essay. The resulting statistic

has a limiting X2 distribution under the null and is robust to condition al and

unconditional heteroskedasticity of unknown form. The CME tests are com­

puted from linear least-squares regressions and they are specifically devised to

detect departures from the null hypothesis in the direction of the alternative.

The comput able statistic of interest for the CME test is a Q x 1 vector:

T

_T-1 L \7,saJ-L/~)'Êt (2.19) t=l

----where \7,saJ-Lt(c5~) is the score vector of the conditional mean parameters

of the alternative model (the supers cri pt a indicates the alternative model)

and Êt is the residual from the model fitted under the null hypothesis. If the

alternative model has nothing more to add regarding the data than the model

true under the null, the covariance in (2.19) should be zero.

As for the implementation of the test, a robust procedure is suggested

that orthogonalizes the \7,sa J-L/~) with respect to the \7 ,snull J-Lt (ifil). The test

performed here closely mirror the one described in procedure 3.1 in Wooldridge

(1990) and is as follows:

Step 0: Obtain the QML estimates of c5 under the null and under the alter-

93

native. These estimators are VT-consistent as required for the test. Obtain

the relative score matrices. Get the vector Êt from the model under the null.

Step 1: Run the multivariate regression

\78af.lJ~) on \78nullf.lt(~I) and call the Q x 1 vector of residuals ft (2.20)

Step 2: Run the regression

(2.21)

Under the null, the statistic T R~ = T - SSR, where R~ is the unbalanced

R2 and SSR is the sum of square residuals, is asymptotically x~.The CME

test can be extended to the multivariate case. Since all the assumptions hold

in a multivariate context, the only modification required to perform the test is

found in the set of regressors in Step 2 which should now include the residual

for aH the assets simultaneously in the model and consequently in the degrees

of freedom of the CME statistic.

For the purpose of this essay Step 2 then becomes the regression

N

1 on I:: fi/ft (2.22) i=l

where N is the number of assets in the system of equations. Accordingly,

the CME statistic results asymptotically x~ x N with Q x N degrees of freedom,

equal to the number of parameters tested times the number of assets in the

system.

This procedure is particularly useful for testing the components of the risk

premium when the price of each factor is time-varying. For instance, taking

into consideration the price of market risk as estimated in a model with no

94

asymmetry and the priee of market risk as estimated in a model that includes

a constant asymmetry parameter, the CME test formally assesses the following

questions. Is

J:(noO)c (noO)( ) u l,t OVt ri,t, r m,t (2.23)

dnoO)USTP')C (noO) ( ) +ul,3 t oVt ri,t, r m,t

a weIl specified priee of risk in the conditional mean equation for a model

that do es not include asymmetric response in the second moments?18 Or is

the alternative model, which includes a constant parameter for asymmetry Bi

for each asset, a priee of risk that better fits the data? The same question can

be asked for aIl the priee of risk jointly and inference can be drawn regarding

the importanee of asymmetries of second moments for pricing.

18Both the 8 and the covariances are estimates: the omitted hat notation should not create

any confusion.

95

Chapter 3

ASSESSING THE QUALITY OF VOLATILITY,

INTERVAL, AND DENSITY FORECASTS FROM OTC

CURRENCY OPTIONS

Peter Christoffersen Stefano 11azzotta

Abstract. Finaneial deeision makers often eonsider the information in eurreney option valuations when making assessments about future exehange rates. The purpose of this paper is to systematieally assess the quality of op­tion based volatility, interval and density foreeasts. We use a unique dataset eonsisting of over 10 years of daily data on over-the--eounter eurreney option priees. We find that the implied volatilities explain a large share of the vari­ation in realized volatility. Finally, we find that wide--range interval and den­sity foreeasts are often misspecified whereas intermediate interval foreeasts are specified better.

JEL Classification: G13, G14, C22, C53. Keywords: FX, Volatility, Interval, Density, Foreeasting.

96

3.1 Introduction and Background

Financial decision makers often consider the forward-looking information in

currency option valuations when making assessments about future develop­

ments in foreign exchange rates.! Option implied volatilities can be used as

forecasts of realized volatility and interval and density forecasts can be ex­

tracted from strangles and risk-reversals. The purpose of this paper is to assess

the quality of such volatility, interval and density forecasts. Our work is based

on a very unique database consisting of more than ten years of daily quotes on

European currency options from the OTe market.2 The OTe quotes include

at-the-money implied volatilities, strangles and risk-reversals on the dollar,

yen and pound per euro3 as well as on the yen per dollar. From this data

we have constructed daily 1-month interval and density forecasts using the

methodology in Malz (1997).

The main findings of the paper are as follows: First and foremost, we find

that the OTe implied volatilities explain a larger share of the variation in

realized volatility than has been found in previous studies. Second, we find

that wide-range interval forecasts are often misspecified whereas narrow inter­

val forecasts are weIl specified. Third, we find that the option-based density

forecasts are rejected in general. Graphical inspection and formaI tests of the

density forecasts suggest that while the sources of rejections vary from cur­

rency to currency misspecification of the distribution tails is a common source

of error.

Several early contributions use market-based options data with mixed re­

sults. Beckers (1981) finds that not all available information is refiected in the

ISee for example Bank for International Settlements (2003), Bank of England (2000),

International Monetary Fund (2002), and OECD (1999). 2The OTC volatilities used in this paper were provided by Citibank N.A 3Prior to January 1, 1999 these were denoted in DEM.

97

current option price and question the efficiency of the option markets. Canina

and Figlewski (1993) find implied volatility to be a poor forecast of subsequent

realized volatility. Lamoureux and Lastrapes (1993) provide evidence against

restrictions of option pricing models which assume that variance risk is not

priced. Jorion (1995) finds that statistical models of volatility based on re­

turns are outperformed by implied volatility forecasts even when the former

are given the advantage of ex post in sample parameter estimation. He also

finds evidence of bias.

More recently, Christensen and Prabhala (1998) using longer time series

and non overlapping data find that implied volatility outperforms past volatil­

ity in forecasting future volatility. Fleming (1998) finds that implied volatility

dominates the historical volatility in terms of ex ante forecasting power and

suggests that a linear model which corrects for the implied volatility's bias can

provide a useful market-based estimator of conditional volatility. Blair, Poon,

and Taylor (2001), find that nearly aH relevant information is provided by the

VIX index and there is not much incremental information in high-frequency

index returns. Neely (2003) finds that econometric projections supplement im­

plied volatility in out-of-sample forecasting and delta hedging. He also provides

sorne explanations for the bias and inefficiency pointing to autocorrelation and

measurement error in implied volatility.

In work concurrent with ours, Pong, Shackleton, Taylor and Xu (2004)

find that high-frequency historical forecasts are superior to implied volatilities

using OTC data for horizons up to one week. Covrig and Low (2003) use OTC

data to find that quoted implied volatility subsumes the information content

of historicaHy based forecasts at shorter horizons, and the former is as good

as the latter at longer horizons.

Our paper contributes in two areas of the literature. First, to our knowl­

edge, the ernpirical performance of option-based interval and density forecasts

98

has not been systematically explored so far. Second, while there is a consid­

erable literature on implied volatility forecasts from market-traded options,

OTe data have only reeently been employed.

One of our contributions consists of analyzing a unique dataset of OTe

European foreign currency options which turns out to have impressive volatility

prediction properties. OTe options are quoted daily with a fixed maturity

(say one month) whereas market-traded options have rolling maturities which

in turn complicate their use in fixed-horizon volatility forecasting.

In addition to volatility forecasts we evaluate option-based interval and den­

sity forecasts which are widely used by practitioners but which have not been

systematically assessed so far. OTe options are quoted daily with fixed mon­

eyness in contrast with market-traded options which have fixed strike priees

and thus time-varying moneyness as the spot priee changes. This time-varying

moneyness complicates the use of market-traded options for interval and den­

sity forecasting in that the effective support of the distribution is changing

over time. Finally, the trading volume in OTe options is often much larger

than in the corresponding market traded contracts which in turn is likely to

render the OTe quotes more reliable for information extraction.

The remainder of the paper is structured as follows. Section 2 defines the

competing volatility forecasts we consider and describes the framework for

volatility forecast evaluation. It also presents results on the option-implied

and historical return-based volatility forecasts of realized volatility. Section

3 suggests a method for evaluating interval forecasts from option priees and

present results from this method. Section 4 suggests methods for evaluating

density forecasts from option priees and present results from these methods.

Finally, Section 5 introduces the following essay and presents potential areas

for future research.

99

3.2 Volatility Forecast Evaluation

3.2.1 The Forecasting Object of Interest

In order to evaluate the informational content of the volatilities implied from

currency options, we define the realized4 future volatility for the next h days

to be

(3.1)

in annualized terms, whereRt+i = ln(St+d St+i-l) is the FX spot return on

day t + i. This realized volatility will be our forecasting object of interest in

this section.5

3.2.2 Volatility Forecasts

We will consider four competing forecasts of realized volatility. First and most

importantly we consider the implied volatility from at-the-money currency

options with maturity h, where h is either 1 month or 3 months corresponding

to roughly 21 and 63 trading days respectively. Denote this options-implied

volatility by (J{~. ,

The other three volatility forecasts are derived from historical FX returns

only. The simplest possible forecast is the historical h-day volatility, defined

as h

HV 252 ~ 2 RV (J t,h = h ~ Rt-h+i = (J t-h,h

i=l

(3.2)

4See Andersen, T., T. Bollerslev, F. X. Diebold, and P. Labys, 2003. 5 Later on we will consider realized volatilities calculated from 30-minute rather than daily

returns.

100

The historical volatility is a simple equal weighted average of past squared

returns.

We can instead consider volatilities that apply an exponential weighting

scheme putting progressively less weight on distant observations. The simplest

such volatility is the Exponential Smoother or RiskMetrics volatility, where

daily variance evolves as

(Xl

0-;+1 = (1 - À) L Ài-1 R;-i+1 = Ào-; + (1 - À) R; (3.3)

i=1

Following JP Morgan we simply fix À = 0.94 for all the daily FX returns.

The fact that the coefficients on past variance and past squared returns sum

to one makes this model akin to a random walk in variance. The annualized

forecast for h-day volatility is therefore sim ply

(3.4)

Finally we consider a simple, symmetric GARCH(l,l) model, where the

daily variance evolves as

(3.5)

In contrast with the RiskMetrics model, the GARCH model implies a non­

constant term structure of volatility. The unconditional variance in the model

can be computed as

A2 W () =----

1-a-,B (3.6)

The conditional variance for day t + h can be derived as

(3.7)

101

And the annualized GARCH volatility forecast for day t + 1 through t + h

is thus h

GH 252 '"'" A 2 ( (3)i-l (A 2 A 2) (J t,h = h L (J + a + (J t+ 1 - (J

i=l

(3.8)

The GARCH model will have a downward sloping volatility term structure

when the current variance is above the long horizon variance and vice versa.6

Figure 3.1 shows the spot rates of the four FX rates analyzed in this paper.

Prior to the euro introduction in 1999 we observe FX options denoted against

the Deutschmark (DEM) and we will therefore work with the DEM spot rates

prior to the euro introduction as weIl. Prior to January 1, 1999 we use DEM

options to forecast DEM volatility and afterwards we use EUR options to fore­

cast EUR volatility. Descriptive statistics of the exchange rates and volatilities

are reported in end of chapter Appendix.

The five volatility specifications including the realized volatility are plotted

in Figures 3.2-3.5. The left columns shows the 1-month volatility and the

right column the 3-month volatility. Notice that the RiskMetrics volatilities in

Figure 3.4 are identical for 1-month and 3-month maturities as the random­

walk nature of this specification implies a fiat volatility term structure.

3.2.3 Predictability Regressions

We are now ready to assess the quality of the different volatility forecasts.

This will be done in simple linear predictability regressions. We first run four

6The GAReR model contains parameters which must be estimated. We do this on rolling

lO-year samples starting in January 1982 and using QMLE. Each year we forecast volatility

one-year out-of-sample before updating the estimation sample by another calendar year of

daily returns. The euro volatility forecasts are constructed using synthetic euro rates in the

period prior to the introduction of the euro.

102

univariate regressions for each currency

O"fi: = a + bO"{ h + ~ h' for j = IV, HV, RM, CH (3.9) , "

The purpose of these univariate regressions is to assess the fit through the

adjusted R2 and to check how close the estimates of a are to 0 and how close

the estimate of b are to 1. Bollerslev and Zhou (2003)1 point out that if

the volatility risk is prieed in the options markets then we should expect to

find a positive intereept and a slope less than one in the above regression. In

a standard stochastic volatility set up, it can be shown that if the priee of

volatility risk is zero, the proeess followed by the volatility is identical under

the objective and the risk neutral measures. In such a case there would be

no bias. However, the volatility risk premium is generally estimated to be

negative which in turn implies that the volatility proeess under the risk neutral

measure will have higher drift. This is also consistent with the fact that implied

volatilities are empirically found to be upward biased estimates of the objective

volatility.

Aside these considerations, for someone using implied volatility in the real

time monitoring of FX movements, the intereept and slope coefficients are

informative of the size of the bias and efficiency respectively of the forecasts.

In addition we will run three bivariate regressions including the implied

volatility forecast as well as each of the three return-based volatility forecasts

in turn. Thus we have

RV b IV j IV,j O"t,h = a + O"t,h + CO"t,h + Et,h , for j = HV, RM, CH (3.10)

The purpose of the bivariate regressions is to assess if the return-based

volatility forecasts add anything to the market-based forecasts implied from

currency options.

7See also Bandi and Perron (2003), Chernov (2003), Bates (2002), and Benzoni (2001)

103

Finally, we run a regression including all the four volatility forecasts in the

same equation. The purpose of this regression is to assess the relative merits

of the different volatility forecasts.

We will run aIl regressions for h = 21 and 63 corresponding to the 1-month

and 3-month option maturities. We will also run aIl regressions in levels of

volatility as above as well as in logarithms.8 Due to the volatilities being

strictly positive, the log specification may have error terms, which are better

behaved than those from the level regressions.

3.2.4 Volatility Forecast Evaluation Results

Tables 3.3 and 3.4 report the regression point estimates as weIl as standard

errors corrected for heteroskedasticity and auto correlation. Throughout this

paper we apply GMM using the Newey-West9 weighting matrix with a pre­

specified bandwidth equal to 21 days for the 1-month horizon (Table 3.3) and

63 days for the 3-month horizon (Table 3.4). The bandwidth is chosen as

to eliminate the influence of the auto correlation induced by the overlapping

observation. We also report the regression fit using the adjusted R2•

Several strong and interesting empirical regularities emerge. First, the

regression fit is very good in all cases. Jorion (1995) reports R2 in the region

0.10-0.15 for the USD/JPY, USD/DEM and USD/CHF using implied volatility

forecasts. We get instead R2 of 0.30-0.38 for the 1-month maturity and 0.16-

0.35 for the 3-month maturity case. Second, comparing the R2 across the

univariate forecast regressions we see that the implied volatility is the best

volatility forecast. This result holds across currencies and horizons.

Third, comparing the slope estimates across the bivariate forecast regres-

8Results of the logarithm regressions are not very diffrent from those of the regressions

in levels and are not reported to conserve space.

9See Newey and West (1987).

104

si ons where the implied volatility forecast is included along with each of the

other three forecasts, the implied volatility al ways has the highest slope. Thus,

in the cases when GARCR has a higher slope in the univariate regression the

bivariate regressions including the IV and GARCR forecasts always assign a

larger slope to the IV forecast. The fact that GARCR-based forecasts some­

times have a slope closer to one than do the implied volatility forecasts is not

surprising given the priee of volatility risk argument in Bollerslev and Zhou

(2003) and others lO . Nevertheless, it is interesting to note that the R2 is higher

for the implied volatility forecasts even in the cases where its slope is lower

than that of the GARCR-based forecasts.

Fourth, comparing the slope estimates across the multivariate forecast

regressions where all four forecasts are included simultaneously the implied

volatility has the highest slope. This result holds across currencies and hori­

zons. Fifth, comparing across the horizon forecasts it appears perhaps not

surprisingly that the 1-month forecasts have higher R2 than the 3-month fore­

casts. Finally, the slope coefficient is often insignificantly different from one for

the IV forecasts, and its intereept is often insignificantly different from zero.

Tables 3.5 and 3.7 contain the same set of regressions as Tables 3.3 and

3.4, but now run on the euro sample (i.e. post January 1, 1999) only, and

furthermore relying on 3D-minute intraday returns rather than daily returns

to compute the one and three month realized volatilities. We also report the

euro sample estimates using daily data in Table 3.6 and 3.8.

The objective of Tables 3.5 and 3.7 is to see if the post-euro sample is differ­

ent from the full sample period which straddles the introduction, and further­

more to assess the value of using high-frequency returns in volatility forecast

evaluation. The theoretical benefits of doing so have been documented in An-

lOWhether volatility risk is priced is of course an empirical question: sorne of our results

indirectly support the conjecture that volatility risk is priced in the currency options markets.

105

dersen and Bollerslev (1998) and Andersen, Bollerslev and Meddahi (2003)

who show that the R2 in the regressions we run will be significantly higher

when proxying for true volatility using an intraday rather than daily return­

based volatility measure. As pointed out by Alizadeh, Brandt and Diebold

(2002), and Brandt and Diebold (2004) this theoretical benefit may in prac­

tice be outweighed by market microstructure noise, but relying on 30-minutes

returns in very liquid markets as we do here should mitigate these problems.

The results in Table 3.5 and 3.7 are broadly similar to those from the full

sample but using high-frequency returns does lead to sorne new interesting

findings. First, for the three euro cross currencies the regression fit is typically

much better now. Due to the obvious structural break in 1999 this is perhaps

not surprising. But it is still interesting that we now get R2 as high as 65% in

the univariate regressions. Note that the R2 for the 3-month JPY /USD case

is now slightly lower than before. It is therefore not simply the case the FX

volatility has become more predictable as of late.

Second, comparing the R2 across the univariate forecast regressions the

implied volatility is typically the best volatility forecast. The exception is the

EUR/ JPY rate. Third, comparing the slope estimates across the bivariate and

multivariate forecast regressions the implied volatility typically has the highest

slope. It is interesting that the simple historical realized volatility forecast now

sometimes has the highest slopeY This result is exclusively due to the use of

high frequency data as it is easy to infer from the comparison of Table 3.5-3.7

to 3.6-3.8. The added accuracy in this forecast from the intraday returns is

thus evident.

In or der to assess the importance of the choice of estimation period we

llThe historical volatility forecast could potentially be improved further by estimating a

slope coefficient thus allowing for mean reversion in the forecast.

106

have also run the same regressions using in sample GARCH estimates. 12 AI­

though the explanatory power as measured by the adjusted R2 of the in sample

GARCH is substantially higher compared to that of the out of sample regres­

sions, in most cases the change does not affect the results in the 1-month

regressions for any currency both in the full sample and in the post 1999 sam­

pIe. The same is true for the 3-month horizon with the only exception of the

GBP in the full sample: here the in sample GARCH get the highest coefficient

in the multivariate regression.

In summary, we find strong evidence that the implied volatility from FX

options has substantial predictive power in forecasting future realized volatility

at the one and three month horizons. The predictability is particularly strong

for the euro cross rates in the recent period. In spite of the potential bias from

volatility risk being priced in the options, the regression slope on the volatility

forecasts are often quite close to one.

Perhaps the most striking finding in Tables 3.3 to 3.7 is the high level of

R2 found in the implied volatility regressions. It appears that the volatility

implied in the OTC options offer rather precise forecasts. We conjecture that

the so-called telescoping bias arising from the rolling-maturity structure of

market-traded options (see e.g. Christensen, Hansen, and Prabhala, 2001)

could be part of the reason. Furthermore, the fact that OTC options are

quoted daily with a fixed moneyness, as opposed to a fixed strike price, which

ensures that the options used for volatility forecasting are exactly at-the-money

each day. Finally, the large volume of transaction in OTC currency options

compared with market traded options may offer additional explanation. 13

12Results are not shown here to conserve space.

13Note that the quotes are from the book of one single large dealer. and thus could be poen-

tially affected by its inventory policy. This deos not seem to affect volatility predictability,

though.

107

3.2.5 Bias and Efficiency

To study the merit of each correlation forecasts with regard to the relative

efficiency and bias we perform a Mincer-Zarnowitz (1969) decomposition of

the MSE into bias squared, inefficiency, and random (or residual) variation. 14

The decomposition is as foIlows: M SE = [E[y] - E[YW + (1 - ,B)2Var(f)) + (1 - R2)Var(y), where y is the variable of interest, in our case the realized

volatility, and f) is each volatility forecast in turn. From the regression of y on

f) and a constant, we obtain the slope coefficient ,B and the regression fit, R2•

The Mincer-Zarnowitz procedure is run for each currency and for each of the

currency forecasts. Table 3.9 in end of chapter Appendix reports the MSE's

in absolute value and their percentage decomposition of the total MSE into

bias squared, inefficiency, and residual variation for the entire sample period.

Table 3.10 in end of chapter Appendix reports the decompositions for the

period foIlowing the introduction of the single currency.

The tables distinctly show the pattern of the trade off between bias and

efficiency for aIl the currencies and aIl the sample periods. The absolute mag­

nitude of the MSE confirms that implied volatility is the best forecast in almost

aIl cases. In addition, at the 1-month horizon, the squared bias is generally

higher for the implied volatility than is for aIl the other volatility forecasts,

with few exceptions in which GARCH is more biased. At the 3-month horizon,

bias in the GARCH volatility becomes more severe, becoming larger than that

of implied volatility in a number of cases. The historical volatility and Risk­

Metrics volatility appear to be rather inefficient but substantially less biased

than the other two forecasts.

In conclusion, the Mincer-Zarnowitz decomposition shows that although

implied volatility is slightly biased, it is generally the best forecast, from both

14We thank an anonymous referee and the thesis committee for pointing us in this direc­

tion.

108

the total forecasting error perspective and the efficiency perspective. In the

following section we study the performance of one-month interval forecasts

calculated from option prices and forward rates.

3.3 Interval Forecast Evaluation

The information in currency options may be useful not only for volatility fore­

casting but for spot rate distribution forecasting more generaIly. In the follow­

ing sections we abstract from the difference between risk neutral and objective

distributions. The empirical question we want to ask is: "How weIl can risk

neutral intervals and densities computed using standard methodologies fore­

cast physical interval and densities". The legitimacy of the question stem from

the fact that financial decision makers often consider the information in cur­

rency option valuations when making assessments about future exchange rates

without worrying about this important theoretical difference. In addition, this

pragmatic approach can be justified by considering that for currencies the risk

premium, i.e. the conditional mean, which would largely determine the dif­

ference between risk neutral and physical, may not be as important as the

higher order moments and particularly the conditional variance, especially at

the short horizons we consider here. 15 In other words, the tests in the following

sections can be considered as joint tests of the methodology used to extract

densities and intervals under the additional hypothesis that the objective and

risk neutral distributions are not very different.

These tests may have low power with respect to generic alternative hypothe­

ses but they can help assessing whether certain specific pieces of information

have been duly taken into account in the construction of these intervals and

15It is also the case that there is no methodology to transform risk neutral distributions into

their objective counterparts without making several, possibly very restrictive assumptions.

109

densities. Rejection may come from the presence of sorne currency risk pre­

mium: this situation will shift apart the mean under the two probabilistic

measures. Rejection could also come from methodological and data short com­

ing in the construction of the interval and densities: this is likely to be the

case for the widest intervals and the tails of the distributions when they are

based on the extrapolation of market data.

In substance, the results of these tests should be seen as suggestion for

improvement of the prevalent methodologies and caveats with regard to how

much trust should one put in these forecasts.

The intervals are constructed from the option-implied densities which in

turn are calculated using the estimation method in Malz (1997). Malz (1997)

proposes a simplified proeedure to extract risk neutral densities from options

that exploits the conventions of the over the counter options markets rather

than priees from eentralized exchanges. For FX options this approach is widely

used by large institutions. Although there are alternative approaches for risk­

neutral density extraction, such as the semi-non parametric approach of Ait­

Sahalia and Duarte (2002) or that based on arbitrage free priees filtering of

Ioffe (2004), they cannot be applied to our dataset as they need a larger number

of contracts with different strikes.

The typical implementation of the Malz method on the contrary needs only

the priees of the strangle16 and the risk reversaI, with deltas of 0.25 and 0.75,

and the at-the-money implied volatility (delta ~ 0.5).

At these strikes individual options and combinations are heavily traded,

and henee the priees quoted tend to be quite reliable. This is sometimes seen

as an advantage sinee the seemingly larger amount of information from options

16Recall that a strangle is a combination of an out-of the-money long call and an out-of

the-money long put, with the strike of the call1arger than the strike of the put and. A risk

reversaI is a combination of the same long call but a short put instead.

110

with several strikes available from exchange traded option must be balanced

with the unreliability of quotes of thinly traded contracts, stale quotes and

suspected violations of put calI parity.

The Malz approximation of the implied volatility function is exactly equal

to the observed implied volatility for the observed three values of the delta and

does not imply violations of the no arbitrage bounds for other contracts. The

procedure entails a quadratic approximation of the volatility smile which takes

into account the well-know departures from the log normality assumption of

the Black and Scholes model (e.g. fat tails, skewness). The functional form

of the approximated volatility function captures the essential features of the

smile: the ATM volatility provides information about the level of volatility, the

risk reversaI and the strangle capture the slope and the curvature, respectively.

The interpolated approximation of the smile is then used to compute a

continuous option price as a function of the strike. The classical result17 in

Bredeen and Litzenberger (1978) is the final step needed to extract the risk

neutral density.

We have computed conditional interval forecasts for the {0.45, 0.55} prob­

ability interval, as well as the {0.35,0.65}, {0.25,0.75}, {0.15,0.85}, and the

{0.05,0.95} intervals. These forecasts for the 10 and 90 per cent are shown

in Figure 3.7. Notice that the intervals for the GBP IDEM look excessively

jagged in a large part of the pre euro sample.

3.3.1 Interval Evaluation Methodology

We now set out to evaluate the usefulness of the interval forecasts following the

methodology developed first in Christoffersen (1998). Let the generic interval

17This famous result shows that the second derivative with respect to the strike of the

price of a call is the discounted risk neutral density.

111

forecast be defined as

{Lt,h(PL), Ut,h(PU)} (3.11)

where PL and Pu are the percent ages associated with the lower and upper

conditional quantiles making up the interval forecast.

Consider now the indicator variable defined as

(3.12)

Then if the interval forecast is correctly calibrated, we must have that

(3.13)

where X t denotes a vector of information variables (and functions thereof)

available on day t. If the interval forecast is correctly calibrated then the

expected outcome of the future FX rate falling out si de the predicted interval

must be a constant equal to the pre-specified interval probability p.

This hypothesis will be tested in a logistic regression set up. Under the

alternative hypothesis we have

It,h - P = a + bXt + Et,h (3.14)

and the null hypothesis corresponds to the restrictions

a=b=O (3.15)

Running these regressions on daily data we again have to worry about

overlapping observations, which we allow for using GMM estimation.

112

3.3.2 Interval Evaluation Results

Table 3.11 shows the results for the logit regression-based tests of the interval

forecasts. The interval forecasts for the {0.45, 0.55}, {0.35, 0.55}, {0.25,0.75},

{0.15, 0.85}, and the {0.05, 0.95} intervals are denoted by the probability of an

observation outside the interval, i.e. p = .90, .70, .50, .30 and .10 respectively.

We refer to these outside observations as hits. The zer%ne hit sequence (less

its expected value p) is regressed on a constant, the 21-day lagged hit and

the 21-day lagged I-month implied volatility. The lagged hit is included to

capture any dependence in the outside observations. The implied volatility

is included to assess if it is incorporated optimaIly in the construction of the

interval forecast. If the interval forecast is correctly specified then the intercept

and slopes should aIl be equal to zero. Table 3.11 reports coefficient estimates

along with t-statistics again calculated using GMM. The "Average Rit" entry

in each subsection of the table should be equal to p. It is reported along with

the t-statistic from the test that the average hit rate indeed equals p. AlI

t-statistics larger than two in absolute value are denoted in boldface type. We

also include Wald tests of the joint hypothesis that aIl the estimated coefficients

are zero.

The results in Table 3.11 can be summarized as foIlows. First, for the

EUR/GBP (third column) the average hit rate is significantly different from

the pre-specified p for almost aIl the intervals. The jagged pound intervals

evident from Figure 3.7 are probably the culprit here. Second, for the other

three FX rates, the average hit rate is often not significantly different from the

pre-specified p for the narrow intervals. The wide-range intervals (with outside

probability .10 and .3) are aIl rejected. It thus appears that the interval forecast

have the hardest time forecasting the tails and the very center of the spot rate

distribution.

113

Third, notice that very few regression slopes are significant in the JPY /EUR

case. No dependence in the hit sequence is apparent and the information in

implied volatilities seems to be used optimally in this case. The slope on the

21-days lagged implied volatility is most often found to be significantly neg­

ative. This indicates that the hits tend to occur when the implied volatility

was relatively low on the day the forecast was made. If the intervals had been

using the implied volatility information optimally then no dependence should

be found between the current implied volatility and the subsequent realization

of the hit sequence.

Table 3.12 reports the interval forecast evaluation results using data from

the euro sample only. The results are now somewhat different and can be

summarized as follows. The average hit rate is rejected across all the four

FX rates for the widest and narrowest intervals. Again, it appears that the

option implied densities have trouble capturing the tails and the center of the

distribution. For three out of four FX rates generally the outside hit frequency

is lower than it should be, thus in these cases the wide-range option-implied

intervals are too wide on average.

Second, in general the pound intervals are better calibrated in the euro

sample than before. Third, the JPY /USD interval is now the most poorly

calibrated interval.

In summary we find that the option-implied interval forecast for the euro

cross rates perform fairly in the post January 1, 1999 sample. The exception

is the forecasts for the widest intervals, which tend to be too wide on average.

The option-implied densities apparently have trouble capturing the tail behav­

ior of the spot rate distributions. The rejection of widest intervals and thus

misspecification of the tails of the density forecasts should perhaps not come

as a surprise. The density tails are estimated on the basis of an extrapolation

of the volatility smile from the values for which option price information is

114

available (that is for deltas equal to .25, .50, and .75). It appears that this ex­

trapolation could be improved. We will pursue the topic of density forecasting

in more detail in the next section.

3.4 Density Forecast Evaluation

The option-implied interval forecasts analyzed above are constructed from the

implied density, which contains much more information than the intervals

alone. We would therefore like to evaluate the appropriateness of these den­

sity forecasts in their own right. Doing so is likely to yield sorne insights into

the poor performance of the widest interval forecasts, which was noted above.

We start off by outlining the general idea behind density forecast evaluation

developed first by Berkowitz (2001).18

Let Ft,h (8) and ft,h (8) denote the cumulative and probability density func­

tion forecasts made on day t for the FX spot rate on day t + h. We can then

define the so-called probability transform variable as

St+h

Ut,h - [00 ft,h(U)du - Ft,h (8t+h) (3.16)

The transform variable captures the probability of obtaining a spot rate

lower than the realization where the probability is calculated using the density

forecast. Figure 3.6 shows a plot of the probabilities. The probability will take

on values in the interval [0, 1]. If the density forecast is correctly calibrated

then we should not be able to predict the value of the probability transform

variable Ut,h using information available at time t. That is, we should not be

able to forecast the probability of getting a value sm aller than the realization.

Moreover, if the density forecast is a good forecast of the true probability

18 800 also Diebold, Gunther and Tay (1998) and Diebold, Hahn and Tay (1999).

115

distribution then the estimated probability will be uniformly distributed on

the [0, 1] interval.

3.4.1 Graphical Density Forecast Evaluation

Figure 3.8 assesses the unconditional distribution of the probability transform

variable Ut,h for each spot rate through a simple histogram. If the density

fore cast is correctly calibrated then each of the histograms should be roughly

fiat and a random 10% of the 31 bars should fall outside the two horizontal

lines delimiting the 90% confidence band.

It appears that the histograms dis play certain systematic differences from

the uniform distribution. Notice in particular that the JPY lEUR histogram

(top right panel) shows a systematically declining shape moving from left to

right. This is indicative of the forecasted mean spot rate being wrong. There

are too many observations where the realized spot rate lies in the left side

of the forecasted distribution (and generates a Ut,h less than 0.5) and vice

versa. In the USD lEUR case (top left panel) it appears that there are not

enough observations in the two extremes, which suggests that the forecasted

density has tails, which are too fat. This finding matches Table 3.11 where we

found that the widest intervals were too wide for the USD lEUR. Finally, the

JPY IUSD distribution (bottom right panel) appears to be misspecified in the

right tail.

For certain purposes, including statistical testing, it is more convenient to

work with normally distributed rather than uniform variables for which the

bounded support may cause technical difficulties. As suggested by Berkowitz

(2001)19 we can use the standard normal inverse cumulative density function

19See also Diebold, Gunther and Tay (1998) and Diebold, Hahn and Tay (1999).

116

to transform the uniform probability transform to a normal transform variable

(3.17)

If the implied density forecast is to be useful for forecasting the physical

density, it must be the case that the distribution of Ut,h is uniformly distributed

and independent of any variable X t observed at time t. Consequently the

normal transform variable must be normally distributed and also independent

of all variables observed at time t.

This is must be true under the null hypothesis regardless of the shape of

the data generating process which can be, and usually is not normal.

Figure 3.9 assesses the unconditional normality of the normal transforms

by plotting the histograms with a normal distribution superimposed.20 The

normal histograms typically confirm the findings in Figure 3.8 but also add

new insights. While it appeared in Figure 3.8 that the GBP lEUR had fairly

random deviations from the uniform distribution, it now appears that the

normal transform is systematically skewed compared with the superimposed

normal distribution.

While the graphical evidence in Figures 3.8 and 3.9 is quite informative

of the potential deficiencies in the option implied density forecasts, it may be

interesting to formally test the hypothesis of the normal transforms following

the standard normal distribution. We do this below.

3.4.2 Tests of the Unconditional Distribution

We first want to test the simple hypothesis that the normal transform vari­

ables are unconditionally normally distributed. Basically, we want to test if

the histograms in Figure 3.9 are significantly different from the superimposed

20The superimposed normal distribution functions have different heights due to the dif­

ferent number of observations available for each currency.

117

normal distribution. The unconditional normal hypothesis can be tested using

the first four moment conditions

(3.18)

We still need to allow for auto correlation arising from the overlap in the

data and so we estimate the following simple system of regressions

Zt,h = al + E~~~

Z; h - 1 = a2 + E~2~ , ,

Z 3 (3) th = a3 + Et h , ,

Z 4 (4) t h - 3 = a4 + Et h , ,

(3.19)

using GMM and test that each coefficient is zero individually as well as

the joint test that they are all zero jointly.21 In each case we allow for 21

day overlap in the daily observations. The results of these tests are reported

in Tables 3.13 and 3.15. Table 3.13 tests for unconditional normality on the

entire sample and Table 3.15 restricts attention to the post 1999 period.

Table 3.13 shows that while only a few of the individu al moments are found

to be significantly different from the normal counterpart, the joint (Wald) test

that all moments match the normal distribution is rejected strongly in three

cases and weakly in the case of the JPY jUSD. The post 1999 results are very

similar. Now the Wald test strongly rejects all four density forecasts. We thus

find fairly strong evidence overall to reject the option-implied density forecasts

using simple unconditional tests.

In order to focus attention on the performance of the density forecasts in

the tails of the distribution, we report QQ-plots of the normal transform vari­

ables in Figure 3.10. QQ-plots display the empirical quantile of the observed

21 See Bontemps and Meddahi (2002) for related testing procedures.

118

normal transform variable against the theoretical quantile from the normal

distribution. If the distribution of the normal transform is truly normal then

the QQ-plot should be close to the 45-degree line.

Figure 3.10 shows that the left tail is fit poorly in the case of the dollar,

and that the right tail is fit poorly in the case of the pound and the JPY jUSD.

In the case of the EURjUSD there are too few small observations in the data,

which is evidence that the option implied density has a left tail that is too

thick. The EURjGBP has too many large observations indicating that the

right tail of the density forecast is too thin. In the JPY jUSD case the right

tail appears to be too thick. These findings are also evident from Figure 3.8.

Rejecting the unconditional normality of the normal transform variables is

of course important, but it do es not offer much constructive input into how

the option-implied density forecasts can be improved upon. The conditional

normal distribution testing we turn to now is more useful in this regard.

3.4.3 Tests of the Conditional Distribution

We would like to know why the densities are rejected, and specifically if the

construction of the densities from the options data can be improved somehow.

To this end we want to conduct tests of the conditional distribution of the

normal transform variable. Is it possible to predict the realization of the time

t + h normal transform variable using information available at time t? If so

then this information is not used optimally in the construction of the density

forecast.

The condition al hypothesis can be tested using the generic moment condi­

tions

E [Zt,hiI (Xt )] = 0, E [Z~hfz (Xt )] = 1

E [Zt,hh (Xt )] = 0, E [Zthf4 (Xt )] = 3

119

(3.20)

Choosing particular moment functions and variables, these conditions can

be implemented in a regression setup as follows

Z b Z b IV (1) t,h = al + 11 t-h,h + 120" t + Et,h

2 2 b (IV) 2 (2) Zt,h - 1 = a2 + b21 Z t _ h,h + 22 O"t + Et,h

Z 3 b Z3 b (IV) 3 (3) t,h = a3 + 31 t-h,h + 32 0" t + Et,h

4 b Z4 b (Iv)4 (4) Zt,h - 3 = a4 + 41 t-h,h + 42 O"t + Et,h

(3.21)

where we include the lagged power of the normal transform as weIl as the

power of the current implied volatility as regressors. We can now test that the

regression coefficients are zero.

Table 3.14 shows the estimation results of the regression systems for the

four exchange rates. In line with previous results we find that the information

in the implied volatility is not used optimally in the construction of the option­

implied density forecast for the GBP /EUR.

Table 3.16 shows the regressions from Table 3.14 run only on the euro

sample. Comparing the two tables, it is evident that the clear rejection of the

pound density forecasts in Table 3.14 is largely due to problems in the pre-euro

sample. Restricting attention to the euro sample there is more evidence on the

implied volatility being misspecified in the JPY /USD rate. Looking across

Tables 3.14 and 3.16 we see that the Wald test of aIl coefficients being zero

is strongly rejected for aIl four FX rates in both samples. It would therefore

seem possible in general to improve upon the option-implied density forecasts

studied here.

120

3.4.4 Density Slices Tests

The inspection of the uniform histograms reveals that the densities have prob­

lems particularly in the tails. In order to formally detect from which slice

of distribution the problem may be originating we extend upon the testing

methodology proposed by Berkowitz (2001).

In particular let a, b E {O, 1} be two real numbers such that a ~ b. From

the properties of the uniform distribution follow that if the density forecast

is weIl specified aIl Ut,h E {O, 1} are uniformly distributed on the sub-interval

{ a, b}. We can than define another random variable

Yi,h (Ut,h - a)/(b - a)

Yi,h E {0,1}

w hich is uniformly distri buted on the interval {O, 1}.

(3.22)

(3.23)

We can again use the inverse Gaussian transform to test for the specification

of the density forecast slice of interest by redefining

(3.24)

For the density forecast to be weIl specified in the interval {a, b}, the con­

dition that Yi,h is uniformly distributed is necessary but not sufficient. For

instance, a particular slice {a, b} E {O, 1}, Ut,h could be uniformly distributed,

but there could still be too few or too many observations falling in that parti cu­

lar interval {a, b}. A further necessary condition is that the coverage is correct.

This corresponds to the requirement that the proportion of the Ut,h E {a, b} is

exactly equal to b - a.

These requirements can be translated into moment conditions, which can

be jointly tested in a GMM framework. To do so we define

{

0, if Ut+h E {a, b} I th =

, 1, if not

And consider the following moments

E [Zt,hfI (Xt )] = 0, E [Z;'hh (Xt )] = 1

E [Zt,hh (Xt)] = 0, E [zthf4 (Xt )] = 3

E [It,h] = b - a

121

(3.25)

(3.26)

Choosing particular moment functions and variables these conditions can

be implemented in a regression system setup as follows. For the unconditional

case we have

Z (1) t,h = al + Et,h

Z 2 (2) t h - 1 = a2 + Et h , ,

Z 3 (3) th = a3 + Et h , ,

Z 4 (4) t h - 3 = a4 + Et h , ,

(3.27)

The estimation of the GMM system is done in such way as to specify the

last condition as a logistic regression. The joint null hypothesis can be tested

with a Wald test that al = a2 = a3 = a4 = 0, and ea5 /(1 + ea5) = b - a.

The conditional test mirrors exactly the test for the entire distribution in the

system (3.21) with the only addition of the coverage equation, i.e. ast equation

in the system.

One appealing feature of the slice test is that can help to pin point the

source of the density forecast misspecification.

122

We implement the test on three slices that are particularly relevant to our

investigation: the left tail, up to a theoretical probability mass of .25, the

central slice, from .25 to .75, with a theoretical mass of .5, and the right tail

with a theoretical mass of .25. The results of unconditional and conditional

tests are reported in Table 3.17-3.22

The results of the slices test confirm that the tails are misspecified across

the board for all the densities, although for different reasons. They also show

however sorne diversity of results for the central slice depending on the cur­

rency. The conditional test cannot reject that both the central slice of the

distribution of the USD lEUR and JPY IUSD are well specified.

3.5 Conclusion

We have presented evidence on the usefulness of the information in over-the­

counter currency options for forecasting various aspects of the distribution of

exchange rate movements. We focused on three aspects of spot rate forecasting,

namely, volatility forecasting, interval forecasting, and distribution forecasting.

While other papers have pursued volatility forecasting in manners similar to

ours we believe to be the first to systematically investigate the properties

of option-based interval and density forecasts. Furthermore, we are sorne of

the first to investigate long time series of volatilities from over-the-counter

options, which we find to be remarkably powerful for volatility forecasting. We

conjecture that reasons for this important finding are likely to be 1) the so­

called telescoping bias arising from rolling maturities in market-traded options

is not an issue in the OTe options, 2) the time-varying moneyness in market­

traded options, and 3) the volume of trades done over-the-counter is much

larger than the exchange trading volume for currency options.

Our other findings can be summarized as follows. First, the implied volatil-

123

ities from currency options typically offer predictions that explain much more

of the variation in realized volatility than do volatility forecasts based on his­

torical returns only. This ranking is however sometimes reversed when histor­

ical volatility forecasts are constructed from intraday returns. Second, when

combining implied volatility forecasts with return-based forecasts, the latter

typically reeeive very litt le weight. Third, in terms of interval forecasting on

the entire 1992-2003 sample, the option-implied intervals are useful for the

JPY /EUR but rejected for the other three currencies in the study. Fourth, fo­

cusing on the euro sample, the option-implied interval forecasts are generally

well specified. Two notable exceptions are the widest-range intervals with 90%

coverage and the JPY /USD intervals in general. The 90% intervals tend to be

too wide due to the misspecification of the tails of the forecast distribution.

Fifth, when evaluating the entire implied density forecasts these are generally

rejected. The graphical evidenee again suggests that the tails in the distribu­

tion are typically misspecified. We thus conclude that the information implied

in option pricing is useful for volatility forecasting and for interval forecasting

as long as the interest is confined to intervals with coverage in the 50-70%

range.

The rejection of the widest intervals and the complete density forecast is of

course interesting and warrants further scrutiny. The potential reasons are at

least fourfold. First, the option contracts used may not have extreme enough

strike priees to be useful for constructing accurate distribution tails. Second,

the information in options could be used sub-optimally in the density esti­

mates. Third, we could be rejecting the densities because eertain information

available at the time of the forecasts is not incorporated in the option priees

used to construct the densities, i.e. option market inefficiencies. Fourth, the

risk premium considerations, which were abstracted from in this paper could be

important enough to reject the risk-neutral density forecasts considered. The

124

misspecification of the mean in the case of the JPY lEUR rate suggests that

an omitted risk premium could be the cul prit in that case. For the other three

currencies, however, Figure 3.10 suggests that the culprit is tail misspecifica­

tion, which is likely to arise from the lack of information on deep in-the-money

and deep out-of-the-money options.

We round off the essay by listing sorne promising directions for future re­

search. First, policy makers may be interested in assessing speculative pres­

sures on a given ex change rate. The option implied densities can be used in

this regard by constructing daily option-implied probabilities of say a 3% ap­

preciation or depreciation during the next month. Second, the accuracy of

the left and right tail interval forecast could be analyzed separately in order to

gain further insight on the probability of a sizable appreciation or depreciation.

Third, relying on the triangular arbitrage condition linking the JPY lEUR, the

USD lEUR, and the JPY IUSD, one can construct option implied covariances

and correlations from the option implied volatilities. These implied covariances

can then be used to forecast realized covariances and correlation as done for

volatilities. This is the topic of the following essay.

125

3.A Tables and Figures

USD I-Month 3-Month

RV IV HV RM RV IV HV RM IV 0.554 IV 0.459

HV 0.452 0.790 HV 0.376 0.818 RM 0.473 0.844 0.945 RM 0.424 0.797 0.869

GARCH 0.471 0.793 0.915 0.922 GARCH 0.437 0.723 0.753 0.903

JPY I-Month 3-Month

RV IV HV RM RV IV HV RM IV 0.609 IV 0.591

HV 0.558 0.821 HV 0.572 0.866 RM 0.571 0.871 0.952 RM 0.571 0.837 0.892

GARCH 0.567 0.845 0.943 0.951 GARCH 0.537 0.798 0.792 0.926

GBP I-Month 3-Month

RV IV HV RM RV IV HV RM IV 0.585 IV 0.398

HV 0.482 0.812 HV 0.362 0.781 RM 0.484 0.847 0.956 RM 0.390 0.791 0.879

GARCH 0.498 0.780 0.915 0.894 GARCH 0.374 0.699 0.684 0.888

JPY/USD I-Month 3-Month

RV IV HV RM RV IV HV RM IV 0.569 IV 0.519

HV 0.597 0.793 HV 0.617 0.801 RM 0.619 0.837 0.960 RM 0.619 0.782 0.913

GARCH 0.605 0.789 0.908 0.905 GARCH 0.549 0.691 0.741 0.890

TABLE 3.1. Pairwise Correlations of Foreign Exchange Volatility. Full sample.

126

USD I-Month 3-Month

RV IV HV RM RV IV HV RM IV 0.628 IV 0.645

HV 0.479 0.762 HV 0.535 0.835 RM 0.501 0.820 0.941 RM 0.504 0.824 0.873

GAReH 0.493 0.836 0.945 0.974 GAReH 0.507 0.836 0.874 0.973

JPY I-Month 3-Month

RV IV HV RM RV IV HV RM IV 0.676 IV 0.677

HV 0.608 0.792 HV 0.632 0.850 RM 0.617 0.845 0.953 RM 0.621 0.843 0.895

GAReH 0.580 0.816 0.940 0.951 GAReH 0.562 0.788 0.792 0.947

GBP I-Month 3-Month

RV IV HV RM RV IV HV RM IV 0.669 IV 0.774

HV 0.524 0.801 HV 0.624 0.825 RM 0.550 0.839 0.956 RM 0.610 0.834 0.887

GAReH 0.532 0.808 0.943 0.938 GAReH 0.559 0.777 0.768 0.935

JPY/USD I-Month 3-Month

RV IV HV RM RV IV HV RM IV 0.506 IV 0.338

HV 0.845 0.747 HV 0.806 0.766 RM 0.843 0.811 0.978 RM 0.829 0.745 0.957

GAReH 0.714 0.798 0.930 0.926 GAReH 0.637 0.706 0.800 0.909

TABLE 3.2. Pairwise Correlations of Foreign Exchange Volatility. Post 1999.

127

USD JPY Int. IV HV RM GH Adj R2 Int. IV HV RM GH Adj R2

2.031 0.785 0.307 0.773 0.897 0.370 0.984 0.096 1.019 0.094

5.787 0.455 0.207 4.894 0.563 0.315 0.729 0.073 0.841 0.078

4.872 0.536 0.228 4.071 0.627 0.330 0.873 0.086 0.888 0.082

1.846 0.789 0.223 3.965 0.645 0.322 1.320 0.123 0.863 0.079

2.104 0.735 0.045 0.307 1.306 0.670 0.187 0.381 0.970 0.120 0.081 0.976 0.143 0.113

2.065 0.747 0.036 0.307 1.226 0.668 0.193 0.378 0.964 0.133 0.111 0.949 0.146 0.129

1.458 0.683 0.152 0.310 1.092 0.669 0.207 0.380 1.247 0.121 0.157 0.979 0.141 0.120

0.845 0.734 0.006 -0.137 0.283 0.311 1.244 0.669 0.168 -0.064 0.090 0.381 1.617 0.132 0.145 0.209 0.268 0.953 0.145 0.170 0.176 0.166

GBP JPYjUSD Int. IV HV RM GH Adj R2 Int. IV HV RM GH Adj R2

1.654 0.749 0.342 0.838 0.876 0.324 0.589 0.072 1.566 0.151

4.152 0.465 0.217 5.028 0.537 0.286 0.631 0.071 1.231 0.123

3.735 0.513 0.218 4.262 0.599 0.302 0.757 0.087 1.315 0.130

3.219 0.582 0.235 1.206 0.851 0.287 0.621 0.068 1.958 0.181

1.639 0.769 -0.018 0.342 1.556 0.593 0.231 0.343 0.563 0.127 0.118 1.240 0.180 0.180

1.627 0.847 -0.098 0.344 1.521 0.558 0.268 0.342 0.581 0.156 0.157 1.255 0.179 0.192

1.552 0.661 0.104 0.345 -0.124 0.586 0.376 0.345 0.606 0.095 0.098 1.936 0.163 0.271

1.259 0.816 0.011 -0.347 0.319 0.359 0.643 0.541 0.089 0.043 0.227 0.346 0.544 0.145 0.117 0.163 0.140 2.032 0.181 0.220 0.167 0.289

TABLE 3.3. 1-Month Volatility Level Predictability Regressions. Full Sample.

128

USD JPY Int. IV HV RM GH Adj R2 Int. IV HV RM GH Adj R2

3.308 0.674 0.210 0.808 0.911 0.349 1.341 0.123 1.829 0.153

6.445 0.398 0.150 4.653 0.589 0.333 1.094 0.095 1.283 0.106

6.399 0.405 0.189 5.206 0.548 0.332 0.893 0.079 1.103 0.086

2.145 0.780 0.199 5.536 0.543 0.288 1.603 0.145 1.032 0.079

3.361 0.645 0.024 0.210 1.770 0.578 0.253 0.365 1.353 0.220 0.154 1.758 0.224 0.153

3.860 0.457 0.172 0.222 1.954 0.565 0.253 0.371 1.320 0.183 0.120 1.788 0.220 0.119

1.538 0.422 0.412 0.237 1.413 0.692 0.180 0.361 1.551 0.164 0.204 1.905 0.234 0.119

1.128 0.513 -0.120 0.017 0.459 0.239 2.107 0.501 0.110 0.198 -0.003 0.372 2.245 0.218 0.151 0.190 0.333 1.765 0.251 0.158 0.236 0.226

GBP JPYjUSD Int. IV HV RM GH Adj R2 Int. IV HV RM GH Adj R2

3.811 0.510 0.158 1.526 0.821 0.269 1.743 0.195 2.667 0.254

5.247 0.337 0.112 5.598 0.493 0.235 1.289 0.139 1.396 0.141

5.279 0.335 0.134 5.750 0.484 0.262 1.172 0.127 0.962 0.099

4.980 0.384 0.125 0.827 0.879 0.229 1.152 0.129 1.656 0.149

3.818 0.461 0.049 0.159 2.207 0.572 0.199 0.283 1.735 0.204 0.123 2.568 0.278 0.119

3.945 0.375 0.121 0.164 2.654 0.476 0.262 0.299 1.745 0.215 0.077 2.452 0.269 0.085

3.657 0.374 0.162 0.170 -0.488 0.563 0.429 0.298 1.732 0.218 0.057 2.213 0.259 0.131

3.649 0.375 0.002 -0.005 0.166 0.169 1.055 0.473 0.032 0.125 0.239 0.301 1.736 0.212 0.170 0.113 0.065 2.622 0.278 0.130 0.122 0.159

TABLE 3.4. 3-Month Volatility Level Predictability Regressions. Full Sample.

129

USD JPY Int. IV RV RM GR Adj R2 Int. IV RV RM GR Adj R2

2.763 0.668 0.525 1.487 0.888 0.541 0.721 0.063 0.903 0.067

3.628 0.643 0.411 2.744 0.779 0.582 0.885 0.085 0.997 0.080

4.948 0.535 0.326 3.460 0.797 0.518 0.844 0.080 1.193 0.107

2.821 0.746 0.329 2.357 0.868 0.467 1.166 0.112 1.652 0.140

2.757 0.664 0.005 0.524 1.693 0.340 0.524 0.603 0.744 0.116 0.123 0.849 0.152 0.149

2.649 0.619 0.067 0.527 1.478 0.528 0.392 0.579 0.685 0.091 0.070 0.884 0.160 0.174

2.282 0.608 0.116 0.528 0.538 0.618 0.365 0.575 0.738 0.089 0.095 1.009 0.137 0.180

1.881 0.639 -0.035 -0.110 0.268 0.528 1.123 0.229 0.444 0.046 0.206 0.616 0.841 0.122 0.126 0.096 0.138 0.941 0.159 0.144 0.195 0.197

GBP JPYjUSD Int. IV RV RM GR Adj R2 Int. IV RV RM GR Adj R2

1.971 0.816 0.648 3.850 0.529 0.330 0.586 0.073 0.995 0.094

1.701 0.803 0.641 4.502 0.532 0.285 0.785 0.094 1.054 0.116

3.651 0.657 0.394 5.156 0.489 0.238 0.796 0.098 1.066 0.125

3.480 0.667 0.393 2.744 0.679 0.231 0.807 0.096 1.796 0.183

1.593 0.442 0.396 0.669 3.605 0.383 0.190 0.341 0.659 0.142 0.171 1.039 0.128 0.132

2.093 0.902 -0.108 0.651 3.403 0.411 0.188 0.349 0.613 0.131 0.121 1.055 0.102 0.112

1.933 0.801 0.021 0.647 2.242 0.411 0.284 0.354 0.630 0.093 0.087 1.481 0.089 0.155

1.420 0.526 0.537 -0.631 0.398 0.699 2.215 0.367 0.085 -0.036 0.285 0.354 0.645 0.127 0.193 0.194 0.119 1.783 0.124 0.148 0.175 0.255

TABLE 3.5. 1-Month Volatility Level Predictability Regressions. High Fre-quency. Post 1999.

130

USD JPY Iut. IV HV RM GH Adj R2 Iut. IV HV RM GH Adj R2

0.969 0.870 0.385 -1.292 1.078 0.490 1.385 0.124 1.481 0.121

5.566 0.477 0.225 4.538 0.627 0.374 1.035 0.090 1.280 0.099

4.715 0.553 0.240 3.571 0.695 0.386 1.171 0.099 1.331 0.099

2.710 0.744 0.234 3.151 0.713 0.341 1.542 0.134 1.550 0.110

0.979 0.861 0.008 0.385 -0.879 0.913 0.134 0.495 1.398 0.172 0.099 1.505 0.185 0.106

0.936 0.921 -0.050 0.385 -1.071 0.970 0.090 0.491 1.397 0.205 0.137 1.540 0.233 0.151

1.287 0.981 -0.145 0.387 -1.280 1.048 0.029 0.490 1.365 0.226 0.204 1.488 0.213 0.136

3.134 1.008 0.286 0.146 -0.780 0.397 -0.361 1.004 0.378 -0.042 -0.323 0.502 1.435 0.243 0.239 0.137 0.464 1.417 0.237 0.139 0.257 0.156

GBP JPYjUSD Iut. IV HV RM GH Adj R2 Iut. IV HV RM GH Adj R2

0.224 0.879 0.486 5.051 0.429 0.127 0.992 0.122 1.712 0.161

3.521 0.533 0.281 6.846 0.298 0.086 0.767 0.103 1.051 0.110

2.790 0.620 0.310 6.673 0.312 0.071 0.841 0.108 1.188 0.121

2.647 0.630 0.290 4.750 0.467 0.087 0.911 0.113 1.695 0.160

0.118 0.984 -0.101 0.489 4.950 0.343 0.106 0.132 1.023 0.177 0.103 1.694 0.204 0.127

0.170 1.001 -0.125 0.489 5.012 0.411 0.024 0.126 1.012 0.209 0.145 1.664 0.231 0.164

0.254 0.956 -0.086 0.488 4.530 0.363 0.116 0.128 1.011 0.177 0.118 1.722 0.222 0.204

0.076 0.996 -0.081 -0.093 0.064 0.489 4.966 0.397 0.285 -0.317 0.073 0.139 1.040 0.207 0.168 0.232 0.179 1.708 0.228 0.159 0.215 0.254

TABLE 3.6. 1-Month Volatility Level Predictability Regressions. Daily. Post 1999.

131

USD JPY Int. IV RV RM GR Adj R2 Int. IV RV RM GR Adj R2

2.986 0.641 0.442 -0.240 1.019 0.571 1.275 0.103 1.714 0.114

3.617 0.640 0.370 1.002 0.896 0.674 1.578 0.145 1.205 0.096

6.166 0.412 0.246 4.003 0.747 0.499 1.047 0.093 1.650 0.135

3.372 0.699 0.247 1.216 0.937 0.415 1.742 0.165 2.722 0.208

2.622 0.493 0.198 0.453 0.133 0.247 0.722 0.681 1.412 0.171 0.206 1.347 0.254 0.220

2.985 0.636 0.006 0.442 0.238 0.723 0.281 0.593 1.271 0.171 0.135 1.801 0.253 0.204

2.830 0.623 0.037 0.442 -1.225 0.828 0.278 0.587 1.502 0.163 0.217 1.650 0.200 0.217

1.784 0.516 0.247 -0.179 0.186 0.456 -1.472 0.154 0.733 -0.167 0.374 0.691 1.853 0.195 0.208 0.157 0.220 1.417 0.296 0.209 0.166 0.180

GBP JPY/USD Int. IV RV RM GR Adj R2 Int. IV RV RM GR Adj R2

1.707 0.839 0.624 4.664 0.441 0.232 0.794 0.101 1.055 0.097

1.984 0.762 0.549 5.601 0.407 0.170 1.107 0.128 1.202 0.123

4.806 0.510 0.270 5.974 0.396 0.203 1.081 0.109 1.017 0.116

4.278 0.569 0.249 1.517 0.757 0.193 1.246 0.129 2.118 0.201

1.491 0.662 0.191 0.630 4.439 0.367 0.107 0.236 0.862 0.145 0.136 1.180 0.107 0.123

1.821 1.184 -0.386 0.673 4.210 0.300 0.221 0.270 0.733 0.233 0.183 1.056 0.118 0.143

2.238 1.019 -0.258 0.645 1.496 0.313 0.430 0.274 0.753 0.152 0.126 2.069 0.108 0.239

0.269 0.839 0.579 -1.010 0.525 0.730 1.709 0.264 0.069 0.028 0.373 0.275 0.983 0.191 0.176 0.331 0.202 1.692 0.115 0.115 0.164 0.184

TABLE 3.7. 3-Month Volatility Level Predictability Regressions. High Fre-quency. Post 1999.

132

USD JPY Int. IV RV RM GR Adj R2 Int. IV RV RM GR Adj R2

0.791 0.881 0.390 -3.104 1.224 0.545 2.116 0.182 2.340 0.170

4.781 0.548 0.259 3.079 0.726 0.420 2.385 0.213 2.053 0.134

5.718 0.464 0.234 4.114 0.656 0.407 1.639 0.144 2.048 0.133

2.660 0.762 0.238 2.799 0.722 0.333 2.457 0.223 2.820 0.179

0.752 0.911 -0.027 0.390 -2.869 1.133 0.071 0.545 1.855 0.292 0.336 2.322 0.285 0.197

0.542 0.985 -0.086 0.392 -3.005 1.196 0.020 0.544 1.874 0.214 0.195 2.406 0.270 0.149

1.123 1.002 -0.160 0.393 -3.114 1.278 -0.050 0.545 2.490 0.217 0.314 2.343 0.240 0.137

1.234 0.979 0.056 -0.011 -0.191 0.392 -2.147 1.169 0.035 0.181 -0.224 0.547 1.995 0.308 0.347 0.152 0.291 2.357 0.317 0.227 0.159 0.087

GBP JPYjUSD Int. IV RV RM GR Adj R2 Int. IV RV RM GR Adj R2

-0.449 0.959 0.611 7.144 0.235 0.043 1.059 0.147 1.654 0.143

2.411 0.669 0.394 9.095 0.069 0.003 0.959 0.117 1.178 0.097

3.058 0.595 0.381 8.471 0.133 0.ü18 0.818 0.105 1.065 0.095

2.225 0.697 0.323 6.677 0.277 0.022 1.109 0.140 1.896 0.159

-0.461 1.012 -0.055 0.611 7.438 0.295 -0.096 0.047 1.074 0.298 0.215 1.607 0.202 0.162

-0.620 1.094 -0.125 0.615 7.103 0.221 0.020 0.042 1.165 0.257 0.155 1.566 0.219 0.168

-0.300 1.064 -0.133 0.615 6.535 0.205 0.085 0.044 1.062 0.218 0.134 1.989 0.210 0.281

-0.548 1.083 0.033 -0.113 -0.043 0.615 7.302 0.259 -0.199 0.129 0.027 0.054 0.991 0.311 0.252 0.171 0.108 1.358 0.226 0.140 0.213 0.212

TABLE 3.8. 3-Month Volatility Level Predictability Regressions. Daily. Post 1999.

133

I-Month 3-Month USD MSE Bias2 Inef!. Residual MSE Bias2 Inef!. Residual

Implied 7.99 1.25 3.16 95.58 6.46 1.43 5.80 92.77 Ristorical 12.08 0.00 27.19 72.81 9.16 0.05 28.80 71.15

RiskMetrics 10.47 0.12 18.18 81.69 9.34 0.00 33.43 66.57 GAReR 9.04 2.80 1.96 95.24 6.34 1.24 1.91 96.85

JPY MSE Bias2 Inef!. Residual MSE Bias2 Inef!. Residual Implied 14.42 1.29 0.76 97.94 11.20 0.49 0.51 99.01

Ristorical 19.76 0.01 21.70 78.29 14.22 0.08 19.53 80.38 RiskMetrics 17.82 0.24 14.83 84.93 15.33 0.00 25.32 74.68

GAReR 17.51 0.01 12.54 87.45 15.98 1.81 21.86 76.33

GBP MSE Bias2 Inef!. Residual MSE Bias2 Inef!. Residual Implied 6.09 2.45 5.38 92.16 5.92 0.29 14.78 84.93

Ristorical 9.25 0.00 26.83 73.17 7.98 0.00 33.01 66.98 RiskMetrics 8.47 0.08 20.15 79.78 8.39 0.06 37.81 62.12

GAReR 7.66 0.02 13.68 86.30 7.29 1.11 26.70 72.19

JPYjUSD MSE Bias2 Inef!. Residual MSE Bias2 Inef!. Residual Implied 14.89 2.31 0.93 96.75 12.19 2.56 1.67 95.77

Ristorical 19.78 0.00 22.96 77.04 16.25 0.02 24.53 75.45 RiskMetrics 17.81 0.15 16.17 83.68 16.61 0.02 28.75 71.23

GAReR 15.66 1.56 1.21 97.23 12.78 2.72 0.55 96.73

TABLE 3.9. MSE and Mincer-Zarnowitz (%) Decomposition. Full sample.

I-Month 3-Month USD MSE Bias2 Inef!. Residual MSE Bias2 Inef!. Residual

Implied 5.14 4.61 1.42 93.97 3.81 8.44 0.86 90.70 Ristorical 8.46 0.00 25.88 74.12 5.34 0.24 17.22 82.53

RiskMetrics 7.34 0.11 16.90 82.99 6.28 0.07 26.57 73.36 GAReR 6.37 0.06 3.27 96.67 4.69 0.10 1.91 98.00

JPY MSE Bias2 Inef!. Residual MSE Bias2 Inef!. Residual Implied 11.89 2.22 0.04 97.75 10.03 3.02 0.15 96.83

Ristorical 17.07 0.36 18.30 81.34 12.94 3.83 10.59 85.59 RiskMetrics 15.64 1.24 11.47 87.29 14.03 1.73 17.52 80.75

GAReR 16.41 2.78 8.25 88.97 14.96 8.08 7.69 84.23

GBP MSE Bias2 Inef!. Residual MSE Bias2 Inef!. Residual Implied 4.49 12.58 2.52 84.90 2.67 21.22 0.53 78.25

Ristorical 6.78 0.00 23.71 76.29 3.84 0.38 14.29 85.32 RiskMetrics 5.87 0.16 15.00 84.84 4.36 0.05 22.76 77.20

GAReR 5.91 0.89 12.67 86.43 4.05 0.51 8.67 90.82

JPYjUSD MSE Bias2 Inef!. Residual MSE Bias2 Inef!. Residual Implied 8.82 22.55 18.67 58.77 9.63 31.58 29.38 39.04

Ristorical 8.50 0.49 30.52 69.00 8.36 4.56 44.61 50.82 RiskMetrics 7.84 1.49 24.17 74.34 8.16 2.65 46.19 51.17

GAReR 7.33 13.90 6.11 79.98 7.01 32.79 7.45 59.76

TABLE 3.10. MSE and Mincer-Zarnowitz (%) Decomposition. Post 1999.

USD JPY GBP JPY /USD p = .90 Estimate t-stat Estimate t-stat Estimate t-stat Estimate t-stat Constant 2.628 5.366 2.1084.113 3.154 6.450 3.579 7.239

Lag hit -0.050 -0.201 -0.057 -0.221 0.328 1.349 -0.210 -0.689 1 month IV -0.048 -1.412 0.022 0.621 -0.133 -3.059 -0.090 -2.567 Average Hit 0.887 -1.362 0.911 1.259 0.912 1.231 0.912 1.389

Stats Wald Test 481.5312

p-val 0.00

Stats p-val 469.570.0000

Stats p-val Stats 369.32 0.0000 524.5237

p-val 0.00

p=.70 Estimate t-stat Estimate t-stat Estimate t-stat Estimate t-stat Constant 2.119 4.746 1.1572.580 2.741 7.155 2.214 4.575

Lag hit -0.363 -2.308 -0.075 -0.471 -0.106 -0.650 -0.030 -0.198 1 month IV -0.101-2.701 -0.011-0.342 -0.183-4.450 -0.103-2.694 Average Hit 0.681 -0.932 0.724 1.227 0.755 2.689 0.727 1.382

Wald Test Stats 75.17

p-val 0.00

Stats p-val 97.41 0.00

Stats 121.64

p-val 0.00

Stats p-val 117.24 0.00

p=.50 Estimate t-stat Estimate t-stat Estimate t-stat Estimate t-stat Constant 1.289 2.575 0.496 1.126 2.138 5.280 1.201 2.702

Lag hit -0.313 -2.058 -0.088 -0.589 0.049 0.326 -0.019 -0.126 1 month IV -0.106 -2.403 -0.026 -0.771 -0.230 -5.099 -0.094 -2.508 Average Hit 0.494 -0.232 0.534 1.283 0.570 2.521 0.526 1.020

Stats p-val Stats p-val Stats p-val Stats p-val Wald Test 10.29 0.02 2.46 0.48 30.65 0.00 8.18 0.04

p=.30 Estimate t-stat Estimate t-stat Estimate t-stat Estimate t-stat Constant 0.740 1.149 -0.133 -0.253 1.204 2.676 0.515 0.909

Lag hit -0.491 -2.560 -0.127 -0.628 0.662 3.384 -0.170 -0.774 1 month IV -0.149 -2.501 -0.055 -1.282 -0.253 -4.772 -0.125 -2.502 Average Hit 0.271 -1.196 0.306 0.234 0.370 2.293 0.284 -0.617

Stats p-val Stats p-val Stats p-val Stats p-val Wald Test 78.32 0.00 42.28 0.00 56.28 0.00 55.30 0.00

p=.10 Estimate t-stat Estimate t-stat Estimate t-stat Estimate t-stat Constant -0.777 -0.798 -1.684 -1.957 0.255 0.415 -1.720 -1.330

Lag hit -2.611-2.474 0.001 0.002 1.561 3.919 -0.609 -0.854 1 month IV -0.168 -1.913 -0.046 -0.611 -0.296 -3.782 -0.081 -0.698 Average Hit 0.066 -2.628 0.097 -0.139 0.166 2.424 0.065 -2.411

Stats p-val Stats p-val Stats p-val Stats p-val Wald Test 187.58 0.00 119.00 0.00 115.54 0.00 130.13 0.00

TABLE 3.11. Interval Logit Regressions. Full Sample.

134

135

USD JPY GBP JPYjUSD p = .90 Estimate t-stat Estimate t-stat Estimate t-stat Estimate t-stat Constant 1.840 2.003 0.454 0.575 1.961 2.695 4.779 6.955

Lag hit 0.142 0.337 0.078 0.234 -0.032 -0.085 -0.407 -1.167 1 month IV 0.012 0.168 0.129 2.110 0.014 0.190 -0.176 -4.327 Average Rit 0.895 -0.364 0.894 -0.391 0.889 -0.653 0.910 0.781

Stats p-val Stats p-val Stats p-val Stats p-val Wald Test 248.96 0.00 196.97 0.00 155.36 0.00 248.07 0.00

p=.70 Estimate t-stat Estimate t-stat Estimate t-stat Estimate t-stat Constant 1.983 2.057 -0.085 -0.106 1.447 2.488 3.630 4.418

Lag hit -0.192 -0.736 -0.112 -0.508 -0.496 -2.181 -0.516 -1.984 1 month IV -0.104 -1.274 0.076 1.327 -0.039 -0.536 -0.204 -3.371 Average Rit 0.673 -0.782 0.687 -0.404 0.689 -0.370 0.702 0.060

Stats p-val Stats p-val Stats p-val Stats p-val Wald Test 19.99 0.00 30.14 0.00 37.45 0.00 40.29 0.00

p=.50 Estimate t-stat Estimate t-stat Estimate t-stat Estimate t-stat Constant 1.359 1.224 -0.618 -0.782 0.200 0.286 2.290 2.962

Lag hit -0.203 -0.867 -0.128 -0.609 -0.388 -1.906 -0.287 -1.116 1 month IV -0.117 -1.183 0.046 0.795 -0.036 -0.439 -0.177 -2.783 Average Rit 0.488 -0.280 0.474 -0.622 0.441 -1.610 0.521 0.520

Stats p-val Stats p-val Stats p-val Stats p-val Wald Test 2.47 0.48 1.33 0.72 5.53 0.14 8.86 0.03

p=.30 Estimate t-stat Estimate t-stat Estimate t-stat Estimate t-stat Constant -0.001 -0.001 -1.351 -1.436 -1.438 -1.322 2.300 2.076

Lag hit -0.026 -0.089 -0.073 -0.230 -0.568 -1.786 -0.491 -1.619 1 month IV -0.086 -0.684 0.022 0.335 0.012 0.102 -0.290 -2.921 Average Rit 0.278 -0.539 0.245 -1.279 0.201 -3.287 0.260 -1.020

Stats p-val Stats p-val Stats p-val Stats p-val Wald Test 22.02 0.00 22.26 0.00 57.68 0.00 32.64 0.00

p=.10 Estimate t-stat Estimate t-stat Estimate t-stat Estimate t-stat Constant -3.659 -1.613 -3.552 -2.146 -4.342 -2.082 0.444 0.158

Lag hit -100.301-221.183 0.650 0.831 -100.609 -167.009 -100.238 -154.415 1 month IV 0.067 0.343 0.044 0.328 0.044 0.190 -0.353 -1.285 Average Rit 0.046 -4.436 0.048 -2.577 0.019 -11.347 0.032 -6.065

Wald Test 121313.20 0.00 60.33 0.00 65046.61 0.00 62950.95 0.00

TABLE 3.12. Interval Logit Regressions. Post 1999.

136

USD JPY GBP JPY/USD Estimate t-stat Estimate t-stat Estimate t-stat Estimate t-stat

Mean 0.072 1.018 -0.297 -4.031 -0.024 -0.278 -0.040 -0.525 Var -0.201 -3.284 -0.070 -0.809 0.343 2.244 -0.073 -0.838

Skew 0.163 0.732 -0.033 -0.120 0.490 1.511 -0.359 -1.243 Kurt -0.299 -0.727 0.180 0.297 1.153 1.247 0.031 0.043

Stats p-val Stats p-val Stats p-val Stats p-val Wald-test 50.56 0.00 64.02 0.00 29.61 0.00 7.49 0.11

TABLE 3.13. GMM Test for Unconditional Normality. Full Sample.

USD JPY GBP JPY/USD Estimate t-stat Estimate t-stat Estimate t-stat Estimate t-stat

Const 0.621 1.962 -0.346 -1.271 0.780 2.762 -0.088 -0.307 Lag LHS 0.128 2.295 0.145 2.573 0.304 4.328 0.120 1.911

IMIV(-21) -0.050 -1.861 0.010 0.448 -0.095 -3.115 0.006 0.247

Const 0.030 0.197 0.128 0.726 0.854 3.635 0.160 0.929 Lag LHS -0.122 -3.540 -0.023 -0.463 0.332 3.150 -0.011 -0.266

IMIV(-21)2 -0.002 -2.046 -0.001 -1.154 -0.009 -4.003 -0.002 -1.751

Const 0.605 1.563 -0.239 -0.680 0.860 2.083 -0.365 -0.956 Lag LHS 0.021 0.911 0.093 2.185 0.328 2.665 0.0682.239

IMIV(-21)3 0.000 -1.674 0.000 1.069 -0.001 -2.258 0.000 0.413

Const 0.165 0.279 0.334 0.610 1.675 1.861 0.170 0.214 Lag LHS -0.047 -2.084 -0.001 -0.048 0.312 2.152 -0.014 -0.833

IMIV(-21)4 0.000 -1.815 0.000 -0.541 0.000 -2.710 0.000 -1.208

Stats p-val Stats p-val Stats p-val Stats p-val Wald-test 106.61 0.00 157.65 0.00 118.43 0.00 50.72 0.00

TABLE 3.14. GMM Test for Condition al Normality. Full Sample.

137

USD JPY GBP JPYjUSD Estimate t-stat Estimate t-stat Estimate t-stat Estimate t-stat

Mean -0.017 -0.147 -0.017 -0.159 -0.047 -0.508 -0.032 -0.294 Var -0.230 -2.723 -0.217 -1. 797 -0.392 -5.851 -0.256 -3.218

Skew 0.244 0.816 0.370 0.844 0.237 0.782 0.089 0.302 Kurt -0.693 -1.839 -0.136 -0.140 -0.685 -1.450 -0.772 -1.888

Stats p-val Stats p-val Stats p-val Stats p-val Wald-test 32.09 0.00 26.15 0.00 308.11 0.00 38.12 0.00

TABLE 3.15. GMM Test for Unconditional Normality. Post 1999.

USD JPY GBP JPYjUSD Estimate t-stat Estimate t-stat Estimate t-stat Estimate t-stat

Const 1.035 1.608 0.562 1.349 0.440 1.077 -0.016 -0.034 Lag LHS 0.191 1.874 0.089 0.820 0.056 0.663 0.076 0.749

1MIV(-21) -0.093 -1.580 -0.044 -1.331 -0.057 -1.196 -0.001 -0.038

Const -0.172 -0.505 -0.330 -1.370 -0.422 -2.056 0.187 1.118 Lag LHS -0.025 -0.444 0.024 0.283 -0.075 -1.240 -0.115 -1.925

1MIV(-21)2 0.000 -0.177 0.001 0.519 0.000 -0.089 -0.003 -3.589

Const 0.856 1.110 0.307 0.563 0.507 0.938 0.016 0.033 Lag LHS 0.111 1.633 0.138 1.487 0.052 0.655 0.076 1.116

1MIV(-21)3 0.000 -0.766 0.000 0.058 0.000 -0.640 0.000 0.242

Const -1.172 -1.305 -0.636 -0.832 -0.869 -1.142 -0.145 -0.263 Lag LHS -0.032 -0.820 0.040 0.692 -0.031 -0.631 -0.065 -1.914

1MIV(-21)4 0.000 0.612 0.000 0.591 0.000 0.156 0.000 -2.871

Stats p-val Stats p-val Stats p-val Stats p-val Wald-test 81.38 0.00 169.79 0.00 439.99 0.00 77.57 0.00

TABLE 3.16. GMM Test for Conditional Normality. Post 1999.

Mean Var

Skew Kurt

USD JPY GBP JPYjUSD Estimate t-stat Estimate t-stat Estimate t-stat Estimate t-stat

0.301 3.831 -0.036 -0.421 -0.025 -0.262 0.015 0.140 -0.292 -4.780 0.055 0.475 0.089 0.629 0.069 0.413 0.143 0.522 -0.219 -0.714 -0.451-1.229 -0.359 -0.813

-0.235 -0.566 0.345 0.508 0.401 0.473 0.558 0.543

Estimate p-val Estimate p-val Estimate p-val Estimate p-val Coverage 0.217 0.00 0.357 0.00 0.287 0.00 0.264 0.00

Stats p-val Stats p-val Stats p-val Stats p-val Wald-test 723.02 0.00 156.09 0.00 288.62 0.00 262.72 0.00

138

TABLE 3.17. Unconditional Normality Test for the Density Forecast Between o and .25.

USD JPY GBP JPYjUSD Estimate t-stat Estimate t-stat Estimate t-stat Estimate t-stat

Const -0.351 -0.801 -0.578 -1.074 -0.692 -1.270 0.581 0.657 Lag LHS -0.144 -1.574 -0.035 -0.415 0.2423.204 0.003 0.026

1MIV(-21) 0.083 2.153 0.045 1.030 0.053 0.961 -0.052 -0.576

Const -0.484 -2.445 0.679 1.542 0.941 1.116 -0.763 -1.968 Lag LHS -0.164 -1.259 -0.012 -0.164 -0.009 -0.115 -0.011-0.171

1MIV(-21)2 0.002 0.968 -0.005 -2.029 -0.008-1.059 0.0062.040

Const -0.057 -0.103 -1.121 -1.170 -2.166-1.304 -0.175 -0.205 Lag LHS 0.033 0.219 -0.061 -1.323 0.060 0.914 -0.030 -0.761

1MIV(-21)3 0.001 3.140 0.000 1.145 0.001 0.995 0.000 -0.306

Const -0.249 -0.309 2.092 1.154 4.185 1.164 -1.189-0.785 Lag LHS -0.002 -0.015 -0.029 -0.768 -0.038 -0.691 -0.027 -0.783

IMIV(-21)4 0.000 1.541 0.000 -2.031 0.000-1.328 0.000 1.340

Estimate p-val Estimate p-val Estimate p-val Estimate p-val Coverage 0.217 0.00 0.357 0.00 0.287 0.00 0.264 0.00

Stats p-val Stats p-val Stats p-val Stats p-val Wald-test 694.07 0.00 221.47 0.00 369.98 0.00 1019.75 0.00

TABLE 3.18. Conditional Normality Test for the Density Forecast Between 0 and .25.

139

USD JPY GBP JPY/USD Estimate t-stat Estimate t-stat Estimate t-stat Estimate t-stat

Mean -0.055 -0.979 -0.137 -2.375 -0.189 -2.934 0.024 0.413 Var -0.066 -1.445 0.050 0.900 0.034 0.535 0.072 1.314

Skew 0.142 0.801 0.188 1.139 0.061 0.340 0.058 0.357 Kurt 0.223 0.580 0.061 0.190 -0.031 -0.094 -0.102 -0.339

Estimate p-val Estimate p-val Estimate p-val Estimate p-val Coverage 0.515 0.56 0.477 0.38 0.453 0.08 0.475 0.32

Stats p-val Stats p-val Stats p-val Stats p-val Wald-test 20.75 0.00 87.05 0.00 82.90 0.00 13.77 0.02

TABLE 3.19. Unconditional Normality Test for the Density Forecast Between 0.25 and. 75.

USD JPY GBP JPY/USD Estimate t-stat Estimate t-stat Estimate t-stat Estimate t-stat

Const 0.080 0.201 0.315 1.182 0.499 1.292 0.374 1.361 Lag LHS -0.005 -0.098 -0.063 -0.924 0.050 0.906 -0.093-1.598

1MIV(-21) -0.012 -0.377 -0.036 -1.623 -0.098 -2.350 -0.026-1.235

Const 0.004 0.023 -0.322 -2.130 0.350 1.910 0.280 1.866 Lag LHS -0.033 -1.237 0.053 0.889 0.001 0.022 0.060 1.435

1MIV(-21)2 0.000 -0.394 0.003 2.713 -0.002 -1.487 -0.001-1.204

Const 0.279 0.714 0.613 2.169 0.100 0.241 0.269 1.012 Lag LHS 0.010 0.428 -0.009 -0.126 0.033 0.728 -0.041-0.924

1MIV(-21)3 0.000 -0.875 0.000 -1.655 -0.001 -1.683 0.000 -1.615

Const 0.010 0.018 -0.530 -0.962 0.826 1.443 0.440 0.859 Lag LHS -0.031-2.251 0.072 0.933 -0.010 -0.271 0.014 0.764

1MIV(-21)4 0.000 0.074 0.000 1.785 0.000 -1.862 0.000 -1.165

Estimate p-val Estimate p-val Estimate p-val Estimate p-val Coverage 0.515 0.56 0.477 0.38 0.453 0.08 0.475 0.32

Stats p-val Stats p-val Stats p-val Stats p-val Wald-test 20.75 0.08 49.12 0.00 92.26 0.00 15.04 0.31

TABLE 3.20. Conditional Normality Test for the Density Forecast Between 0.25 and .75.

140

USD JPY GBP JPY/USD Estimate t-stat Estimate t-stat Estimate t-stat Estimate t-stat

Mean -0.075 -0.978 -0.212 -1.893 0.256 1.844 -0.280 -3.476 Var -0.205 -2.610 -0.231 -1.755 0.6372.504 -0.309 -4.462

Skew 0.010 0.036 0.403 0.866 0.729 1.791 -0.213 -0.633 Kurt 0.306 0.462 -0.129 -0.125 0.663 0.761 0.232 0.394

Estimate p-val Estimate p-val Estimate p-val Estimate p-val Coverage 0.268 0.00 0.166 0.00 0.260 0.00 0.261 0.00

Stats p-val Stats p-val Stats p-val Stats p-val Wald-test 316.66 0.00 730.00 0.00 316.94 0.00 664.42 0.00

TABLE 3.21. Unconditional Normality Test for the Density Forecast Between 0.75 and 1.

USD JPY GBP JPY /USD Estimate t-stat Estimate t-stat Estimate t-stat Estimate t-stat

Const 0.455 0.845 -1.667 -1.818 1.245 2.968 0.205 0.258 Lag LHS -0.221-2.937 0.031 0.163 0.377 2.264 0.103 1.021

1MIV(-21) -0.051 -0.931 0.149 1.608 -0.122-2.078 -0.022-0.275

Const -0.578 -3.380 -1.035 -1.845 1.171 2.377 -0.394 -0.891 Lag LHS 0.010 0.193 0.018 0.150 0.298 1.603 -0.137-1.674

1MIV(-21)2 0.003 2.281 0.010 1.690 -0.003 -0.556 0.003 0.827

Const 0.127 0.233 -2.203 -1.790 1.989 2.977 0.338 0.247 Lag LHS -0.072 -0.982 -0.093 -0.623 0.431 2.067 0.000 -0.005

1MIV(-21)3 0.000 -0.515 0.002 1.870 -0.001-2.054 0.001 0.426

Const -0.307 -0.332 -4.195 -2.108 3.140 2.147 0.228 0.127 Lag LHS 0.018 0.726 -0.038 -0.267 0.331 1.396 -0.104-1.693

1MIV(-21)4 0.000 0.969 0.000 1.908 0.000 -1.885 0.000 1.122

Estimate p-val Estimate p-val Estimate p-val Estimate p-val Coverage 0.268 0.00 0.166 0.00 0.260 0.00 0.261 0.00

Stats p-val Stats p-val Stats p-val Stats p-val Wald-test 446.29 0.00 723.36 0.00 328.29 0.00 504.71 0.00

TABLE 3.22. Conditional Normality Test for the Density Forecast Between 0.75 and 1.

141

Daily Returns USD IDEM JPY IDEM JPY IDEM JPY IUSD

Mean -8.25E-05 -4. 11E-05 2.21E-05 -3.57E-05 Std. Dev. 0.00712 0.00732 0.00538 0.00788 Skewness -0.110 -0.876 0.653 -0.855 Kurtosis 5.084 10.346 9.267 10.962

Jarque-Bera 305.6291 3783.162 2836.112 4615.042 Observations 1670 1592 1661 1670

Squared Daily Returns USD IDEM JPY IDEM JPY IDEM JPY IUSD

Mean 5.07E-05 5.35E-05 2.89E-05 6.21E-05 Std. Dev. 0.00010 0.00016 0.00008 0.00020 Skewness 6.129 11.998 20.852 14.721 Kurtosis 57.905 194.273 634.957 328.912

Jarque-Bera 220217.8 2465015 27760096 7451371 Observations 1670 1592 1661 1670

TABLE 3.23. Descriptive Statistics.

Daily Returns Pairwise Correlations USDIDEM JPY/DEM GBP/DEM

JPY IDEM 0.301 GBP IDEM 0.348 0.087 JPY IUSD -0.467 0.228 -0.203

Squared Daily Returns Pairwise Correlations USD IDEM JPY IDEM GBP IDEM

JPY IDEM 0.155 GBP/DEM 0.124 0.073 JPY IUSD 0.215 0.326 0.045

TABLE 3.24. Descriptive Statistics. Correlation.

Daily Returns USD lEUR JPY/EUR GBP/EUR JPU/USD

Mean -0.000141 -0.000118 -0.000117 2.24E-05 Std. Dev. 0.00692 0.00841 0.00504 0.00666 Skewness 0.422 0.046 0.347 -0.149 Kurtosis 4.512 6.107 4.379 4.581

Jarque-Bera 128.6449 414.6895 102.3379 111.1164 Observations 1030 1030 1030 1030

Squared Daily Returns USD/EUR JPY/EUR GBP/EUR JPU/USD

Mean 4.79E-05 7.07E-05 2.54E-05 4.43E-05 Std. Dev. 0.00009 0.00016 0.00005 0.00008 Skewness 8.472 8.871 4.300 4.836 Kurtosis 139.912 128.608 28.362 35.773

Jarque-Bera 816789.6 690617.9 30778.29 50109.36 Observations 1030 1030 1030 1030

TABLE 3.25. Descriptive Statistics. Post 1999.

Daily Returns Pairwise Correlations USD lEUR JPY lEUR GBP lEUR

JPY lEUR 0.638 GBP lEUR 0.704 0.487 JPY IUSD -0.234 0.600 -0.117

Squared Daily Returns Pairwise Correlations USD lEUR JPY lEUR GBP lEUR

JPY lEUR 0.610 GBP lEUR 0.421 0.212 JPY IUSD 0.134 0.488 0.011

TABLE 3.26. Descriptive Statistics. Correlation Post 1999.

142

143

USD I-Month 3-Month

RV IV HV RM GARCH RV IV HV RM GARCH Mean 10.64 10.94 10.69 10.81 11.17 10.75 11.07 10.90 10.81 11.06

Std. Dev. 3.33 2.34 3.36 3.00 2.01 2.74 1.88 2.72 3.00 1.60 Skewness 1.32 0.90 1.29 1.18 1.43 0.90 0.44 0.94 1.18 1.29 Kurtosis 5.36 4.59 5.19 4.82 6.59 4.08 3.03 3.76 4.82 6.42

Jarque-Bera 1508.0 662.1 1378.7 1072.3 2545.0 530.9 87.1 497.7 1072.3 2207.6 Observations 2893 2748 2893 2893 2893 2893 2748 2893 2893 2893

JPY I-Month 3-Month

RV IV HV RM GARCH RV IV HV RM GARCH Mean 11.21 11.69 11.29 11.46 11.28 11.41 11.73 11.59 11.46 10.92

Std. Dev. 4.72 3.21 4.71 4.32 4.16 4.13 2.68 4.01 4.32 4.07 Skewness 1.17 0.61 1.16 0.96 1.21 0.52 0.44 0.59 0.96 1.02 Kurtosis 4.80 3.87 4.76 4.39 5.25 2.84 2.96 2.83 4.39 4.48

Jarque-Bera 1044.6 256.0 1019.4 679.7 1316.5 131.3 88.3 169.6 679.7 769.1 Observations 2893 2748 2893 2893 2893 2893 2748 2893 2893 2893

GBP I-Month 3-Month

RV IV HV RM GARCH RV IV HV RM GARCH Mean 7.68 8.13 7.66 7.75 7.73 7.83 8.04 7.82 7.75 7.56

Std. Dev. 2.95 2.28 2.97 2.71 2.45 2.47 1.91 2.48 2.71 2.27 Skewness 1.69 0.35 1.68 1.42 2.20 1.01 -0.07 1.03 1.42 2.34 Kurtosis 9.09 3.21 8.99 7.05 13.35 4.83 2.85 4.80 7.05 14.99

Jarque-Bera 5839.9 61.9 5672.6 2945.0 15229.3 894.5 4.7 902.8 2945.0 19959.6 Observations 2893 2747 2893 2893 2893 2893 2747 2893 2893 2893

JPYjUSD I-Month 3-Month

RV IV HV RM GARCH RV IV HV RM GARCH Mean 9.72 11.47 9.72 9.91 11.07 10.03 11.66 10.03 9.91 11.49

Std. Dev. 4.82 3.00 4.82 4.44 2.96 4.20 2.53 4.20 4.44 2.19 Skewness 0.84 1.26 0.84 0.61 3.11 0.23 1.04 0.23 0.61 3.62 Kurtosis 7.22 6.29 7.22 7.29 20.14 6.45 4.39 6.45 7.29 24.95

Jarque-Bera 5813.4 1970.8 5813.4 5643.1 43425.5 3402.8 713.4 3402.8 5643.1 69790.1 Observations 6766 2748 6766 6786 3135 6724 2748 6724 6786 3135

TABLE 3.27. Foreign Exchange Volatility - Descriptive Statistics.

144

.76 1.2

.72

1.1

.68

.64 1.0

.60 0.9

.56

.52 0.8 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003

-USDDEM - USDEUR

90 140

85 130

80

\ J~fl l.fIyd~ 120

75 110

70

100 65

VA~~ 60 90

55 80 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003

-JPYDEM -JPYEUR

.48 .72

.46

f"\ .44 .68

.42

.40 '",j~ .64

.38

.36 .60

.34

.32 .56 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003

-GBPDEM -GBPEUR

150

140

130

120

110

100

90

1994 1996 1998 2000 2002

-JPYUSD

FIGURE 3.1. Foreign Echange Spot Rates, Pre and Post Euro Introduction.

145

40 40

30 30

20 20

10 10

0 0 92 93 94 95 96 97 98 99 00 01 02 92 93 94 95 96 97 98 99 00 01 02

-USD 1M -USD3M

40 40

30

~~~ 30

20 20

10 10

0 0 92 93 94 95 96 97 98 99 00 01 02 92 93 94 95 96 97 98 99 00 01 02

-JPY 1M -JPY3M

40 40

30 30

20 20

10 ~ 10

0 0 92 93 94 95 96 97 98 99 00 01 02 92 93 94 95 96 97 98 99 00 01 02

-GBP 1M -GBP3M

40 40

30 30

20 20

10 10

0 0 92 93 94 95 96 97 98 99 00 01 02 92 93 94 95 96 97 98 99 00 01 02

-JPYUSD 1M -JPYUSD3M

FIGURE 3.2. Implied Volatility Annualized. 1-month (left), 3-month (right).

40r----------------------,

30

20

10

O~~~~~~~~~~~~

-USD 1M

40r----------------------,

30

20

10

O~~~~~~~~~~~~

-JPV 1M

40r----------------------,

30

20

10

O~~~~~~~~~~~~

-GBP 1M

40~---------------------,

30

20

10

O~~~~~~~~~~~~

-JPVUSD 1M

40,----------------------,

30

20

10

O~~~~~~~~~~~~ 92 93 94 95 96 97 98 99 00 01 02

-USD3M

40,----------------------,

30

20

10

O~~~~~~~~~~~~ 92 93 94 95 96 97 98 99 00 01 02

-JPV3M

40r----------------------,

30

20

10

O~~~~~~~~~~~~ 92 93 94 95 96 97 98 99 00 01 02

-GBP3M

40.----------------------,

30

20

10

O~~~~~~~~~~~~ 92 93 94 95 96 97 98 99 00 01 02

-JPVUSD 3M

146

FIGURE 3.3. Historica1 Vo1atility Annua1ized. 1-month (left), 3-month (right).

147

40~----------------~

30

20

10

92 93 94 95 96 97 98 99 00 01 02

-USD 1M and 3M

40~----------------~

30

20

10

92 93 94 95 96 97 98 99 00 01 02

-JPY 1 M and 3M

40r------------------,

30

20

10

92 93 94 95 96 97 98 99 00 01 02

-GBP 1 M and 3M

40r-----------.------,

30

20

10

92 93 94 95 96 97 98 99 00 01 02

-JPYUSD 1M and 3M

FIGURE 3.4. RiskMetrics Volatility Annualized. 1-month (left), 3-month (right).

40~--------------------~

30

20

10

O~~~~~~~~~~~~ 92 93 94 95 96 97 98 99 00 01 02

-USD 1M

40~--------------------~

30

20

10

O~~~~~~~~~~~~

-JPY 1M

40~--------------------~

30

20

10

O~~~~~~~~~~~~

-GBP 1M

40~------------.-------~

30

20

1 0 ~-..\ ... -'..,."., ...

O~~~~~~~~~~~~ 92 93 94 95 96 97 98 99 00 01 02

-JPYUSD 1M

40~--------------------~

30

20

10

O~~~~~~~~~~~~ 92 93 94 95 96 97 98 99 00 01 02

-USD3M

40.---------------------~

30

20

10

O~~~~~~~~~~~~ 92 93 94 95 96 97 98 99 00 01 02

-JPY3M

40~--------------------~

30

20

10

O~~~~~~~~~~~~ 92 93 94 95 96 97 98 99 00 01 02

-GBP3M

40~--------------------~

30

20

92 93 94 95 96 97 98 99 00 01 02

-JPYUSD 3M

148

FIGURE 3.5. Garch Volatility Annualized.1-month (left) , 3-month (right).

1.0-rr---;--T-r-,---------,---,

0.8

0.6

0.4

0.2

0.0+......,...,rr'~".....,...,_,_.-+ ... ~...,..-'f_,_-h-"..~,.-I 1992 1993 1994 1995 1996 1997 1998

- USD Probability - Pre Euro Intoduction

1.0-ro---;--------:---------,

0.8

~!I\ 0.6

0.4

0.2

\~ 0.0 1»

1992 1993 1994 1995 1996 1997 1998

- JPY Probability - Pre Euro Intoduction

1.°TT"'---.-... ...... nr-.-""TTrT'1rT-.,------,

0.8

0.6

0.4

0.2

0.0 +......,...,~,....,..'r-....... "t"~.,..y+-+...,.,r-.u\-,.-"..~--' 1992 1993 1994 1995 1996 1997 1998

- GBP Probability - Pre Euro Intoduction

1.0-r--------y----,--,------,

0.8

0.6

0.4

0.2

O.O-h~f_r_~_h-'r-... ~.....,....,...,~,...~"T'""-."J~ 1992 1993 1994 1995 1996 1997 1998

- USD_JPY Probability - Pre Euro Intoduction

1.0-,--------,--------;---:-,

~ 0.8

0.6

0.4

0.2 ~

- USD Probability - Post Euro Intoductlon

1.0-,-------....... ----.,-----,

0.8

0.6

0.4 i

0.2 ~ 0.0

1999

1.0

0.8

0.6

0.4

0.2

0.0 1999

2000 2001 2002

- JPY Probability - Post Euro Intoduction

2000 2001

- GBP Probability - Post Euro Intoduction

2003

1.0-r-----------;-----,

0.8

0.6

0.4

02

O.O+-.~.;..,..=T""'"~~,....,~=_,..,.,.~..,I,-,~..._/ 1999 2000 2001 2002 2003

- USD_JPY Probability - Post Euro Intoduction

FIGURE 3.6. Risk Neutral Probabilities.

149

.85,-----------------,

.80

.75

.70

.65

.60

.55

.50

USD/DEM

.45-h...._,~~..._~....,....,~..,.~...._,~~..._~...J

1992 1993 1994 1995 1996 1997 1998

100,---------------,

JPY/DEM 90

80

70

60

50-h...._,~~..._~....,....,-..,.~...._,~~..._~...J

1992 1993 1994 1995 1996 1997 1998

1992 1993 1994 1995 1996 1997 1998

160~-------------70--,

150

140

130

120

110

100

90

80

JPYUSD

70-h...._,r_~..._~__,_~..,.~~r_~..._~...J

1992 1993 1994 1995 1996 1997 1998

150

1.2-,-----------------,

1.1

1.0

0.9

0.8

0.7,....~~~J"TT"~~......,.~~~"T'~~~.-f 1999 2000 2001 2002 2003

150,---------------,

140

130

120

110

100

90

2000 2001 2002 2003

.72-.--:-----------------,

.68

.64

.60

.56

2000 2001 2002 2003

150,-----------------,

140

130

120

110

100

2000 2001 2002 2003

FIGURE 3.7. Interval Forecasts, Pre and Post Euro introduction. 10% and 90% intervals.

151

USD JPY 200 200

150 150

100 100

50 50

0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8

GBP JPY/USD 200 200

150 150

100

50

0.2 0.4 0.6 0.8

FIGURE 3.8. Histogram of Probability Transforms with 90% Confidence Band.

USD 300r---------~----------_,

250

200

150

100

50

o '-----=----5 o 5

GBP 300r---------~----------_,

250

200

150

100

50

0'------5 o 5

JPV 300r---------~----------_.

250

200

150

100

50

0"----­-5 o 5

JPV/USD ~Or----------------------.

250

200

150

100

50

0'----­-5 o 5

152

FIGURE 3.9. Histogram of Normal Transforms with Normal Distribution Im­posed.

153

USD JPY 4 4

",.

2 2

0 0

-2 -2 .. -4 -4

-4 -2 0 2 4 -4 -2 0 2 4

GBP JPY/USD 6 4

4 ..

2

2

0

0

-2 -2

-4 -4 -4 -2 0 2 4 -4 -2 0 2 4

FIGURE 3_10_ QQplots of Normal Trasnforms Variables.

154

Chapter 4

FOREIGN EXCHANGE OPTION AND RETURNS

BASED CORRELATION FORECASTS: EVALUATION

AND Two ApPLICATIONS

OIli Castrén Stefano 11azzotta

Abstract. We compare correlation forecasts from a dataset consisting of over 10 years of daily data on over-the-counter (OTC) currency option priees to a set of return-based correlation measures and assess the relative quality of the correlation forecasts. We find that while the predictive power of implied corre­lation is not always superior to that of returns based correlations measures, it tends to provide the most consistent results across currencies. Predictions that use both implied and returns-based correlations generate the highest adjusted R2s, explaining up to 42 per eent of the realized correlations. We then ap­ply the correlation forecasts to two policy-relevant topics, to pro duce scenario analyses for the euro effective exchange rate index, and to analyze the impact on cross-currency co-movement of interventions on the JPY /USD exchange rate.

JEL Classification: F31, F37, G15. Keywords: Correlation forecasts, Currency Options Data, Effective Ex­

change Rate.

155

4.1 Introduction

The purpose of this study is to investigate the extent to which it is possible

to use returns based measures and foreign exchange options based measures

to predict the correlation between bilateral exchange rates. In particular, we

study whether the forward-Iooking information contained in the OTe currency

options data can provide good forecasts of the future realized correlation be­

tween exchange rates by themselves or in addition to various correlation fore­

casts derived from returns based measures. Armed with the results from the

correlation forecast analysis, we then illustrate two different applications of the

methodology for policy related purposes.

There is ample anecdotal evidence that over time, certain currency pairs

tend to move in tandem. In other words, when one of the two exchange

rates appreciates (depreciates), the other tends to follow a similar pattern.

In economic terms, these patterns are interesting from several points of view.

First, the reason why two currency pairs show a positive correlation over time

could be that their dynamics is driven by the same economic fundamentals.

Second, a sudden fall in a historically stable correlation relationship could be

indicative of attempts by policy makers to try to influence the dynamics of some

particular exchange rate. Third, a set of correlations among several exchange

rates could provide an idea about which currencies are facing excess demand

in the foreign exchange market. And fourth, if we have a reliable forecast of

the correlation relationship between, say, the euro and the currencies of two

or more euro area major trading partner economies, then the impact of an

assumed future movement in one of the bilateral exchange rates on the future

movements in the other bilateral exchange rates can be assessed using these

correlation forecasts. For a central bank that uses exchange rates mainly as an

indicator for future infiationary risks, it is important to have a forecast of as

156

many of the bilateral exchange rates entering into the effective exchange rate

basket as possible. Forecasts of correlation provide one way of expanding the

information on future developments received from individu al bilateral exchange

rates.

There is a substantial literature investigating the informational content

of options in relation to asset price returns. Several early contributions use

market-based options data with mixed results to investigate conditional sec­

ond moments, but they almost invariably concentrate on volatility rather than

correlation. Beckers (1981) finds that not aH available information is refiected

in the current option price and questions the efficiency of the option markets.

Canina and Figlewski (1993) find that implied volatility is a poor forecast

of subsequent realized volatility. Lamoureux and Lastrapes (1993) provide

evidence against restrictions of option pricing models which assume that vari­

ance risk is not priced. However, Jorion (1995) finds that statistical models

of volatility based on returns are dominated by implied volatility forecasts

even when the former are given the advantage of ex post in sample parameter

estimation. He also finds evidence of bias. More recently, Christensen and

Prabhala (1998) use longer time series and non-overlapping data and find that

implied volatility outperforms past volatility in forecasting future volatility.

Fleming (1998) finds that implied volatility dominates historical volatility in

terms of ex ante forecasting power and suggests that a linear model which

corrects for the bias present in implied volatility forecasts can provide a use­

fuI market-based estimator of conditional volatility. Blair, Poon, and Taylor

(2001) find that nearly aH relevant information is provided by the VIX in­

dex and there is not much incremental information in high-frequency index

returns. Neely (2003) finds that econometric projections supplement implied

volatility in out-of-sample forecasting and delta hedging. He also provides sorne

explanations for the bias and inefficiency pointing to autocorrelation and mea-

157

surement errors in implied volatility. Pong, Shackleton, Taylor and Xu (2004)

find that high-frequency historie al forecasts are superior to implied volatilities

using OTC data for horizons up to one week. Covrig and Low (2003) use OTC

data to find that quoted implied volatility subsumes the information content

of historically based forecasts at shorter horizons, while the former is as good

as the latter at longer horizons. Finally, Christoffersen and Mazzotta (2004)

systematically assess the quality of option based volatility, interval and density

forecasts for the major currencies 1992-2003. They find that implied volatilities

explain a large share of the variation in realized volatility and that wide-range

interval and density forecasts are often misspecified whereas narrow interval

forecasts are specified better.

It is of course striking that aH of the above studies investigate options

informational content with regard to volatility forecasts. Studies investigat­

ing exchange rate correlations implied by market data are, on the contrary,

rather sparse. 1 The contributions perhaps closest related to our work are

Siegel (1997), Campa and Chang (199S) and Lopez and Walter (2000), who

specifically focus on exchange rate correlations. Campa and Chang find that

implied correlation among the DEM/USD, USD/JPY and DEM/JPY currency

pairs from January 19S9 to May 1995 outperform alternative forecasts at one­

month and three-month horizons. In addition, they find that when included

in joint forecast regressions, implied correlation always incrementally improves

the performance of other forecasts.

In this study, we extend upon the results by Campa and Chang by looking

at several other currencies in a larger sample that also covers the first five

years of the single European currency. In particular, we focus our attention

1 However, there exists a more generous literature in correlations among stock and bond

markets. Good reviews of such studies are provided Kroner and Ng (1998) and Cappiello,

Engle and Sheppard (2003).

158

on the correlations between the following exchange rate pairs: USD /EUR -

JPY/EUR; USD/EUR - GBP/EUR; GBP/EUR - JPY/EUR; USD/GBP -

JPY/GBP; USD/JPY - GBP/JPY; USD/EUR - PLN/EUR; and USD/EUR

- CZK/EUR.2 Our sample starts in January 1992 and ends in March 2004,

except for the Polish zloty and the Czech koruna currency pairs for which the

sample period commences at January 2001. Prior to the launch of the euro in

January 1999, we use data on D-mark currency pairs. This is reflected in our

estimations in that all regressions are run in two samples, the full sample and

the post-January 1999 sample. In the case of the full sample the notation, for

simplicity, refers only to the euro.

We find that the implied correlation calculated from currency options prices

shows predictive power for the future realized correlation among all currency

pairs except the GBP /EUR-JPY /EUR. Rowever, for the exchange rate pairs

that show correlation predictability, implied correlation is not the only one that

pro duces good forecasts. Both GARCR and RiskMetrics correlation forecasts

show substantial predictive power. In substance, the two types of correlations

forecasts seem to ni cely complement each other in that the best forecasts are

often produced when implied and return-based correlations are used jointly.

The highest adjusted R2 is almost invariably obtained from the encompassing

(multivariate) regressions. This result is in contrast with the findings in Campa

and Chang (1998). The total predictability obtained using a combination of

forecasts ranges from 18 to 38 per cent for the entire sample and from 20 to

42 per cent for the post-January 1999 sample.

To shed further light on the relative merits of the various correlation fore­

casts we perform a Mincer-Zarnowitz decomposition of the forecasting error.

We find that different measures of correlation have different informational con-

2The choice of the particular correlation pairs is partially dictated by data availability

on the currency options, as will be discussed in more detail below.

159

tent and therefore they tend to provide the best forecasts when used jointly.

After assessing the relative forecasting properties of the various methodolo­

gies, we apply the correlations measures on two policy relevant cases. In the

first study, the correlation forecasts are employed to generate scenario anal­

ysis for the euro effective exchange rate conditional on assumptions on the

future evolution of the JPY /USD exchange rate. In the second case, we study

whether the interventions by the Japanese authorities on the JPY jUSD ex­

change rate in the 1990s and 2000s have affected the patterns of co-movement

among the JPY /EUR and USD /EUR exchange rates.

The rest of this study is organized as follows. Section 2 introduces the

framework in which the various correlation measures will be analyzed. Section

3 specifies the estimated equations and the reports the results. Section 4

presents the two applications and Section 5 concludes.

4.2 Correlation Forecast Evaluation

4.2.1 Data issues

The currency options data used in this study consists of 1-month implied

volatilities on a large number of exchange rates, obtained from Citigroup. Tra­

ditionally, the bulk of trading in options is on OTC basis and not at centralized

futures/options exchanges. Christensen, Hansen and Prabhala (2001) argue

that in terms of forecasting properties, OTC options data could be of superior

quality relative to exchange traded options. This is because OTC prices are

quoted daily with fixed "moneyness" (the distance between the forward rate

and the option's strike price) in contrast with market-traded options, which

have fixed strike prices and thus time-varying moneyness as the forward ex­

change rate changes. Moreover, the trading volume in OTC options is often

much larger than in the corresponding market traded contracts. The underly-

160

ing liquidity on OTC quotes is therefore deeper, which makes the OTC quotes

a more reliable source for information extraction. The fact that the currency

options market is heavily concentrated on a few global players does that the

liquidity problems can be reduced further if data from these institutions is

available. Citigroup has a significant market share both in options on major

exchange rates as weIl as on the emerging currencies.

4.2.2 The Forecasting Object of Interest

The methodologies we adopt for this study are in several ways similar to

those used to investigate volatility predictability from OTC currency options

in Christoffersen and Mazzotta (2004), with some major differences. The par­

ticular object of interest of our study is forecasting the realized future sample

correlation of an exchange rate pair over the horizon of the following h = 21

trading days.

There exists substantial literature regarding the use of realized volatility

as a measure of equity and foreign exchange variability (see e.g. Andersen

and Bollerslev (1998) and Andersen et al. (2001a, 2001b, 2003)). The com­

mon thread of this literature is the idea that one can sum squared log returns

at a frequency higher than that of interest to obtain a measure of the real­

ized quadratic variation over the frequency of interest. For instance, one can

compute the monthly variance as the sum of squared daily log returns or the

daily variance as the sum of intraday squared log returns. In this theoretical

framework, by increasing the sampling frequency it is possible to construct ex

post realized volatility measures for the integrated latent volatilities that are

asymptotically free of measurement error. In practice, the benefit of increasing

the frequency is offset by the microstructure noise which is invariably included

in the observed market quotes.

One approach commonly taken is to strike a balance between the horizon

161

of interest and the number of sub-periods in which such horizon is divided for

the purpose of computing the squared returns. In the case of daily variance

estimates, whereas early work suggests using 5-minute returns more recent

contributions indicate that 30-minute returns (i.e. about 16-18 data points

per trading day) provide a measure of daily volatility relatively robust to mi­

crostructure noise. In our case, since we want a measure of monthly correlation,

the sum of own and cross products of demeaned3 daily log return over the 21

trading days can be considered a sufficiently robust measure of monthly re­

alized co-variation. The measure of correlation we obtain is nothing but the

ex-post sample correlation over the next 21 trading days. FolIowing the con­

ventions established in the above mentioned literature, we calI this measure

"realized correlation", henceforth RC.

We define RC for the next h days as follows

(4.1)

where

CYR,j = 1 I:h 2 - R·· h i=l ],Ht

(4.2)

and

(4.3)

3 Although asymptotically the mean should be irrelevant, and in practice is very close to

zero almost always, in the case of correlation it is a good empirical practice to subtract the

sample mean from each 21-day sample to constrain the realized correlation to be between

minus one and one.

162

are the FX spot return of exchange rate SIon day t + i. This object of

forecasting is just the sample correlation of forex log returns over the next 21

days.

The plots of the foreign exchange rates are shown in Figures 4.1-4.2 and

aU correlation measures are illustrated in Figures 4.3-4.10 in end of chapter

Appendix (note that we have labeUed the realized correlation as "historical

correlation" as the latter is simply a lagged realized correlation as will be ex­

plained in more detail below). The correlation charts show that on daily basis,

the measures are very volatile. In particular, it seems that the correlations be­

tween the USD lEUR and JPY lEUR currency pairs, between the USD lEUR

and GBP lEUR currency pairs, between the USD/GBP and JPY IGBP cur­

rency pairs, and between the USD 1 JPY and GBP 1 JPY currency pairs have

fluctuated in the positive territ ory most of the time. Moreover, the positive

correlation seems to be higher in the post-euro sub sample.

4.2.3 The Measures of Correlation

To forecast future realized correlation, four alternative correlation measures

are applied. First, we calculate the implied correlation from options implied

volatility. To do so it is neeessary to assume that in addition to the Black and

Scholes model also the triangular parity condition between exchange rate cross

rates holds.

Being based on options data, implied correlation provides a forward-Iooking

perspective to the analysis of co-movements between currency pairs. Because

exchange rate options provide information on the currency options market's

uncertainty about the priee of one currency in terms of another, with three

currencies and options on each of the possible exchange rate pairings we can

derive an estimate of the market's expected future, or implied, correlation

between any two of the exchange rates. To put it in another way, implied

163

correlation represents the degree of co-movement between two currencies using

a third currency as a numeraire.

The implied correlations are derived using the well-known Black-Scholes

pricing formula as weIl as exploiting the arbitrage condition on currencies.

The Black-Scholes formula allows one to derive implied volatilities for the un­

derlying asset. The no-arbitrage condition provides, given the proportional

changes in returns of two exchange rates, RI and R2' the proportional change

in the return of a third exchange rate R3 simply as R3 = RI - R 2. lt then

follows that

(4.4)

whereby it is straightforward to derive the implied correlation (le) between

RI and R2 knowing Var(Rl), Var(R2 ), and Var(R3).4 The implied correlation

is then defined as

(J2 + (J2 - (J2

(R R )IC _ l,t 2,t 3,t P l, 2 t,h - 2

(Jl,t(J2,t (4.5)

square of the implied volatility on each of the currency pairs. The implied

correlation for a particular date can then be calculated simply by inserting

values for the implied volatilities in the equation.5

Bollerslev and Zhou (2003) point out that if the volatility risk is priced

in the options markets then implied volatility is a biased predictor of realized

4See Malz (1997), Butler and Cooper (1997) and Brandt and Diebold (2003) for further

details. 5Whether the no arbitrage condition holds or not, especially for less liquid currencies, is

an empirical question. We cannot find substantial differences with respect to deviations from

the no arbitrage conditon between major currencies and currencies of acceding countries.

164

volatility.6 In fact, implied volatilities are often empiricaIly found to be upward

biased estimates of the objective volatility. In a standard stochastic volatility

set up, it can be shown that if the price of volatility risk is zero, the process

foIlowed by the volatility is identical under the objective and the risk neutral

measures. In such a case there would be no bias. However, the volatility risk

premium is generaIly estimated to be negative which in turn implies that the

volatility pro cess under the risk neutral measure will have higher drift. These

theoretical arguments do apply to the computation of implied correlation as

weIl. However, because such a potential bias could affect aIl variances used in

the computation ofthe implied correlation in (4.5), it is not clear at priori that

the bias for implied correlations is a problem as severe as it is for volatilities.

We will show below that bias is indeed present in correlations computed from

options.

The other three volatility forecasts are derived from historical FX returns

only. The simplest possible forecast is the historical h-day correlation, defined

as

HC(1,2) RC(1,2) Pt,h = Pt-h,h (4.6)

The historical correlation is simply the lagged realized correlation. Alterna­

tively, we can consider second moments that apply an exponential weighting

scheme putting progressively less weight on distant observations. The sim­

plest correlation measure using such a scheme is the Exponential Smoother or

RiskMetrics correlation. Daily variance and covariance then evolve as

6See also Bandi and Perron (2003), Chernov (2003), and Bates (2002).

165

CXl

G-(I),t+1 = (1 - À) L Ài-

I Rî,t-i+1 = ÀG-(I),t + (1 - À) Rî,t (4.7)

i=1 00

G-(1,2),HI = (1 - À) L Ài-

I R I ,t-i+I R 2,t-i+1 = ÀG-(1,2),t + (1 - À) R I ,tR 2,t

i=1

Following JP Morgan we simply fix À = 0.94 for aIl the daily FX returns.

The forecast for h-day correlation is therefore

-2 RM(j,k) _ O"(j,k),HI

PHI - - -O"j,t+IO"k,t+1

(4.8)

The third estimate for correlation based on past exchange rate returns

that is considered here is the GARCH correlation. The GARCH methodology

permits the calculation of time-varying second moments for the universe of

assets that are considered by the researcher. According to this approach,

variances and correlations are conditional on a time-varying information set

that allows one to update the estimated second moments at each point in

time when new information becomes available. We have adopted a bivariate

GARCH model where Rt is defined as the vector of returns

(4.9)

We assume that Rt follows a GARCH process7

(4.10)

In (4.10) Ct is an identical and independently distributed vector sequence

with mean zero and unit variance. The conditional covariance H t evolves

according to a diagonal BEKK GARCH process

7See Engle and Kroner (1995) for further details.

166

Ht = On' + BHt_lB' + ARI t-lR2 t-lA' , , (4.11)

where

2 x 2, A, B = 2 x 2 diagonal, 0 = 2 x 2 lower triangular

H(l, l)t = variance of exchange rate 1 at time t

H(2,2)t = variance of ex change rate 2 at time t

H(1,2)t = covariance of currency 1 and currency 2 at time t

The next day GARCH correlation is thus defined as

(R R )GARCH _ H(1,2)Hl

P l, 2 Hl -y'H(l, l)Hl y'H(2, 2)t+l

In contrast to the RiskMetrics model, which implies a random walk volatil­

ity process, to forecast the 21 days ahead correlation with GARCH it is nec­

essary to consider the mean reversion of the model and iteratively forecast

variances and covariances. The computations to obtain the GARCH correla­

tion forecasts are detailed and the plots of the GARCH correlations (GC) for

the various exchange rate pairs are found in theA ppendix at the end of chap­

ter. The plots are substantially smoother than those obtained from historical

correlations.

4.3 Correlation Forecast Evaluation Methodology and

Results

To compare the forecasting capability of the different correlation measures, we

run simple linear predictability regressions. These are carried out in-sample, by

167

using different windows for the realized correlation (the left-hand side variable)

and for the right-hand side variables. In other words, we assess how various

estimates of monthly exchange rate correlations have in the past predicted

realized correlation one month ahead in time. More specifically, the following

univariate regressions are first run for each correlation

Re .. Pt,h = a + b~,h + C~,h ( 4.12)

for j = JG, HG, GG

These univariate regressions8 serve to assess the fit through the adjusted R2

and to check how close the estimates of a are to 0 and how close the estimates

of b are to 1. In addition, bivariate regressions are performed, including the

implied correlation and the two return-based forecasts in turn, as follows:

Re Je' Je' Pt,h = a + bpt,h + C~,h + Ct,h,J

for j = HG,GG

These bivariate regressions shed sorne light into whether the return-based

correlation forecasts add anything to the market-based forecast implied from

currencyoptions. FinaIly, a regression will be run including aIl three correla­

tion forecasts in the same equation, in order to asses the relative merits of the

different correlation forecasts.

The results are reported in Tables 4.3-4.6 in Appendix where both regres­

sion point estimates as weIl as standard errors corrected for heteroskedastic­

ity and autocorrelation, using GMM, are included. The robust Newey-West

8See e.g. Fleming et al. (1995)

168

weighting matrix9 with a pre-specified bandwidth equal to 21 days is applied.

The regression fit is reported using adjusted R2• Tables 4.5 and 4.6 in Ap­

pendix include the same regressions than Table 4.3 and 4.4, but now using the

sample period beginning from January 1999.

We find that correlation between foreign exchange pairs is predictable to

a substantial extent. The adjusted R2 of the GMM regressions10 ranges from

18 to 38 per cent for the entire sample and from 20 to 42 per cent for the

post-January 1999 sample. Rowever, for the exchange rate pairs that show

correlation predictability, implied correlation is only in a few cases the best

univariate forecast. Both GARCR and RiskMetrics correlation forecasts show

considerable predictive power, too.

When comparing these results with predictability regressions for volatility

forecasts, one difference we find is, therefore, that information from currency

options priees does not always seem to be as helpful in predicting correlation

as it is in predicting volatility. Returns based measures sometimes perform

better than correlation measures based on options data. We note however

that the return based measures also sometimes perform very poorly. This is in

contrast with the implied correlation, which seems to be more consistent as it

shows less variability in the predictive power from one pair of exchange rates to

the other. In substance, the two types of correlations forecasts seem to ni cely

complement each other. The best forecasts obtain when return based measures

are used jointly with market based measures, as the highest adjusted R2 is

almost invariably obtained from the encompassing (multivariate) regressions.

For the entire sample implied correlation and GARCR correlation generally

show good predictive power and typically outperform historical correlations.

9See Newey and West (1987)

lOFor the technicalities regarding the GMM implementation refer to Christoffersen Maz-

zotta (2004), i.e. the essay 2 in this thesis.

169

Implied and GARCR correlations between the most important currency pairs

from the euro area perspective, i.e. the correlations between the USD lEUR;

GBP lEUR and the USD lEUR; JPY lEUR exchange rates, provide reliable

forecasts of future correlation. They can thus be useful in assessing near-term

future inflationary risks that originate from exchange rate movements. Per­

haps surprisingly, in the post-1999 sample the best forecasts are RiskMetrics

and implied correlation, both winning the race in 3 out of 7 cases. It is possible

that RiskMetrics displays a better ability to model the extremely high persis­

tence of typical forex correlations. Rowever, we conjecture that the fact that

RiskMetrics outperform GARCR may be due to the choice of the adjusted R2

as the metric to determine the best forecastY We leave an in-depth analysis

of this and related issues for future research.

4.3.1 Efficiency and Bias

To study the merit of each correlation forecasts with regard to the relative ef­

ficiency and bias we perform a Mincer-Zarnowitz (1969) decomposition of the

MSE into bias squared, inefficiency and random variation. 12 The decomposi­

tion is as follows: MSE = [E[y]- E[Y]]2 + (1- ,B)2Var(y) + (1- R2)Var(y),

where y is the variable of interest, in our case the realized correlation, and

y is each correlation forecast in turn. From the regression of y on y and a

constant, we obtain the slope coefficient ,B and the regression fit, R2 . The

Mincer-Zarnowitz regressions are run for each of the currency pairs and for

each of the currency forecasts. Table 4.7 in end of chapter Appendix reports

the MSE's in absolute value and their decomposition into bias squared, in­

efficiency, and residual variation, in percent age of the total MSE. It appears

Il For the importance of the 10ss function see e.g. Christoffersen and Jacobs (2004).

12We thank an anonymous referee and the thesis committee for pointing us in this direc-

tion.

170

that bias is generally higher for the implied correlation than is for all the other

correlation forecasts, with the only exception of the RiskMetrics correlation for

the USD lEUR - JPY lEUR pair for the entire sample. In the same sample, his­

torical correlation is shown to be the least efficient of the correlation forecasts.

In the post 1999 sample however, implied correlation bias becomes less of an

issue, almost disappearing for the USDIEUR - CBP lEUR and CBP lEUR

- JPY lEUR pairs. A notable exception to this pattern is the USD 1 JPY -

CBP 1 JPY implied correlation bias which almost doubles to 47.29 per cent. In

the post 1999 sample the historical correlation is shown to be rather inefficient

but substantially unbiased. RiskMetrics correlation appears to be somewhat

inefficient for sorne currency pair and biased for others. CARCH often perform

better than the other forecasts under one measure, but not the other.

In summary, although in general implied correlation from options is a more

efficient but biased forecast and return based measures are less biased but also

less efficient, the ranking does not hold for all the currency pairs in both sample

periods. In other words, the decomposition reinforces the idea that different

measures of correlation may have different informational content and therefore

they may contribute to provide the best forecasts when used jointly.

4.4 Two Applications of Correlation Forecasts

Measures of correlation were above shown to provide effective forecasts of fu­

ture realized correlation. A question that arises from the practical perspective

is then whether such measures can contribute to enhance our understanding

on exchange rate developments beyond the simple co-movement among various

bilateral exchange rates. In this section we propose and illustrate two applica­

tions where correlation forecasts can be useful when monitoring and assessing

exchange rate developments.

171

4.4.1 Scenario Analysis for the Euro Nominal Effective Exchange

Rate Index

The nominal effective exchange rate (NEER) index of a currency is commonly

calculated as a weighed average whereby the various bilateral exchange rates

of the most important trading partner currencies are aggregated using the

respective trade shares as weights. The resulting index would then better

refiect the possible future infiationary risks originating from ex change rate

movements in so far as diverging movements of bilateral exchange rates would

partially cancel each other out. Many central banks therefore use the NEER

among indicators of medium-term risks to price stability. In addition, the

price-defiated real effective exchange rates (REERs) provide an insight to the

economy's overall price competitiveness in the medium to long term.

In the context of forward-Iooking monetary policy, various scenarios for the

likely future developments of the NEER index could prove useful in assessing

the risks to a given baseline model. Due to the known near-impossibility of

forecasting bilateral exchange rates it should be c1ear that assessing the future

level of an index that consists of a large number of bilateral rates should,

if anything, multiply the difficulty of the task. However, by using measures

of correlation it is, in principle, possible to construct consistent scenarios for

future movements in a NEER index conditional on an assumption of a future

change in one bilateral exchange rate only.

As an example, we take the euro nominal effective exchange rate index

with the narrow group of trading partner currencies, calculated by the ECB. 13

Since the weights in the euro NEER are rather concentrated on the currencies

of the three largest trading partner countries of the euro area (the United

States, the UK and Japan), we analyze how the changes in these currencies,

13 A detailed overview of the methodology used to calculate the euro effective exchange

rate indices is provided by Baldorini, Makrydakis and Thimann (2002).

172

conditional on an assumed movement in another major world exchange rate,

the JPY jUSD rate, are reflected in the NEER index. We consider here the

sample period starting from January 1999 only. To this end, we exploit the

property of conditional expectation under bivariate normal distribution that

can be written as follows

(4.13)

i = USDjEUR,JPYjEUR,GBPjEUR

In (4.13), the left-hand side captures the level expected to be realized at

time t + 1 of the bilateral exchange rate of the euro against the dollar, the

pound or the yen (Xi)' given an assumption 1) made at time t about the level

of the JPY jUSD exchange rate (Y) to be realized at t + 1. The right-hand side

of expression (4.13) shows how this conditional expectation on Xi differs from

the unconditional expectation of that exchange rate that is provided at time t

by the t + 1 horizon forward exchange rate Et(Xi,t + 1).14 In particular, under

the horizon of 1 month, the spread between the assumed future level 1) of the

JPY jUSD exchange rate and the 1-month forward JPY jUSD rate Et (Yt+1)

is multiplied by the forecast correlation between the JPY jUSD and the rele­

vant bilateral euro exchange rate, scaled by the ratio of forecast volatilities.

After having calculated the conditional expectations for the three main euro

bilateral exchange rates, the conditional expectation of the NEER index can

be calculated by multiplying the former with the relevant trade weights, and

aggregating across currencies.15

14Under the same assumption, the conditional variance could be calculated simply as

Vart(Xi,t+l IYi+! = "J) = (1 - PXi,y,t)criï,t. 15Note that since the calculation of the expectation of the euro NE ER requires as input the

173

In the end of chapter Appendix, we run regressions à la Fama and find

that the conditional expectations on the bilateral USD lEUR, JPY lEUR or

GBP/EUR exchange rates as calculated using equation (4.13) produce esti­

mates that outperform the forecasts provided by the forward exchange rates.

We can now construct a framework for scenario analysis on the euro NEER

index. To this end, the particular question we want to ask is the following.

What is the impact on the expectation of the euro NEER one-month ahead,

given that the Japanese yen is expected to appreciate by 10% against the US

dollar over one month's horizon? Clearly, since the measures of correlation are

time-varying the impact on the euro NEER of an expected yen appreciation

against the US dollar vary across different dates. For instance, a scenario where

the euro NEER would be expected to move significantly following an expected

10% move in the JPY IUSD rate would presuppose that the euro would be ex­

pected to move in the same direction against aIl three major currencies. 16 In

that case, the USD 1 JPY rate would need to be positively correlated against an

three major bilateral euro exchange rates. Table 4.1 illustrates the scenarios

on the bilateral euro exchange rates and on the euro NEER for four selected

dates using GARCR correlation forecasts.

The forecast co-movements of the various bilateral euro exchange rates

condition al on the assumed 10% appreciation of the yen vis-à-vis the US dol­

lar vary substantially across episodes. This is also refiected by the fact that

correlation between the GBP JEUR and the JPY jUSD exchange rates, which do not enter

the same exchange rate "triangle", the correlation forecasts using the irnplied correlation

approach cannot be used for this exercise. 16The results have to be qualified in so far as the three main currencies "only" represent

sorne 70% of the weight in euro NE ER basket. In the calculations it is assurned that the

other bilateral rates do not change, although sorne of thern could be rather sensitive to

rnovernents in the JPY jUSD rate.

174

Assumption: 10% JPY appreciation against USD in 1 month USD/EUR GBP/EUR JPY/EUR Euro NEER

27 Sep 2000 -7.21% -2.95% -22.6% -6.24% 21 Jan 2002 0.47% -1.01% -11.88% -2.01% 22 Jul 2002 6.89% 0.67% -0.85% 1.71% 12 Dec 2003 2.48% 1.58% -1.37% 0.76%

TABLE 4.1. Scenarios for the euro exchange rates one month ahead (GAReR correlation). Positive (negative) reading denotes euro appreciation (deprecia­tion).

the euro NEER depreciates in sorne occasions, while it appreciates in others.

Therefore, expectations on a stronger yen against the US dollar could con­

tribute to higher or lower expected import prices and inflationary pressures in

the euro area, depending on the particular correlation configuration in the FX

market at the time when the scenario is conducted.

Looking at the conditional expectations of the bilateral rates, a general

observation is that the conditional expectations on the movements in the euro

bilateral exchange rates have changed over time. In particular, there is a

tendency from expected euro weakness against the US dollar and the pound

towards expected euro strength as a response to the assumed 10% apprecia­

tion of the yen against the US dollar. Moreover, there is a tendency from a

sharp towards more moderate projected future euro depreciation against the

yen. What could be the factors contributing to the constellation during the

early years of the single currency whereby an appreciation of the yen against

the dollar would have contributed to a st ronger dollar against the euro, rather

than to a general weakness of the US currency? Soon after its launch in Jan­

uary 1999, the euro entered a protracted period of broad-based depreciation

that by fall of 2000 was considered to have brought the single currency out

of line of the underlying fundamentals. The euro exchange rates subsequently

stabilized but remained weak throughout 2001. From 2002 Q2 onwards the US

dollar started depreciating against all major currencies amid growing concerns

175

regarding the large US current account deficit. This seems to have changed

also the correlations that measure the interplay among the various bilateral ex­

change rates and, consequently, the conditional expectations regarding future

movements in the euro NEER as a response to a hypothetical yen appreciation

vis-à-vis the US currency. Finally, throughout 2003 the Japanese authorities

markedly increased the intervention activity to retard the pace of yen appre­

ciation against the US dollar. In that context, a sudden switch in policy to

"tolerate" a 10% appreciation of the yen could have been seen as reducing

the pressure on the euro to appreciate against the US currency. This would

explain the conditional expectation indicating a more moderate appreciation

of the euro relative to the US dollar than was the case in mid-2002.

4.4.2 Exchange Rate Intervention and Correlation Among Cross­

Rates

In the 1990s and in the early 2000s, the activist policy by Japanese authorities

to protect the price competitiveness of Japanese exporters by preventing ex­

cessive yen appreciation against the US dollar was often an important factor

affecting C3 exchange rate dynamics.17

How is foreign exchange market intervention supposed to affect exchange

rates and their cross-rates? According to the standard monetary or portfolio

balance approach to interventions, an increased supply of a currency (or bonds

denominated in that currency) in the context of an open market operation

should imply a depreciation of that currency against aIl other currencies in

the market until the equilibrium is restored. For example, an intervention

operation by the Japanese authorities where the yen is sold against the US

dollar should imply a depreciation of the yen not only against the US dollar

17 See Castrén (2004) and!to (2002) for analyses of the Japanese interventions using official

Japanese intervention data.

176

but also against the euro, the pound and so on. Conversely, the purchase

of US dollars should exert a general upward pressure on the US currency in

the market. Therefore, a yen-selling intervention against the US dollar should,

ceteris paribus, contribute to a weaker yen and a st ronger US dollar also against

the euro.

However, as argued by Sarno and Taylor (20001), the daily trading vol­

umes in the foreign exchange markets are so large that even relatively sizeable

interventions are unlikely to affect the levels of major currencies through the

monetary or the portfolio channels. On the other hand, if the interventions are

repeated and follow a systematic strategy, possibly combined with oral com­

munication, they are likely to affect the market's expectations regarding the

"desired" level of the USD / JPY rate. In such a constellation, the adjustment

pressures in the FX market are likely to be channelled increasingly through

currency pairs that are not actively managed. Following the previous example,

with the USD / JPY rate "managed" by systematic intervention any pressure

on the US dollar to depreciate - for instance due to the large US current ac­

count deficit - would imply that the euro would be expected to appreciate over

time both against the dollar and, due to the interventions on the JPY /USD

rate, against the yen. If these hypotheses were correct, the implications of

interventions should demonstrate themselves in increased correlation between

the cross rates.18

We will augment our earlier correlation forecast regressions by incorporat­

ing a variable that measures the daily purchases of Japanese yen carried out

18BIS (2004) reports evidence from Reuters and EBS trading systems suggests that in

2002-2004, there was a marked reduction in absolute trading volumes in the JPY /USD

exchange rate while the absolute volumes on the USD/EUR and the USD/GBP exchange

rates sharply increased. The period incorporates sorne of the most pronounced episodes of

interventions by the BoJ that could have reduced the traders' appetite to take large yen

positions.

177

by the Bank of Japan in the FX market between April 1992 and March 2004.

Our goal is to analyze whether data on the interventions on the JPY IUSD

exchange rate can improve the forecasts of correlation between the USD lEUR

and JPY lEUR exchange rates. In other words, we want to find out whether

interventions can work as an additional explanatory factor for realized corre­

lation between the two cross rates of the particular exchange rate that is the

focus of the market operation. The particular equation we estimate is

Re· . Pt h = a + bPi h + cI NTt + ~ h , , , (4.14)

for j = HC,RM,GC,IC (4.15)

The regressions serve to assess whether the coefficients of the intervention

variable are positive and significant and whether the adjusted R2 improves

relative to standard correlation forecast equations.

The results are summarized in Table 4.2. The regressions show that the

variable measuring the BoJ yen-purchasing interventions reeeives the negative

and statistically significant coefficient all regressions. The interpretation of the

negative coefficient means that yen-selling interventions (almost all observa­

tions in the data set were yen sales) have a positive impact on the forecasts

of future realized correlation. In all cases, the adjusted R2s improve; the in­

crease is particularly marked in the case of implied correlation forecast (15% in

the full sam pIe). Henee, an intervention strategy that aims at systematically

stabilizing a particular exchange rate over time could increase the expected fu­

ture co-movement among its cross exchange rates as reflected by the currency

options priees.

178

Full sample Post euro sample Corr. Interv. R'2 Corr. Interv. R'2

0.747* 0.205

.0924* 0.359

Implied (0.105) (0.113) 0.745* -0.28* 0.920* -0.013* (0.067) (0.053)

0.220 (0.069) (0.044)

0.365

0.564* 0.314

0.583* 0.343

(0.053) (0.076) Historical

0.561 * -0.197* 0.326

0.581* -0.013* 0.349

(0.037) (0.045) (0.050) (0.044) 0.884*

0.235 1.687*

0.382 (0.078) (0.094)

RiskMetrics 0.871* -0.022*

0.242 1.163* -0.011*

0.387 (0.079) (0.046) (0.093) (0.044) 0.858*

0.329 0.834*

0.362 (0.066) (0.094)

GARCH 0.854* -0.020

0.341 0.832* -0.014*

0.370 (0.049) (0.045) (0.094) (0.045)

TABLE 4.2. Japanese interventions on JPY /USD and forecasts of correlation between USD/EUR and JPY /EUR (standard errors in parenthesis).

4.5 Concluding Remarks

The various estimations of correlation between the major bilateral exchange

rates show distinctive fluctuations over time. The correlations generally in­

creased soon after the introduction of the euro, but have more recently re­

turned doser to their longer-term average levels. This development reflects

the episode of broad-based euro depreciation 1999-2000, followed in 2002-early

2003 by euro appreciation that was somewhat more prominent against the US

dollar than against the pound sterling and the Japanese yen.

Regarding the ability to forecast future correlation, implied correlation can

predict up to 36% of future realized correlation. Nevertheless, it is not uni­

vocally the best predictor of future correlation as G ARCH and RiskMetrics

correlations yield occasionally very good predictive power, too. When used

together, implied correlation, GARCH correlation and RiskMetrics correlation

179

are particularly useful in predicting future correlation between the major euro

currency pairs at the one-month horizon. The predictive power seems to have

strengthened after the introduction of the euro.

When applying the estimated measures, we found that using correlation

forecasts to analyze scenarios for effective exchange rates is useful as an ex­

pected movement in one currency pair seems to indicate a very different im­

pact on the effective exchange rate in various points in time. The time-varying

correlation forecasts take into account the market's current perception of the

relative adjustment of various exchange rates as a response to a sudden move­

ment in one major exchange rate. Mapping these bilateral movements into

the NEER index provides conditional forecasts that could be a useful input in

analyzing future infiationary risks. FUrthermore, data on interventions on the

JPY /USD ex change rate improve the ability of implied correlation to forecast

future realized correlation. This suggests that systematic intervention might be

capable of affecting the options market's perception about future co-movement

among the cross-rates of the currency pair that is on the focus of the market

operation.

180

4.A Tables and Figures

18For aIl the pre-1999 period or the full sample period the DEM proxies for the EUR.

181

USD IEUR-JPY lEUR USD IEUR-G BP lEUR Int. IV HV RM GH Adj R2 Int. IV HV RM GH Adj R2 0.045 0.747 0.205 0.009 0.79 0.207 0.052 0.105 0.073 0.124

0.178 0.564 0.314 0.209 0.548 0.295 0.027 0.053 0.034 0.058

0.174 0.874 0.229 0.257 0.563 0.125 0.03 0.112 0.048 0.12

0.095 0.858 0.329 0.093 0.875 0.324 0.029 0.066 0.04 0.079

0.053 0.356 0.446 0.346 0.045 0.384 0.426 0.33 0.04 0.095 0.059 0.065 0.12 0.059

0.028 0.452 0.602 0.282 0.009 0.653 0.216 0.219 0.044 0.103 0.115 0.07 0.126 0.113

0.022 0.263 0.706 0.343 -0.022 0.323 0.708 0.347 0.038 0.096 0.085 0.062 0.12 0.086

0.048 0.29 0.367 -0.55 0.59 0.366 -0.008 0.425 0.182 -0.493 0.765 0.383 0.037 0.094 0.105 0.167 0.165 0.062 0.123 0.11 0.137 0.164

GBP IEUR-JPY lEUR USD/GBP-JPY IGBP Int. IV HV RM GH Adj R2 Int. IV HV RM GH Adj R2 0.021 0.661 0.136 -0.02 0.751 0.203 0.039 0.117 0.049 0.101

0.14 0.351 0.123 0.211 0.315 0.099 0.021 0.063 0.031 0.068

0.133 0.43 0.101 0.166 0.46 0.156 0.022 0.091 0.033 0.075

0.109 0.584 0.171 0.ü78 0.745 0.174 0.021 0.077 0.042 0.107

0.027 0.47 0.232 0.179 -0.011 0.662 0.097 0.209 0.035 0.112 0.063 0.048 0.113 0.067

0.017 0.506 0.259 0.166 0.001 0.56 0.204 0.221 0.036 0.115 0.089 0.046 0.12 0.084

0.024 0.385 0.43 0.206 -0.033 0.517 0.372 0.227 0.033 0.108 0.084 0.047 0.125 0.132

0.034 0.376 -0.044 -0.476 0.945 0.227 -0.059 0.491 -0.19 0.012 0.669 0.234 0.032 0.103 0.096 0.127 0.173 0.05 0.126 0.102 0.141 0.279

TABLE 4.3. Correlation Predictability Regressions. AlI Sample: January 1992 - March 2003.

182

USD/JPY-GBP/JPY USD /EUR-PLZ/EUR Int. IV HV RM GH Adj R2 Int. IV HV RM GH Adj R2 0.075 0.698 0.337 -0.101 1.316 0.285 0.068 0.093 0.139 0.261

0.316 0.406 0.161 0.237 0.496 0.234 0.048 0.075 0.085 0.12

0.221 0.565 0.222 0.143 0.69 0.296 0.057 0.091 0.091 0.143

-0.008 0.982 0.228 -0.409 1.786 0.187 0.095 0.16 0.278 0.519

0.075 0.674 0.031 0.338 -0.047 0.955 0.215 0.306 0.067 0.111 0.075 0.113 0.252 0.149

0.07 0.629 0.092 0.34 -0.016 0.675 0.405 0.318 0.068 0.117 0.097 0.109 0.328 0.236

0.016 0.604 0.219 0.342 -0.324 1.065 0.672 0.3 0.083 0.108 0.158 0.299 0.283 0.67

-0.057 0.603 -0.145 -0.013 0.507 0.346 -0.125 0.647 -0.032 0.388 0.293 0.319 0.107 0.114 0.135 0.154 0.275 0.245 0.342 0.125 0.232 0.595

USD /EUR-CZK/EUR Int. IV HV RM GH Adj R2

-0.132 0.906 0.186 0.082 0.217

0.175 0.109 0.012 0.047 0.116

0.177 0.114 0.007 0.051 0.154

0.16 0.238 0.002 0.081 0.423

-0.134 0.888 0.039 0.186 0.084 0.202 0.086

-0.134 0.897 0.025 0.185 0.087 0.204 0.121

-0.144 0.897 0.089 0.184 0.108 0.209 0.305

-0.131 0.889 0.074 -0.054 -0.008 0.183 0.104 0.205 0.113 0.233 0.568

TABLE 4.4. Correlation Predictability Regressions. AU Sample: January 1992 - March 2003. Continued.

183

USD IEUR-JPY lEUR USD IEUR-G BP lEUR Int. IV RV RM GR Adj R2 Int. IV RV RM GR Adj R2 0.1 0.924 0.359 -0.085 1.139 0.217

0.075 0.113 0.188 0.274

0.254 0.583 0.343 0.34 0.473 0.206 0.052 0.076 0.071 0.096

0.189 1.168 0.382 0.253 0.979 0.218 0.056 0.143 0.088 0.198

0.19 0.834 0.362 0.272 0.692 0.206 0.056 0.094 0.1 0.164

0.097 0.577 0.32 0.411 -0.012 0.728 0.294 0.27 0.062 0.139 0.107 0.145 0.205 0.082

0.091 0.473 0.717 0.417 -0.023 0.661 0.608 0.265 0.062 0.17 0.254 0.14 0.191 0.196

0.088 0.512 0.477 0.406 -0.035 0.713 0.412 0.261 0.062 0.14 0.151 0.151 0.211 0.149

0.093 0.485 0.106 0.55 -0.025 0.418 -0.013 0.67 0.185 0.169 0.075 0.27 0.062 0.154 0.123 0.557 0.307 0.144 0.186 0.173 0.386 0.317

GBP IEUR-JPY lEUR USDIGBP-JPY IGBP Int. IV RV RM GR Adj R2 Int. IV RV RM GR Adj R2 0.222 0.467 0.067 0.077 0.682 0.198 0.069 0.141 0.054 0.119

0.251 0.387 0.146 0.274 0.252 0.063 0.038 0.081 0.043 0.094

0.217 0.581 0.152 0.224 0.382 0.1 0.044 0.127 0.047 0.108

0.218 0.586 0.193 0.153 0.612 0.09 0.041 0.095 0.068 0.18

0.179 0.229 0.334 0.158 0.077 0.67 0.014 0.197 0.057 0.13 0.087 0.054 0.129 0.094

0.167 0.178 0.513 0.16 0.076 0.646 0.041 0.198 0.057 0.138 0.147 0.054 0.139 0.12

0.177 0.131 0.543 0.199 0.08 0.696 -0.026 0.197 0.053 0.117 0.107 0.065 0.144 0.207

0.191 0.147 -0.049 -0.309 0.855 0.204 0.149 0.697 0.026 0.335 -0.61 0.204 0.05 0.12 0.14 0.471 0.423 0.064 0.139 0.109 0.157 0.314

TABLE 4.5. Correlation Predicatability Regressions. Sample: March 1999 -March 2004.

184

USDjJPY-GBP jJPY USD jEUR-PLZjEUR Int. IV RV RM GR Adj R2 Int. IV RV RM GR Adj R2

-0.075 0.856 0.242 -0.128 l.368 0.314 0.106 0.144 0.131 0.248

0.412 0.213 0.043 0.225 0.514 0.267 0.061 0.097 0.075 0.105

0.305 0.404 0.103 0.14 0.694 0.329 0.071 0.115 0.Q78 0.123

0.105 0.771 0.112 -0.456 l.872 0.219 0.118 0.203 0.244 0.458

-0.087 0.943 -0.092 0.247 -0.06 0.965 0.228 0.338 0.108 0.179 0.107 0.107 0.254 0.134

-0.077 0.876 -0.022 0.242 -0.024 0.677 0.416 0.351 0.107 0.201 0.152 0.104 0.316 0.203

-0.068 0.878 -0.042 0.242 -0.358 l.083 0.719 0.332 0.118 0.213 0.289 0.262 0.28 0.611

-0.167 0.79 -0.337 0.192 0.39 0.258 -0.135 0.649 -0.03 0.392 0.302 0.351 0.138 0.216 0.147 0.187 0.425 0.229 0.328 0.125 0.21 0.571

USD jEUR-CZKjEUR Int. IV RV RM GR Adj R2

-0.156 0.956 0.206 0.077 0.21

0.166 0.114 0.012 0.046 0.115

0.166 0.126 0.008 0.05 0.151

0.152 0.233 0.002 0.08 0.423

-0.158 0.94 0.037 0.205 0.079 0.196 0.085

-0.158 0.947 0.027 0.205 0.081 0.197 0.118

-0.165 0.948 0.074 0.204 0.103 0.203 0.304

-0.145 0.937 0.068 -0.02 -0.095 0.203 0.1 0.2 0.114 0.227 0.558

TABLE 4.6. Correlation Predicatability Regressions. Sample: March 1999 -March 2004. Continued.

185

AIl sample Post 1999 USD IEUR-GBP lEUR USD IEUR-GBP lEUR

MSE Bias2 Ineff. Residual MSE Bias2 Ineff. Residual Implied 0.077 16.379 1.513 82.108 0.030 0.061 0.411 99.528 Historical 0.073 0.008 22.189 77.803 0.042 0.040 24.315 75.645 RiskMetrics 0.085 10.684 7.093 82.223 0.091 65.952 0.005 34.043 GARCH 0.056 2.789 0.936 96.275 0.044 24.434 3.717 71.848

USD IEUR-JPY lEUR USD IEUR-JPY lEUR MSE Bias2 Ineff. Residual MSE Bias2 Ineff. Residual

Implied 0.078 7.778 2.656 89.566 0.044 8.144 0.347 91.509 Historical 0.077 0.005 21.491 78.504 0.052 0.089 21.072 78.839 RiskMetrics 0.088 22.581 0.476 76.943 0.101 61.291 0.489 38.220 GARCH 0.062 3.089 1.294 95.618 0.052 22.414 1.704 75.882

USD IEUR-PLZ/EUR USD IEUR-PLZ/EUR MSE Bias2 Ineff. Residual MSE Bias2 Ineff. Residual

Implied 0.071 2.171 2.200 95.629 0.070 1.565 3.166 95.269 Historical 0.095 0.136 23.985 75.878 0.094 0.041 24.650 75.310 RiskMetrics 0.072 0.133 7.875 91.991 0.071 0.049 8.714 91.237 GARCH 0.080 0.358 4.284 95.358 0.081 0.738 5.730 93.532

USD IEUR-CZK/EUR USD IEUR-CZK/EUR MSE Bias2 Ineff. Residual MSE Bias2 Ineff. Residual

Implied 0.082 33.949 0.165 65.885 0.083 35.371 0.035 64.593 Historical 0.124 0.132 46.572 53.295 0.123 0.324 45.211 54.465 RiskMetrics 0.099 0.029 33.026 66.945 0.099 0.001 31.743 68.256 GARCH 0.070 1.932 3.623 94.445 0.071 1.124 3.553 95.323

GBP IEUR-JPY lEUR GBP IEUR-JPY lEUR MSE Bias2 Ineff. Residual MSE Bias2 Ineff. Residual

Implied 0.088 7.151 3.708 89.141 0.065 0.084 8.620 91.296 Historical 0.118 0.001 32.415 67.584 0.077 0.013 29.989 69.998 RiskMetrics 0.099 0.537 16.411 83.052 0.065 9.191 7.749 83.060 GARCH 0.085 1.299 9.349 89.352 0.064 10.222 9.597 80.182

USD/GBP-JPY IGBP USD/GBP-JPY IGBP MSE Bias2 Ineff. Residual MSE Bias2 Ineff. Residual

Implied 0.091 18.372 2.238 79.390 0.051 6.521 4.774 88.705 Historical 0.124 0.007 34.097 65.896 0.085 0.000 37.359 62.641 RiskMetrics 0.096 0.001 20.289 79.711 0.066 0.040 22.535 77.425 GARCH 0.077 0.000 2.405 97.595 0.054 0.612 3.863 95.525

USD 1 JPY-GBP 1 JPY USDI JPY-GBP 1 JPY MSE Bias2 Ineff. Residual MSE Bias2 Ineff. Residual

Implied 0.061 24.794 6.556 68.650 0.065 47.291 0.475 52.235 Historical 0.074 0.009 29.115 70.875 0.070 0.047 38.557 61.396 RiskMetrics 0.057 0.691 14.431 84.877 0.051 0.885 19.967 79.148 GARCH 0.049 0.677 0.010 99.313 0.041 0.901 1.095 98.004

TABLE 4.7. MSE and Mincer Zarnowitz Decomposition ofMSE in Percentage.

186

.76 1.3

.72 1.2

.66 1.1

.64

1.0 .60

.56 0.9

.52 0.6 1992 1993 1994 1995 1996 1997 1996 1999 2000 2001 2002 2003

-USDDEM -USDEUR

90 150

85 140

80 130

75

'\ ~! 120

70 110

65 ~~v 100

60 90

55 60 1992 1993 1994 1995 1996 1997 1996 1999 2000 2001 2002 2003

-JPYDEM -JPYEUR

.46 .76

.46 .72

.44

.42 .66

.40

.36 .64

.36 .60

.34

.32 .56 1992 1993 1994 1995 1996 1997 1996 1999 2000 2001 2002 2003

-GBPDEM -GBPEUR

150

140

130

120

110

100

90

1994 1996 1996 2000 2002

-JPYUSD

FIGURE 4.1. Spot Foreign Exchange Rates.

187

2.0 210

1.9 200

\\~~ 1.8 190

1.7

v/ 180

1.6 170

1.5 160

1.4 150

1.3 140 1999 2000 2001 2002 2003 1999 2000 2001 2002 2003

-USDGBP -JPYGBP

.Q100 .0068

.0095

.0090

.0085

.0080

.0075

.0070 1999 2000 2001 2002 2003 1999 2000 2001 2002 2003

- USDJPY -GBPJPY

5.2 36

35

\ 4.8 34

4.4 33

32

4.0 31

30 3.6

29

3.2 28 1999 2000 2001 2002 2003 1999 2000 2001 2002 2003

- PLZEUR -CZKEUR

4.5 44

4.4

~ 4.3 40

~~ 4.2 36

4.1

~ 4.0

32 3.9

3.8 28

3.7

3.6 24 1999 2000 2001 2002 2003 1999 2000 2001 2002 2003

- PLZUSD -CZKUSD

FIGURE 4.2. Spot Foreign Exchange Rates.

188

0.8 0.8

0.4 0.4

0.0 0.0

-0.4 -0.4

-0.8 -0.8

1992 1993 1994 1995 1996 1997 1998 1992 1993 1994 1995 1996 1997 1998

- usdeurjpyeur - usdeucgbpeur

0.8 0.8

0.4 0.4

0.0 0.0

-0.4 -0.4

-0.8 -0.8

1992 1993 1994 1995 1996 1997 1998 1992 1993 1994 1995 1996 1997 1998

- gbpeur.jpyeur - usdgbpjpygbp

0.8 0.8

0.4 0.4

0.0 0.0

-0.4 -0.4

-0.8 -0.8

1992 1993 1994 1995 1996 1997 1998 1992 1993 1994 1995 1996 1997 1998

- usdjpY..Qbpjpy - usdeucplzeur

0.8

0.4

0.0

-0.4

-0.8

1992 1993 1994 1995 1996 1997 1998

- usdeucczkeur

FIGURE 4.3. Implied Correlation. Pre-January 1999.

0,8

0,4

0,0 tf---t-''--'--'\/-t-fl/--JMHt--tHHIHf---tR~

-0,4

-0,8

0,8

0,4

-0,4

-0,8

0,8

1992 1993 1994 1995 1996 1997 1998

- usdeurjpyeur

1992 1993 1994 1995 1996 1997 1998

- gbpeurjpyeur

0,4

O,Op-ftt----'-...:!LJ'------+----j

-0,4

-0,8

0,8

1992 1993 1994 1995 1998 1997 1998

- usdjpy_gbpjpy

0.4

0,0+----------------1

-0,4

-0,8

1992 1993 1994 1995 1998 1997 1998

- usdeur_czkeur

0,8

OA

O,O-++HH1H-'---lIl-'t--Hw:.-4JI---J-II-+--1

-OA

-0,8

0,8

0.4

1992 1993 1994 1995 1996 1997 1998

- usdeur_9bpeur

O,°-ttl---'li-t-lHI--IIItIJ--IrtIH'--tl--/t--l/--'-ItH-tlI

-OA

-0,8

0.8

1992 1993 1994 1995 1996 1997 1998

- usdgbpjpygbp

0.4

0.0+----------------1

-OA

-0.8

1992 1993 1994 1995 1996 1997 1998

- usdeurJ)lzeur

FIGURE 4.4. Historical Correlation. Pre-January 1999.

189

190

0.8 0.8

0.4 0.4

0.0 0.0

-0.4 -0.4

-0.8 -0.8

1992 1993 1994 1995 1996 1997 1998 1992 1993 1994 1995 1996 1997 1998

- usdeurjpyeur - usdeur--9bpeur

0.8

0.4

0.0

-0.4

-0.8 -0.8

1992 1993 1994 1995 1996 1997 1998 1992 1993 1994 1995 1996 1997 1998

- gbpeurjpyeur - usdgbpjpygbp

0.8 0.8

0.4 0.4

0.0 0.0

-0.4 -0.4

-0.8 -0.8

1992 1993 1994 1995 1996 1997 1998 1992 1993 1994 1995 1996 1997 1998

- usdjpy-gbpjpy - usdeur-plzeur

0.8

0.4

0.0

-0.4

-0.8

1992 1993 1994 1995 1996 1997 1998

- usdeur_czkeur

FIGURE 4.5. RiskMetrics Correlation. Pre-January 1999.

191

O.B 0.8

0.4

0.0

-0.4 -0.4

-O.B -0.8

1992 1993 1994 1995 1996 1997 1998 1992 1993 1994 1995 1996 1997 1998

- usdeurjpyeur - usdeur_gbpeur

0.8 0.8

0.4

0.0

-0.4 -0.4

-0.8 -0.8

1992 1993 1994 1995 1996 1997 1998 1992 1993 1994 1995 1996 1997 1998

- gbpeurjpyeur - usdgbpjpygbp

0.8

0.4

0.0

-0.4 -0.4

-0.8 -0.8

1992 1993 1994 1995 1996 1997 1998 1992 1993 1994 1995 1996 1997 1998

- usdjpy_gbpjpy - usdeur....,plzeur

0.8

0.4

0.0

-0.4

-0.8

1992 1993 1994 1995 1996 1997 1998

- usdeur_czkeur

FIGURE 4.6. GARCR Correlation. Pre-January 1999.

192

0.8

0.4

o.oa----------------1 0.0+-----------------1

·0.4 -0.4

·0.8 -0.8

1999 2000 2001 2002 2003 1999 2000 2001 2002 2003

- usdeurjpyeur - usdeur _9 bpeur

0.8

0.4

O.Oftlf--+---------------1 0.0

-0.4 -0.4

-0.8 -0.8

1999 2000 2001 2002 2003 1999 2000 2001 2002 2003

- gbpeurjpyeur - usdgbpjpygbp

0.8 "'~ ~rtY\~,i ••. 0.4 l(' 11/'" ·n VV' 'yV

0.8

0.4

0.0+----------------1 0.0

-0.4 -0.4

-0.8 -0.8

1999 2000 2001 2002 2003 1999 2000 2001 2002 2003

- usdjpy-gbpjpy - usdeurJ)lzeur

0.8

0.4

0.0+---------------F---1

-0.4

-0.8

1999 2000 2001 2002 2003

- usdeucczkeur

FIGURE 4_7. Implied Correlation. Post-January 1999.

0.8

0.4

O.O+-'---'---------I'--lHH-----j

-0.4

-0.8

-0.4

-0.8

-0.4

-0.8

0.8

0.4

1999

1999

1999

2000 2001 2002 2003

- usdeurjpyeur

2000 2001 2002 2003

- gbpeur.Jpyeur

2000 2001 2002 2003

- usdjpy-gbpjpy

O.O+---------'--t--lHIHlI-II--t-+1'Ht-'t-fH

-0.4

-0.8

1999 2000 2001 2002 2003

- usdeur_czkeur

0.8

0.4

O.O+----------------!I-II

-0.4

-0.8

1999 2000 2001 2002 2003

- usdeur_gbpeur

0.8

0.4

O.O+-IV--+-++---''I--'-W''HI--'+-l

-0.4

-0.8

1999 2000 2001 2002 2003

- usdgbP.Jpygbp

0.8

0.4

O.O+------+----+---+tIHfHI

-0.4

-0.8

1999 2000 2001 2002 2003

- usdeur-plzeur

FIGURE 4.8. Historical Correlation. Post-January 1999.

193

194

0.8 0.8

0.4 0.4

0.0 0.0

-0.4 -0.4

-0.8 -0.8

1999 2000 2001 2002 2003 1999 2000 2001 2002 2003

- usdeur .Jpyeur - usdeur_9bpeur

0.8 0.8

0.4 0.4

0.0 0.0

-0.4 -0.4

-O.B -O.B

1999 2000 2001 2002 2003 1999 2000 2001 2002 2003

- gbpeurjpyeur - usdgbpjpygbp

0.8

0.4

0.0

-0.4 -0.4

-O.B -0.8

1999 2000 2001 2002 2003 1999 2000 2001 2002 2003

- usdjpy-gbpjpy - usdeur-plzeur

O.B

0.4

0.0

-0.4

-O.B

1999 2000 2001 2002 2003

- usdeucczkeur

FIGURE 4.9. RiskMetrics Correlation. Post-January 1999.

195

0.8

0.4

0.0

-0.4 -0.4

-0.8 -0.8

1999 2000 2001 2002 2003 1999 2000 2001 2002 2003

- usdeurjpyeur - usdeur_gbpeur

0.8 0.8

0.4

0.0

-0.4 -0.4

-0.8 -0.8

1999 2000 2001 2002 2003 1999 2000 2001 2002 2003

- gbpeurjpyeur - usdgbpjpygbp

0.8 0.8

0.4 0.4

0.0 0.0

-0.4 -0.4

-0.8 -0.8

1999 2000 2001 2002 2003 1999 2000 2001 2002 2003

- usdjpy_gbpjpy - usdeucplzeur

0.8

0.4

0.0

-0.4

-0.8

1999 2000 2001 2002 2003

- usdeucczkeur

FIGURE 4.10. GARCH Correlation. Post-January 1999.

Chapter 5

CONCLUSIONS AND DIRECTIONS FOR FUTURE

RESEARCH

196

This dissertation presented some advancements on the measurement and use

of conditional second moment of equities and currencies as a measure of risk

for asset pricing and policy purposes in the context of international markets.

The introductory survey examined selected papers from the international

finance literature and from the volatility literature with a focus on the theo­

retical and empirical relationship between first and second unconditional and

conditional moments of domestic and international asset returns.

The first essay investigates the importance of asymmetric volatility when

computing the risk premium of international assets. The results indicate that

conditional second moment asymmetry is significant and time-varying. They

also show that, if the price of risk is time-varying, the world market and foreign

exchange risk premia estimated without allowing for time-varying asymmetry

are misspecified. Furthermore, they imply that asymmetry is more pronounced

when the business condition is such that investors require higher compensation

to bear risk.

Given the empirical relevance for international asset pricing found in this

study, the question of how systematically organize asymmetry from a theoret­

ical point of view gains relevance. This interesting avenue is left for future

research.

In the second essay we have assessed the quality of option based volatility,

interval and density forecasts. We have found that the implied volatilities

explain a large share of the variation in realized volatility. We also find that

wide-range interval and density forecasts are often misspecified whereas narrow

197

interval forecasts are specified better.

In the third essay we have examined whether the information contained in

various measures of correlation among exchange rates can be used to assess

future currency co-movement. We have compared correlation forecasts from

currency option prices to a set of return-based correlation measures. We have

found that while the predictive power of implied correlation is not always

superior to that of returns based correlations measures, it tends to provide the

most consistent results across currencies. Predictions that use both implied

and returns-based correlations generate the highest adjusted R2s, explaining

up to 42 per cent of the realized correlations.

One question related to the second and third essays that remains open is

whether OTe options do have a larger information content as compared to

exchange traded options. This topic is also left for future research.

198

REFERENCES

Adler, Michel, and Bernard Dumas, 1983, International Portfolio Choice and Corporation Finance, The Journal of Finance 38, 925-984.

Alizadeh, Sassan, Michael W. Brandt, and Francis X. Diebold, 2002, Range­Based Estimation of Stochastic Volatility Models, Journal of Finance 57, 1047 - 1091.

Andersen, Torben, and Tim Bollerslev, 1998, Answering the Skeptics: Yes, Standard Volatility Models Do Provide Accurate Forecasts, International Economic Review 39, 885-905.

Andersen, Torben, Tim Bollerslev, Francis X. Diebold, and Heiko Ebens, 2000, The distribution of stock return volatility, NBER Working Paper No. W7933.

Andersen, Torben, Tim Bollerslev, Francis X. Diebold, and Heiko Ebens, 2001, The Distribution of Stock Return Volatility, Journal of Financial Economics 61,43-76.

Andersen, Torben, Tim Bollerslev, Francis X. Diebold, and Paul Labys, 2002, Modeling and Forecasting Realized Volatility, Econometrica 71.

Andersen, Torben, Tim Bollerslev, Francis X. Diebold, and Paul Labys, 2003, Modeling and Forecasting Realized Volatility, Econometrica 71.

Andersen, Torben, Tim Bollerslev, and Nour Meddahi, 2003, Correcting the Errors: A Note on Volatility Forecast Evaluation Based on High-Frequency Data and Realized Volatilities, Working Paper, Duke University.

Ang, Andrew, and Geert Bekaert, 2002, International Asset Allocation with Regime Shifts, Review of Financial Studies 15, 1137-1187.

Avramov, Doron, 2002, Stock Return Predictability and Model Uncertainty, Journal of Financial Economics 64 (3), 423-458.

Babsiri, Mohamed. El, and Jean-Michel Zakoïan, 2001, Contemporaneous Asymmetry in GARCH Pro cesses , Journal of Econometrics 101 (2), 257-294.

Bandi, Federico, and Benoit Perron, 2003, Long Memory and the Relation Between Implied and Realized Volatility, Manuscript, University of Chicago.

199

Bates, David, 2003, Empirical Option Pricing: A Retrospection, Journal of Econometrics pp. 387-404.

Bayraktar, Sema, 1999, International Asset Pricing with Exchange Rate Risk and Market Segmentation, Ph.D. Thesis Drexel University.

Beckers, Stan, 1981, Standard Deviations Implied in Options Prices as Pre­di ct ors of Future Stock Price Volatility, Journal of Banking and Finance 5, 363-81.

Bekaert, Geert, and Campbell Harvey, 1997, Emerging Equity Market Volatil­ity, Journal of Financial Economics 43, 29-77.

Bekaert, Geert, and Guojun Wu, 2000, Asymmetric Volatility and Risk in Equity Markets, Review of Financial Studies 13, 1-42.

Benzoni, Luca, 2001, Pricing Options under Stochastic Volatility: An Empiri­cal Investigation, Working Paper, University of Minnesota.

Berkowitz, Jeremy, 2001, Testing Density Forecasts, Journal of Business and Economic Statistics 19, 465-474.

BIS, 2003, Annual Report, Basle, Working paper, Bank for International Set­tlements.

BIS, 2004, Annual Report, Working paper, Bank oflnternational Settlements.

Black, Fischer, 1976, Studies in Stock Price Volatility Changes, Proceedings of the Business and Economics Statistics Section American Statistical Associ­ation, 177-181.

Blair, Bevan, Ser-Huang Poon, and Stephen J. Taylor, 1981, Forecasting S & P 100 Volatility: The IncrementaI Information Content of Implied Volatilities and High-Frequency Index Returns, Journal of Econometrics 105, 5-26.

BOE, 2000, Quarterly Bulletin, Working paper, February Bank of England.

Bollerslev, Tim, and Jeffrey M. Wooldridge, 1992, Quasi-Maximum Likelihood Estimation and Inference in Dynamic Models with Time-Varying Covari­ances, Econometric Reviews 11 (2), 143-172.

Bollerslev, Tim, and Hao Zhou, 2003, Volatility Puzzles: A Unified Framework for Gauging Return-Volatility Regression, Working paper, Duke University, Department of Economics.

Bontemps, Christian, and Nour Meddahi, 2002, Testing Normality: A GMM Approach, Manuscript, University of Montreal, Department of Economics.

200

Brandt, Michel, and Frank X. Diebold, 2005, A No-Arbitrage Approach to Range-Based Estimation of Return Covariances and Correlations, Journal of Business, forthcoming.

Breeden, Douglas T., and Robert H. Litzenberger, 1978, Prices of State­Contingent Claims Implicit in Option Prices, Journal of Business 51, 621-651.

Brown, Keith C., W.V. Harlow, and Seha M. Tinic, 1988, Risk Aversion, Un­certain Information, and Market Efficiency, Journal of Financial Economics 22, 355-385.

Buldorini, Luca, Stelios Makrydakis, and Christian Thimann, 2002, The Effec­tive Exchange Rates of the Euro, European Central Bank Occasional Paper 2.

Butler, Creon, and Neil Cooper, 1997, Implied Exchange Rate Correlations and Market Perceptions of European Monetary Union, Bank of England Quarterly Bulletin.

Campa, Jose Manuel, and Kevin Chang, 1998, The Forecasting Ability of Correlations Implied in Foreign Exchange Options, Journal of International Money and Finance 17-5, Journal of International Money and Finance.

Campbell, John Y., and Ludger Hentschel, 1992, No News is Good News. An Asymmetric Model of Changing Volatility in Stock Returns, Journal of Financial Economics 31, 281-318.

Canina, Linda, and Stephen Figlewski, 1993, The Informational Content of Implied Volatility, Review of Financial Studies 6, 659-81.

Cappiello, Lorenzo, Robert F. Engle, and Kevin Sheppard, 2003, Asymmetric Dynamics in the Correlations of Global Equity and Bond Returns, WP European Central Bank, and University of California, San Diego.

Castrén, Olli, 2004, Do Options-Implied RND Functions on G3 Currencies Move Around the Times of Interventions on the JPY jUSD Exchange Rate?, Mimeo, ECB.

Chang, Jow-Ran, Vihang Errunza, Ked Hogan, and Mao-Wei Hung, 2005, An Intertemporal International Asset Pricing Model: Theory and Empirical Evidence., Forthcoming in the European Financial Management.

Chernov, Mikhail, 2003, Implied Volatilities as Forecasts of Future Volatility, Time-Varying Risk Premia, and Returns Variability, Manuscript, Columbia University.

201

Christensen, Bent J., and Nagpurnanand R. Prabhala, 2001, The Relation Between Implied and Realized Volatility, Journal of Finaneial Economies 50, 125-50.

Christie, Andrew A., 1982, The Stochastic Behavior of Common Stock Vari­ances Value, Leverage and Interest Rate Effects, Journal of Finaneial Eco­nomies 10 (4), 407-432.

Christoffersen, Peter, 1998, Evaluating Interval Forecasts, International Eco­nomie Review 39, 841- 862.

Christoffersen, Peter, and Kris Jacobs, 2004, The Importance ofthe Loss Func­tion in Option Valuation, Journal of Finaneial Economies 72, 291-318.

Christoffersen, Peter, and Stefano Mazzotta, 2004, The Informational Content of Over-the-Counter Currency Options, European Central Bank, Working Paper 366.

Covrig, Vicentiu, and Buen Sin Low, 2003, The Quality of Volatility Traded on the Over-the-Counter Currency Market: A Multiple Horizons Study, Journal of Futures Markets 23, 261-285.

DeSantis, Giorgio, and Bruno Gerard, 1997, International Asset Pricing and Portfolio Diversification with Time-Varying Risk, Journal of Fianance Vol. 52, 1881-1912.

DeSantis, Giorgio, and Bruno Gerard, 1998, How Big is the Premium for Currency Risk, Journal of Financial Economics 49, 375-412.

Diebold, Fracis X., Todd A. Gunther, and Antony S. Tay, 1998, Evaluating Density Forecasts, with Applications to Financial Risk Management, Inter­national Economic Review 39, 863-883 .

Diebold, Francis X., Jinyong Hahn, and Anthony S. Tay, 1999, Multivariate Density Forecast Evaluation and Calibration in Financial Risk Management: High Frequency Returns on Foreign Exchange, Review of Economies and Btatistics 81, 661-673.

Duffee, Gregory R., 2002, Balance Sheet Explanations for Asymmetric Volatil­ity, WP Haas Behaal of Business University of California, Berkeley.

Dumas, Bernard, and Bruno Solnik, 1995, The World Price of Foreign Ex­change Risk, The Journal of Finance 50, 445-479.

Engle, Robert F., and Kenneth F. Kroner, 1995, Multivariate Simultaneous Generalized ARCH, Econometrie Theary 11, 122-150.

202

Engle, Robert F., and Victor K. Ng, 1993, Measuring and Testing the Impact of News on the Volatility, The Journal of Finance 48,1749-1778.

Engle, Robert F., and Andrew J. Patton, 2001, What Good is a Volatility Model, Quntitative Finance 1, 237-245.

Engle, Robert F., and Kevin Sheppard, 2003, Theoretical and Empirical Prop­erties of Dynamic Conditional Correlation Multivariate GARCH, Working paper UCSD.

Errunza, Vihang, and Etienne Losq, 1985, International Asset Pricing under Mild Segmentation: Theory and Test, Journal of Finance 40 (1), 105-124.

Ferson, Wayne E., 1989, Changes in Expected Security Returns, Risk, and the Level of Interest Rates, The Journal of Finance 44, 1191-1218.

Ferson, Wayne E., Stephen R Foerster, and Donald B. Keim, 1993, General Tests of Latent Variable Models and Mean-Variance Spanning, Journal of Finance 48, 131-156.

Ferson, Wayne E., and Campbell R Harvey, 1993, The Risk and Predictability of International Equity Returns, The Review of Financial Studies 6, 527-566.

Figlewski, Stephen, and Xiaozu Wang, 2000, Is the "Leverage Effect" a Lever­age Effect?, Working Paper.

Fleming, Jeff, 1998, The Quality of Market Volatility Forecasts Implied by S & P 100 Index Option Priees, Journal of Empirical Finance 5, 317-45., Journal of Empirical Finance 5, 317-45.

Ghysels, Eric, Andrew C. Harvey, and Eric Renault, 1996, Stochastic Volatility. (G.S. Maddala and C. R Rao Eds. Handbook of Statitics, Vol 14, Elsevier Science).

Glosten, Lawrence R, Ravi Jaganathan, and David E. Runkle, 1993, On the Relation Between the Expected Value and the Volatility of the Nominal Excess Return on Stocks, Journal of Finance 48 (5), 1779-180l.

Goyal, Amit, and Ivo Welch, 2003, Predicting the Equity Premium with Div­idend Ratios, Management Science 49 (5), 639-654.

Harvey, Campbell R, 1991, The World Price of Covariance Risk, The Journal of Finance 46 (1), 111-157.

Harvey, Campbell R, and Akhtar Siddique, 2000, Conditional Skewness in Asset Pricing Tests, Journal of Finance 55, 1263-1295.

203

Hentschel, Ludger, 1995, AlI In the Family. Nesting Symmetric and Asymmet­ric GARCH Models, Journal of Financial Economics 39, 71-104.

Huang, Chi-Fu, and Robert H Litzenberger, 1988, Foundations for Financial Economics. (Prentice Hall Engelwood Cliffs, New Jersey 07632).

IMF, 2002, Global Financial Stability Report, Working paper, International Monetary Fund.

Ito, Takatoshi, 2002, Is Foreign Exchange Intervention Effective?: The Japanese Experiences in the 1990s, NBER Working Papers 8914 National Bureau of Economic Research, Inc.

Jorion, Philip, 1995, Predicting Volatility in the Foreign Exchange Market, Journal of Finance 50, 507-528.

Karolyi, Andrew G., and Rene Stulz, 2002, Are Financial Assets Priced Lo­cally or Globally?North Holland vol. in the Handbook of the Economics of Finance.

Kroner, Kenneth, and Victor K. Ng, 1998, Modeling Asymmetric Comove­ments of Asset Returns, Review of Financial Studies 11 (4), 817-44.

Lamoureux, Christopher G., and William D. Lastrapes, 1993, Forecasting Stock-Return Variance: Toward an Understanding of Stochastic Implied Volatilities" Review of Financial Studies 6, 293-326.

Longin, François, and Bruno Solnik, 2001, Extreme Correlation of Interna­tional Equity Markets, Journal of Finance 56 (2), 649-676.

Lopez, Jose A., and Christian A. Walter, 2000, Is Implied Correlation Worth Ca1culating? Evidence from Foreign Exchange Option Prices, Journal of Derivatives 7(3).

Malz, Allan M., 1997, Estimating the Probability Distribution of the Future Exchange Rate from Option Prices, Journal of Derivatives Winter, 18-36.

Merton, Robert C., 1973, A Intertemporal Capital Asset Pricing Model, Econo­metrica 41, 867-87.

Mincer, Jacob, and Victor Zarnowitz, 1969, The Evaluation of Economic Fore­casts. (in Economic Forecasts and Expectations, Columbia University Press New York).

Modigliani, Franco, and Merton H. Miller, 1958, The Cost of Capital, Corpo­ration Finance, and the Theory of Investment, American Economic Review 48, 261-297.

204

Neely, Christopher, 2003, Forecasting Foreign Exchange Volatility: Is Implied Volatility the Best We Can Do?, Manuscript, Federal Reserve Bank of St. Louis.

Nelson, Daniel B., 1991, Conditional Heteroskedasticity in Asset Returns: A New Approach, Econometrica 59, 347-370.

Newey, Whitney, and Kenneth West, 1987, A Simple Positive Semi-Definite, Heteroskedasticity and Autocorrelation Consistent Covariance Matrix, Econometrica 55, 703-708.

Ng, David Tat-Chee, 2004, The International CAPM When Expected Returns are Time-Varying, Journal of International Money and Finance 23, 189-230.

OECD, 1999, The Use of Financial Market Indicators by Monetary Authorities, Working paper, Organisation for Economic Co-operation and Development.

Pong, Shiu-Yan, Mark Shackleton, Stephen J. Taylor, and Xinzhong Xu, forth­coming, Forecasting Currency Volatility: A Comparison of Implied Volatil­ities and AR(FI)MA Models, Journal of Banking and Finance.

Sarno, Lucio, and Mark Taylor, 2001, Official Intervention in the Foreign Ex­change Market: Is It Effective, and If so, How Does It Work?, Journal of Economic Literature 39.

Sentana, Enrique, 1995, Quadratic ARCH Models, The Review of Economic Studies 62, 639-661.

Sercu, Piet, 1980, A Generalisation of the International Asset Pricing Model, Revue de l'Association Francaise de Finance 1,91-135.

Siegel, Andrew F., 1997, International Currency Relationship Information Re­vealed by Cross-Option Prices, Journal of Futures Markets 17:4, 369-384.

Solnik, Bruno H., 1974, An Equilibrium Model of the International Capital Market, Journal of Economic Theory 8, 500-524.

Stulz, Rene M., 1981, On the Effects of Barriers to International Investment Rene, The Journal of Finance 36 (4), 923-934.

Veronesi, Pietro, 1999, Stock Market Overreaction to Bad News in Good Times: A Rational Expectations Equilibrium Model, The Review of Fi­nancial Studies 12, 975-1007.

Whitelaw, Robert F., 2000, Stock Market Risk and Return: An Equilibium Approach, The Review of Financial Studies 13 (3), 521-547.

205

Wooldridge, Jeffrey M., 1990, An Encompassing Approach to Conditional Mean Tests with Applications to Testing Nonnested Hypotheses, Journal of Econometries 45, 331-350.

Wu, Guojun, 2001, The Determinants of Asymmetric Volatility, The Review of Financial Studies 14, 837-859.