exchange rate volatility implied from option prices

ASTON UNIVERSITY

MSc IN FINANCE & INVESTMENTS

……………………………

FORECASTING EXCHANGE RATE VOLATILITY USING OPTION-IMPLIED

INFORMATION

Student name: Srdjan Begovic

Candidate number: 816205

Name of Supervisor: Dr. Leonidas Tsiaras

Date of submission: September, 2014

Dissertation Submitted in Partially Fulfillmentof the degree of Master of Science in Finance & Investments

Exchange Rate Volatility Forecasting

Table of Contents

Abstract.................................................................................................................................................... ii

1. Introduction..........................................................................................................................................1

2. Literature review..................................................................................................................................4

3. Methodology........................................................................................................................................8

3.1. GARCH (1, 1)........................................................................................................................8

3.2. Black and Scholes Implied volatility....................................................................................10

3.3. Model-free implied volatility................................................................................................12

3.4. Realized volatility................................................................................................................14

3.5. Evaluation of forecasting accuracy......................................................................................15

4. Data processing and description........................................................................................................17

5. Empirical analysis..............................................................................................................................19

5.1. Forecasted volatility for the out-of-sample period (Jan. 2002 – Dec. 2006)............................19

5.2. Examining incremental information in volatility forecasting models.......................................22

6. Conclusion.........................................................................................................................................30

Reference list.........................................................................................................................................32

Appendix................................................................................................................................................36

A1.1 Volatility smile plot for CAD/USD exchange rate:..................................................................36

A1.2 Plot of strike prices across delta values for CAD/USD exchange rate:...................................37

A1.3 Diagrammatical plot of call prices against options delta values:...........................................38

A1.4 Calculation of the integral.......................................................................................................39

A2.1 Descriptive statistics for forecasts...........................................................................................40

i


Abstract

Forecasting volatility has been always a useful tool in financial world, due to its wide application in

almost every financial and financially-related activity. For decades, volatility forecasts have been used

by individuals engaged in every field of finance as the most important building block of their

investment decision-making. With the aim of achieving more accurate predictions of the future

movements of financial instruments, volatility forecasting has been improved a lot throughout its

history and still represents an interesting field of finance for exploration. However, when it comes to

investing, hedging or portfolio management, a greater challenge presents deciding upon the

forecasting model to rely on. Therefore, this study focuses on predicting the exchange rate volatility

based on three different forecasting models, and then compares the three applying several measures of

forecast accuracy in order to determine the most reliable estimate of the future realized volatility. The

three models are time-series GARCH (1, 1), Black-Scholes implied (BS) and Model-Free implied

(MF), which were applied for the out-of-sample period between January 2002 and December 2006

using weekly data on seven exchange rates. As an approximation of the actual volatility and a

benchmark to which the forecasts are compared in this study is a realized volatility computed by

squared returns.

Once forecasts were obtained, each was evaluated using accuracy measures that are based on errors

between forecasts and realized volatility. In addition, realized volatility was regressed against each

forecast in order to determine and compare their explanatory power and informational efficiency. It

was found that GARCH (1, 1) is the weakest model since the two implied volatility models subsume

its informational content and also possess an incremental information over GARCH (1, 1). The

complex part of the work though was deciding upon informational superiority between BS and MF

implied volatilities since measures of accuracy favor BS model for some exchange rates, while MF is

the dominant one for others. Moreover, the difference between the error statistics of the two models is

very small to be considered significant. Due to the fact that a strict comparison that would take into

account even these small differences won’t help much in determining the most accurate model, the

paper finally applies Diebold-Mariano’s test in order to check the significance of the difference

between the BS and MF implied forecasts. It was found that the null hypothesis which states that the

difference between the two is statistically insignificant, cannot be rejected at 1% level of significance.

Therefore, this study paper concludes that the GARCH (1, 1) is the least accurate estimate of the

future realized volatility, while it is indifferent as to BS and MF implied forecasts which are

significantly better than GARCH (1, 1) but do not differ much between one another. Although, it was

expected that the MF model will be the dominant one, as many studies prove this with the other

financial assets, it seems that results are not very stabile in the case of exchange rates. It was suggested

that the success and failure of the MF implied volatility is driven by the data availability which may

Abstract ii

Exchange Rate Volatility Forecastingeither weaken or strengthen the model, thus this can be a major reason for having such results in this

work.

Table of contents iii


1. Introduction

One of the greatest problems faced by all financial market participants is to predict future for purposes

of making an appropriate investment decisions today. Therefore, human natural inability to predict the

future prevents investors to overcome the uncertainty about future outcomes that may be financially

harmful or rewarding for investments. As for any other case, in financial world, every future outcome

is to some extent dependent on today’s performance while in the same time independent to a much

larger extent since various unrealized factors may affect the future likelihood of targeted outcome.

Although future predictions are supported with various calculations, estimates and assessments, it still

presents an approximation of the actual future that remains unknown.

In general, the point of a concern, when it comes to investing, is the direction of market prices and

thus the future payoff from investment. Over time prices are moving upwards, downwards and

sideways making any short-term or long-term predictions complex. Such variability is known as

volatility and presents a core parameter of interest to all individual investors, financial institutions,

portfolio managers and other market participants. Volatility can be defined as a measure of the extent

to which the asset’s return fluctuates, or disperse from its estimated average. Since increased future

dispersion of security’s returns from the mean increases uncertainty regarding payoff at maturity,

volatility has been always linked to the financial risk. The general rule suggests that the greater the

volatility the greater is the financial risk associated with specific security, asset or a financial

instrument.

However, volatility is not constant, but rather changes over time. To some small extent such a change

in volatility can be explained by various parameters, while most of it still remains inexplicable. For

example, Taylor (2007) suggests that the variance in volatility can be partially explained by major

macroeconomic factors such as inflation and unemployment rate, interest-rates, foreign exchange, and

etc. Besides economic parameters, it is also believed that human psychology drives prices in the

market contributing on such way to their volatilities. An example can be “overconfidence problem”, a

consequence of a previous steady bull market that prevents investors from on-time reaction to

reversals, resulting afterwards in a sudden threat of market crash that leads to massive sales and thus

tremendous fall in prices (Ritter, 2003). However, the unexplained part of change in volatility presents

the uncertainty which is strongly linked to the financial risk associated with all the assets traded.

With the purpose of obtaining some picture about the future price volatility, forecast models for

volatility have become the most valuable practice widely used nowadays. Generating an

approximation of future volatility reduces uncertainty with regards to investing and thus leads to better

allocation of funds invested in portfolio securities, more accurate development of hedging strategies as

well as pricing of securities and other financial instruments.

1


Considering such a close relationship between financial risk and volatility, accurate predictions of the

possible future pattern of returns, volatility of returns, has become the major challenge that involves

lots of complexities and struggles. Till now, many different theories were established and various

volatility forecasting models have been developed. Traditionally, it was believed that historical price

patterns repeat as time passes and that it is therefore possible to predict the future based on past price

performance. From this idea time-series volatility forecasting models were derived using past standard

deviation of returns, so called realized volatility, to predict the future volatility of returns. More

advanced approaches to forecasting volatility have been derived with the use of conditional volatility

which is a future volatility of returns conditional on the information contained in past returns. This has

led to the development of more sophisticated time-series models such as ARCH family models, while

further application of stochastic process resulted in derivation of stochastic volatility models that

account for asymmetry in volatility process. With development of Black and Scholes model for

pricing option contracts, volatility forecasting has advanced to a new level since the model considers

volatility as an immense parameter for determining the fair value of options (Figlewski, 2004).

Namely, all the other parameters involved in Black and Scholes formula, such as strike price, time to

expiration, interest rate and price of the underlying asset, are directly observable, while volatility is the

only one that needs to be predicted. Therefore, carrying this forecasted parameter, options fair values

are reflections of the future expectations of market participants. Such forward-looking volatility is also

known as implied volatility and it has become the basis of most newly developed volatility forecasting

models. However, existence of misspecification errors that caused biasness in BS implied volatility

model have led to new advances which further improved forecasting accuracy and reduced model

exposure to misspecification errors. One such alternative is a model-free implied volatility which

subsumes all the information contained in both time-series and BS implied volatility models by taking

into consideration all possible strike prices for specific option.

Application of volatility forecasting is wide and important. For example it plays a vital role in

financial risk management for banks and other financial institutions since it provides the estimate of

Value-at-Risk (VaR) which these institutions use to determine their capital requirements (Poon and

Granger, 2003). Moreover, prices of derivative securities are based on volatility of underlying asset,

thus it has become more convenient nowadays for dealers to display implied volatilities of options

instead of options prices since they carry more information about the market’s future expectations.

Finally, due to the fact that volatility may negatively impact the economy, the policy makers often rely

on market estimates of volatility as a measure of susceptibility of financial markets and the economy

as a whole.

The purpose of this paper is to evaluate accuracy of three forecasting models that are applied in

calculation of exchange rate volatilities. First two models that will be applied are historical or time-

series and implied volatility models, which comparisons are usually the central topic of majority of

studies, while the third one is relatively new and is known as model-free implied volatility. However,

2


before the methodology is fully explained it is necessary first to assess findings of prior studies that

apply, evaluate and compare various forecasting models.

3


2. Literature review

Earlier studies of implied volatility found that although it contains relevant incremental information

regarding future and is thus more efficient and accurate than time-series models of historical volatility,

it is still biased (Jiang and Tian, 2005). For example, Day and Lewis (1992) evaluated and compared

the informational content of volatility implied from call options on the S&P 100 relative to

informational content of conditional volatility estimates of GARCH and EGARCH. Study finds that

information contained in implied volatility is also reflected in conditional ones, but when in-sample

tests were performed, both estimates failed to explain the stock market volatility. Another study that

confirms such findings is the one conducted by Jorion (1995) which assesses the information content

and forecasting ability of implied volatilities that are derived from options written on foreign currency

futures, relative to that of time-series models. The results have shown that implied volatilities

outperform time-series ones even in case when time-series models have the benefit of “ex post”

parameters, but they are still upward biased forecasts. Xu and Taylor (1995) once again confirmed that

implied volatilities have richer information content and thus greater and accurate predicting power

than historical volatilities in forecasting exchange rate volatility for four different exchange rates

taking a sample within a period from 1985 to 1991. However, studies by Canina and Figlewski (1993),

Lamourex and Lastrapes (1993) and Fleming (1998) suggest that incremental information of implied

volatility doesn’t go much beyond what is contained in historical volatilities and thus indicate the

biasness in forecast of a future realized volatility.

However, recent researches investigated the possibility of improving accuracy of and reducing

biasness in implied volatility models by using high-frequency data. This approach obtained contrary

opinions and conclusions which, based on new results, now suggest that such improved implied

volatility forecasting method provides more accurate forecasts than historical volatility. Pong,

Shackleton, Taylor and Xu (2004) study applies four different models in forecasting exchange rate

volatility for Pound/Dollar, DM/Dollar and Yen/Dollar taking a sample from period between 1994 and

1998 and then compares their forecasting accuracy based on mean-squared error (MSE) criterion and

R2statistics. For models applied, ARFIMA, ARMA (2,1), GARCH (1,1) and option implied volatility,

with one day to three months forecast horizon, MSE finds that ARFIMA and ARMA are much more

effective than implied volatility for short-term forecasting, while volatility implied from options

outperform all models when longer term horizons are considered. As means of R2statistics which

measures the informational content in forecasts for each individual model, it has shown results that are

consistent with MSE findings. Namely, for shorter term horizon ARFIMA and ARMA have

significantly higher R2 values than implied volatility meaning that these models are “information

richer”, while opposite holds in the case of longer term time interval. GARCH model has been

evaluated as the one with the weakest predicting accuracy with both MSE and R2 criterion.

4


This is also supported by the work of West and Cho (1995), which uses weekly data on five different

exchange rates within a time interval between 1973 and 1989 to assess the out-of-sample predicting

ability of GARCH and three other time-series models from ARCH family, finding that none of them is

providing efficient and satisfactory results for a horizons longer than a week. Martens and Zein

(2002), however, find that if historical forecasting approach is improved by considering high-

frequency data and long memory models, it will provide forecasts that are good enough to compete

with and even show better results than implied volatility approach.

In addition, de Andre and Tabak (2001) assess the informational content of dollar-real exchange rate

implied volatility relative to that of a historical volatility. The study paper examines whether implied

volatilities possesses relevant information over the life of the option that is not contained in historical

returns. Applying GMM criterion it was found that implied volatility contains more significant

incremental information and thus provides more accurate forecast relative to results generated by

GARCH (p,q) model or any Moving Average (MA) predictor.

Moreover, Malz (2001) shows that informational content of implied volatility really is valuable since

it can provide an early indication of future market stress. Performing out-of-sample tests in estimating

the future realized volatility for bond, stock and foreign exchange markets, Busch, Christensen and

Nielsen (2009) find that implied volatility involves incremental information for all examined markets,

but also that it is an unbiased forecasting model for foreign exchange and stock markets. Similarly,

Kroner, Kneafsey and Claessen (1995) support the conclusion that volatility implied from the

commodity option prices contains incremental information regarding future realized volatility relative

to historical estimates.

BS Implied volatility forecasts outperform historical volatility not only in forecasting exchange rate

volatility but in forecasting volatilities of other financial instruments, assets, commodities, indexes and

etc. Day and Lewis (1993) study indicated that in out-of-sample tests for predicting the volatility of

futures prices on crude oil, GARCH and EGARCH models fail to provide satisfactory results while

implied volatility has once again shown to be the most convenient approach to apply. Comparing the

informational content of VIX implied volatility to high-frequency index returns in predicting volatility

of S&P 100 index, Blair, Poon and Taylor (2000) find that for both, in-sample and out-of-sample tests,

VIX contains more significant incremental information than the latter one. In the same way, focusing

on S&P100 and NASDAQ100 indexes, Giot (2005) suggests that the VIX contains the greater

informational content relative to forecasts based on GARCH and Risk Metrics. Based on regression

results of some prior studies on volatility forecasting for S&P 500 futures options, Ederington and

Guan (2002) suggest that the implied volatility really dominates the historical volatility forecast since

it incorporates all the information carried by the latter one.

Viewing a credit default swap (CDS) as an out-of-the-money put option, Cao, Yu and Zhong (2009)

examine a possibility of utilizing the volatility implied from put options in pricing CDS. What was

5


found is that implied volatility is much more effective than historical volatility in predicting future

realized volatility and thus in clarifying time-series variability in CDS spreads. Moreover, Szakmary et

al (2003) examines forecasting efficiency of implied volatility using data from 35 futures markets,

estimating that once again implied volatility possesses crucial incremental information regarding

future realized volatility than historical volatility does.

As means of commodities, Egelkraut and Garcia (2006) found that informational content of implied

forward volatility beats that contained in historical volatility, however, forecasting accuracy varies

with commodity’s characteristics. For example, market volatility for soybeans and corn can be well-

predicted throughout the essential growth periods, while market volatility for soybean meal, wheat and

hogs is less predictive. Giot (2002) also indicates that implied volatility for a set of agricultural

commodities overcomes GARCH (1, 1) regarding its informational structure. Similarly, a study of

Benavides (2009), which evaluates the predictive accuracy of time-series ARCH, option implied and

composite (mixture of historical and implied forecasts) models, indicates that implied volatility model

is superior relative to ARCH, but that the composite one is the most accurate. Such finding is also

supported by Manfredo et al (2001), suggesting that there is no significant informational difference

between GARCH (1, 1) and option implied forecasts, but rather, the composite model should be used

as the most reliable one.

Besides comparisons between realized and implied volatility based on an informational content, there

are also studies that evaluate the efficiency of implied volatilities based on whether out-of-money or

at-the-money options were used in computations. For example, Hung-Gay-Fung, Chin-Jen Lie, and

Moreno (1990) assessed the forecasting ability and accuracy of different estimates of the future

exchange rate volatility. It was estimated that the out-of-the-money implied volatility is more precise

than at-the-money implied volatility. Therefore, when forecasting the exchange rate volatility they

suggest that considering out-of-the-money options is more convenient practice. On the other hand

Poon and Granger (2003) argue that at-the-money options are much better solution for deriving

implied volatilities because of increased trading volume of these options, since the increased trading

volume indicates more efficient trading environment and thus less biasness produced by BS model.

However, taking into account only at-the-money options still leads to misspecification errors in BS

model estimates due to the fact that on such way all the relevant information carried in other options is

excluded from forecasting procedure.

Since the Black and Sholes model for pricing options was introduced in 1973, the information carried

by the volatility implied from option prices has been an important subject of examination in majority

of studies. For so long, it was considered as more accurate forecast of future realized volatility,

relative to time-series volatility forecasting models, due to the richer information content, until the

introduction of the model-free implied volatility by Britten-Jones and Neuberger in 2000. Improving

the prior work of Derman and Kani (1994), Dupire (1994) and Rubenstein (1994) that take into

account only deterministic volatility, Britten-Jones and Neuberger (2000) have managed to prove that

6


volatility forecasting doesn’t need a particular process but rather only option prices. Such a model is

not only considered to be more accurate than historical forecasts due to its options-orientated origin,

but is also expected to dominate the BS implied volatility model since it is model-independent and it

considers options at all available strikes (Taylor, Yadav and Zhang, 2007). Studies of Carr and Madan

(1998) and Demeterfi, Derman, Kamal and Zou (1999), Carr and Wu (2003) have tested the

information content of the model-free implied volatility indicating that equilibrium variance swap rate,

which approximates to model-free volatility expectations, is highly significant in determining time-

series realized volatility for all stock indices and most of individual stocks. Jiang and Tian (2005)

indicate that model-free implied volatility expresses significantly higher correlation with the future

realized volatility than both ATM implied or any other historical volatility. Comparing model-free

volatility to historical volatility, estimated by high-frequency returns, for S&P 500, FTSE 100,

Eurodollar futures and short sterling futures, Lynch and Panigirtzoglou (2004) showed that besides

being more powerful than the latter approach, model-free volatility is biased forecast of future realized

volatility. According to Taylor, Zhang and Wang (2010), the model-free implied volatility really

highly correlates with the future realized volatility, but its informational content is not significantly

different from the ATM BS implied volatility due to its sensitivity to limited range of strike prices as

well as to measurement errors incorporated in option prices.

Taking into consideration all the previous studies reviewed and their findings, this paper will once

again recheck their validity by performing volatility forecasting for seven different exchange rates

comparing between GARCH, BS implied and model-free implied volatility forecasts indicating the

most accurate one. Besides the fact that most studies highlight the success, superiority and advantages

of model-free approach, it is still arguable whether this forecasting model will dominate the other two

as exchange rates data is used. Namely, presence of outliers and shortages in data availability may lead

to lots of inaccuracies and measurement errors that will result in a failure of the model-free approach

since the calculation procedure of this model is very sensitive in its own nature and requires a lot of

mathematical assumptions that may harm the informational content of forecasted volatility. Therefore,

greater attention in this study paper will be given to comparison between model-free implied volatility

and BS implied volatility as they are considered to be rivalry models.

7


3. Methodology

3.1. GARCH (1, 1)

Generalized ARCH or so called GARCH model has become a very popular approach in time-series

volatility forecasting due to the fact that it is an extended ARCH model that takes into consideration

not only past squared residuals but also past conditional variance. Its specification as GARCH (1, 1) is

interpreted as taking into account one previous squared residual and one previous conditional variance

(Taylor, 2007). Therefore, forecasting future volatility using GARCH (1, 1) model is to a large extent

dependent on the historical prices pt, returnsrt, conditional variances ht and standardized residuals z t

which are linked by two major equations:

rt=log( p t

pt−1)=μ+h t

1/2 zt(1)

and ht=ω+α ¿

As it can be seen, in order to apply these equations, a derivation ofμ,ω, α and β parameters is required

since they are a vital part of the two formulas. However, obtaining these is not so straightforward, but

rather involves generation of some “temporary values” forμ,ω, α and β, in order to get “model

estimates” for conditional variances which are afterwards readjusted by the maximum log-likelihood

procedure. Most studies perform this part of work using various software packages, usually e-views,

but here GARCH (1, 1) will be computed in excel and each step will be explained in full.

To begin with the “model building”, it is necessary first to define in-sample and out-of sample periods,

since GARCH (1, 1) out-of-sample estimates of conditional variance are dependent on the conditional

variance estimates from in-sample period. In this study, the whole sample ranges from January 1990

till December 2006. However, due to the fact that the provided options data ranges only between

January 2002 and December 2006, this will be chosen as an out-of-sample period for GARCH (1, 1)

so its forecasts during that period can be compared to implied volatility forecasts.

First step towards parameters estimation involves generating sample mean, standard deviation and

unconditional variance using logarithmic returns from the in-sample period:

μ= 1N is

∑i=1

N is

rt ,is(3)

σ= 1N is

∑i=1

N is

(rt ,is−r )(4)

8


σ 2= 1N is

∑i=1

N is

¿¿

Where, N is is a sample size of the in-sample period, while rt ,is and r are returns and a mean return

within same period respectively. Obtaining these three allows derivation of previously mentioned

“temporary values” for the GARCH (1, 1) parameters: 1) conditional mean (μ) is set to be equal to

computed sample mean, 2) alpha (α ¿ and beta¿) parameters are assigned values of 0.06 and 0.92

respectively, as these values are usually used in prior studies and in literature, and finally 3) omega

(ω) is calculated from the unconditional variance formula which has a following form:

σ 2= ω1−α−β

(6)

Since alpha, beta and unconditional variance are already known, computation of omega presents a

very simple equation with one unknown:

ω=σ 2∗(1−(α+ β ))(7)

GARCH (1, 1) equation for conditional variance can be now applied for the in-sample period due to

the fact that all necessary parameters have been estimated. However, in order to compute conditional

variance for one specific date, it is important to have one past variance and a standardized residual.

The in-sample period stars on the 15th of January 1990 and before that date paper considers no past

information, thus the variance for that “starting date” will be computed as a simple unconditional

variance which is just a squared standard deviation. In addition, standardized residual at this starting

point is generated as a ratio of the difference between that date’s return and conditional mean, to

square root of previously computed “starting variance”:

z t=rt−μ

√ht

(8)

Conditional variances for all other dates within in-sample period will be computed by GARCH (1, 1)

equation:

ht=ω+¿

Although most of the necessary building blocks of the GARCH (1, 1) forecasting equation have been

explained, it is still important to readjust μ, ω, α and β parameters for the robust standard errors

computing their actual values. This is done by maximum log-likelihood approach which involves

9


derivation of log-likelihood densities and then maximization of their sum. The log likelihood function

is as follows:

logL=∑t=1

n

lt(10)

While the densities are: lt=−12

[ log (2 π )+ log ( ht )+zt2] (11)

Maximization procedure is performed by changing the values of μ, ω, α and β parameters under

conditions of α ≥ 0.0001 , α+ β ≤ 0.9999 , and σ 2≥ 0. However, Taylor (2005) suggests that the whole

process is more easily conducted if parameters are of the same magnitude. Thus reparametrization of

previously assigned values is required before going any further:

μreparametrized=1000∗μ

α reparametrized=0.06

α +β persistance=0.06+0.92

σ reparametrized2 =10000∗σ2

As the magnitudes of the four parameters (μ,ω, α and β ) have been reparametrized, maximization of

the sum of log-likelihood densities is performed over these so called “optimization parameters”. Once

this is done, in-sample conditional variance has been readjusted for the actual values of the

parameters. Final step in forecasting volatility by GARCH (1, 1) involves computation of out-of-

sample conditional volatility. This is done by simply using the same volatility equation as before, but

the variance computed is now conditional on the variance of last date of the in-sample period.

Conditional volatility estimated for this period is the one that will be compared to the implied

volatility forecasts.

3.2. Black and Scholes Implied volatility

Although time-series models such as previously presented GARCH(1,1) can provide a significant

information regarding future volatility, they are considered as backward looking due to the fact that

they rely on the historical data. However, much better volatility forecasting alternative has been

identified in models that price options. Namely, it is believed that prices of options written on financial

assets show market’s expectations of the future movement of these assets. Therefore, the concept of

volatility that is implied in the observed market prices has become a usual practice used by most

financial analysts when it comes to volatility forecasting since it is forward looking, based on market’s

10


future expectations, and as such contains relevant incremental information relative to time-series

models.

The general idea about implied volatility is based on one of the most sophisticated and simplest option

pricing model, known as Black and Scholes model. Although many studies have proven that the model

is biased due to a number of unrealistic assumptions it takes, it is still very convenient since these

assumptions are really powerful and as such enable analysts to generate important information

regarding future behavior of the asset underlying option. Namely, by assuming that markets are

complete and efficient, without arbitrage opportunities, BS model ensures that a perfect hedge can be

found for any asset traded. Moreover, all the assets are assumed to follow a geometric Brownian

motion (GBM) process, shown below, which also implies a constant volatility and i.i.d lognormal

returns (Alexander, 2001). In the case when option is written on currency, the GBM has a following

formula (Malz, 1997):

d S t=(r−r ¿) S t dt+σ S t dZ (12)

Another important contributor to simplicity of the model, which in the same time leads to biasness, is

the risk-neutral valuation. Its essence lies in the fact that it considers, as an assumption, risk-neutral

world where prices of all the financial assets grow at the same interest rate, known as a risk-free rate.

Finally, it is assumed that there are no transaction costs when trading options, nor taxes or restrictions

on short sales (Macbeth and Merville, 1979). Therefore, with such assumptions the BS model is

considered to be incorrect providing only a rough approximation of the reality. The following are BS

formulae for currency call and put options:

For call option: c=S0e−r f T N ( d1 )−K e−rT N (d2) (13)

For put option: p=K e−rT N (−d2 )−S0 e−rf T N (−d1 ) (14)

As it can be seen, the model incorporates and links price of the underlying currency or spot exchange

rate S0, a strike price K , time to maturity T , and foreign and domestic interest rates r f and r

respectively. N (d1 ) and N (d2) present cumulative standard normal distributions of d1 and d2

computed as follows:

d1=ln( S0

K )+(r−r f +σ2

2 )Tσ √T

, and d2=d1−σ √T (15)

So, the BS formula creates a linkage between the price of option, the volatility of the asset underlying

the option σ , as well as all other previously mentioned parameters that impact the option’s value. The

procedure for generating the implied volatility is very straight forward. Namely, by assuming the

11


random value for the volatility, it is possible to estimate a theoretical price, or so called “model price”

for specific option using BS model. If the option is traded in the market at the same strike and maturity

its market price is observed and compared to the previously computed model price. Usually, there is a

discrepancy between the two, so the question is, what should be the volatility that once plugged into

the BS equation equates the model and market prices of the option. Such volatility presents the

implied volatility and its generation can be either done on iterative basis or using some sophisticated

computer programs. The ease of such procedure lies in the fact that, except volatility that needs to be

estimated, all determinants of the option’s value such as strike price, expiry date, interest rate and

dividends over the life of the option, are directly observable data. However, nowadays, prices of

options are mostly presented in terms of their implied volatilities, thus ATM implied volatility will be

directly obtained from the British Bankers Association online database so there is no need for its

computation in this study.

3.3. Model-free implied volatility

The importance of this recently developed approach in volatility forecasting lies in the fact that it is

not dependent on any particular option pricing model since it is a pure result of no-arbitrage condition.

As mentioned in the literature review, it is expected to be the most accurate one since many studies

have found that it subsumes the information contained in the other two presented models by extracting

some extra information from all strike prices for one specific option. Unfortunately, in reality

availability of strike prices in the market is limited which generally presents an obstacle to easily and

directly conduct this approach. However, various studies apply various methods in order to get as

close as possible to a real-life mathematical approximations of the model-free volatility formula. One

such formula that will be used in this paper is proposed in the study of Jiang and Tian (2005) built on

findings of Britten-Jones and Neuberger (2000), which has the following format:

2 ∫Kmin

Kmax CF (T , K )−Max (0 , F0−K)K2 dK (16)

As it is shown, model-free implied volatility is presented as an integral of option prices over the range

of strikes that goes from minimum to maximum (Jiang and Tian, 2005). The problem of limited

availability of strike prices in the market can be overcome by using a volatility surface as a base for

extracting all possible strike prices. Construction of a volatility surface, which is a plot of implied

volatilities across different strike prices and maturities, allows model-free approach to take into

consideration not only ATM implied volatility like BS model, but OTM one as well. Such a huge pool

of both ATM and OTM implied volatilities can be used to generate a pool of strike prices

corresponding to each implied volatility. Therefore, the first step towards extracting strike prices

involves building volatility surface using the method proposed by Malz (1997). This method is known

in literature as second-degree polynomial interpolation in delta due to the fact that it interpolates

12


between prices of 25-delta risk reversals (rr t), 25-delta strangles (strt), and at-the-money implied

volatilities (atm t) in the delta-volatility space (Beneder and Elkenbracht-Huizing, 2003). The equation

for such interpolation method has the following form:

σ (δ)=b0 atmt +b1 rrt ( δ−0.5 )+b2 str t ¿ (17)

Applying this second-degree polynomial for each date in the examined horizon and across observed

delta values generates what is known as volatility smile (presented in appendix A1.1). The volatility

smile is a plot of the option’s implied volatility against strike prices with the shape of a smile, U-

shape, which is a result of having the lowest implied volatilities at near-the-money (NTM) options and

the highest at out-of-money (OTM) ones (Hull, 2012). However, the smile in this case is plot in delta-

volatility space instead of strike-volatility space. Its equation is formatted such that ATM implied

volatility presents a measure of location for the constructed smile, while risk reversal and strangle

prices determine its skew and a degree of curvature respectively.

Delta here presents the “option’s delta” which is defined as a change in the price of an option when

there is a change in the price of the underlying (Reiswich and Wystup, 2010). Most traders use it as a

hedging strategy as it shows how much of the underlying should be bought or sold in order to hedge

taken position in options. However, in this specific case, when constructing the volatility surface, delta

presents a substitute for moneyness which is a ratio of strike price to asset price. Therefore, the

volatility surface lies in the delta-volatility-time space as implied volatilities will be generated across

different delta values, which in this study vary from 0.01 to 0.99, and across time. The formula for the

option delta has the following functional form:

δ v (S t , T , K , σ , rd , rf ) ≡∂ v ( S t ,T ,K ,σ , rd ,r f )

∂ S t

=e−r f T Φ [ ln( St

K )+(rd−rf +σ 2

2 )Tσ √ T ]

(18)

St−¿ spot exchange rate;

T−¿time to maturity;

K−¿ strike price;

σ−¿implied volatility;

rd−¿ domestic interest rate (USD monthly LIBOR rate);

r f−¿ foreign interest rate (monthly LIBOR rate of any currency other than USD);

Φ−¿standard cumulative normal distribution function

13


However, as this study uses forward prices instead of spot ones, the right hand side of the equation

will be just slightly different:

e−r f T Φ [ ln(F t

K )+( σ2

2 )T

σ √ T ], where F t=S t e( rd−rf ) T (19)

Once implied volatilities are generated with the second-degree polynomial formula, the next goal is

extraction of strike prices for each corresponding implied volatility at each delta, moving on such way

from delta-volatility to delta-strike space. Since it is shown previously that strike price is a function of

delta, the extraction process can be done by rearranging the delta formula solving it for K. However,

this procedure is not very straightforward as it seems, but rather requires few steps. It begins by

directly obtaining a value for the standard normal cumulative distribution function:

δ er f T=Φ [ ln(F t

K )+( σ2

2 )T

σ √ T ]Applying the inverse function of the estimated value, d1 parameter of Black-Scholes formula for

option pricing can be derived:

d1=ln( F t

K )+( σ2

2 )T

σ √T

Finally, this allows for a rearrangement such that the equation above is solved with respect to a strike

price K. The equation has following form:

K=F t e( σ2

2 )T −d1 σ √T (20)

Again, this is done across all deltas and for the whole assessed time horizon getting on such way strike

prices for each date at various delta values. Diagrammatical plot of strike prices across delta values is

provided in appendix A1.2. Once strike prices are extracted, all parameters necessary to obtain call

prices that correspond to these strikes are now available to be plugged into BS call price equation

presented earlier:

c=e−rd T [F ¿¿0 N ( d1 )−KN (d2)]¿

Again, these call prices are generated in the delta-call space, and the plot of call prices against delta

values for a specific date is presented in appendix A1.3. The final step involves computing the term

within the integral, presented in formula (16), across delta values and then summing up these terms

14


generating on such way model-free implied volatility at each date. Explanation regarding integral

computation is presented in the appendix A1.4.

3.4. Realized volatility

In order to evaluate the forecasting ability and informational significance of the three models

examined in this work, it is very important to have an approximation of the actual volatility that will

be used as a benchmark to which the forecasts will be compared. This study paper will use daily

squared returns to proxy the realized volatility, as this approach is widely used in prior studies. Since

weekly data is used and each Monday presents one week, then realized variance on particular Monday

needs to consider all squared returns within that week. Thus the formula for the realized variance for

each date in the sample is as follows:

σ 2=1n∑i=1

n

r t ,i2 (21)

Where n presents the number of days within a week. One limitation to such realized volatility is the

fact that squared returns are too noisy approximation of actual volatility. This will present a small

problem when it comes to model evaluation part since it will lower the explanatory power of each

model when regressed against realized volatility. However, comparisons won’t be based only on

coefficients of determination, but will consider other accuracy measures as well.

3.5. Evaluation of forecasting accuracy

Forecasting volatility is a valuable practice to all market participants as well as to all financial

institutions. For example, future movement of asset prices is of a great importance to all traders and

analysts when pricing options, while exchange rate volatility forecasts play an important role in

deciding upon future operations and interventions of central banks around the world. However, due to

the fact that there are a lot of models for volatility forecasting, an important step towards securing the

reliability of estimations, is to carefully decide upon the model to be used. Therefore, whatever model

applied, it should be evaluated by so called “loss functions” that show the magnitude of mismatch

between forecasted volatilities and realized one (Lopez, 1995). On such way, these loss functions

present sufficiently good benchmarks for volatility forecasts.

In general, Diebold and Lopez (1996) suggest that the common characteristic for all of these forecast

accuracy measures is the fact that they are based on, or in other words, derived from forecast errors or

percent errors which have the following forms respectively:

Forecast errors: ε t+H= y t+H−f t+H

15


Percent errors: pt+k , t=( y t+H −f t+H)/ f t+H

Where, ε t+His the error, or a discrepancy between realized volatility ( y t+H) and the forecast

( f ¿¿ t+H )¿. However, the error equation presented above is just one of various loss function types.

Depending on which loss function is used, there are different types of accuracy measures. Several such

measures will be applied in this study in order to strengthen the analysis:

MSE ( j )= 1N∑i=1

N

¿¿ (22)

RMSE ( j )=√ 1N ∑

i=1

N

¿¿¿ (23)

MAE ( j )= 1N∑i=1

N

|y t+1−f ( j)t+1| (24)

MAPE ( j )= 1N∑i=1

N |y t+1−f ( j)t+1|

y t+1

(25)

HMSE ( j )= 1N∑i=1

N

¿¿ (26)

As it can be seen the first two, equations (22) and (23), are using squared errors as a loss function.

MSE or so called Mean Squared Error is simply an average of squared errors, while RMSE (Root

Mean Squared error) is its square root. Similarly, equations (24) and (25) can be paired based on their

loss functions since either uses absolute errors. The two only differ in a sense that MAE, or Mean

Absolute Error, takes an average of absolute errors, while MAPE averages the proportion of absolute

errors in realized volatility and is thus called Mean Absolute Percentage Error. Final measure of

accuracy HMSE, equation (26), has a bit different loss function relative to others. It presents a

variation of MSE that is adjusted for heteroscedasticity and is thus expected to provide more accurate

results than other measures. The general rule for all accuracy measures presented above suggests that,

the most reliable model will be the one that has the lowest results (Taylor, 2005). This would mean

that the forecasts estimated by that specific model are the closest to realized volatility.

Besides evaluations based on loss functions, the empirical analysis will be strengthen further by

regression analysis. Regression analysis is a very useful tool when it comes to evaluation of the

informational content of volatility forecasting models. Parameters of interest are slope coefficients,

which show whether there is any correlation between realized volatility and forecasts, and coefficient

of determination that evaluates the explanatory power of forecasting models. This study will apply two

types of regression: 1) univariate – which regresses realized volatility against each forecasting model

16


separately, and 2) encompassing – that involves more than one volatility model in a regression.

Univariate and encompassing regressions have the following form:

Univariate regression: y t+1=β j , 0+β j ,1 f t+1+εt (27)

Encompassing regression: y t+1=β j , 0+β1GARCH (1 , 1 )+β2 f IV +εt (28)

The main objective of univariate regression is to derive slope coefficients (beta coefficients) with the

aim of testing correlation of each model with realized volatility and their biasness. In addition,

coefficients of determination for each model generated from this regression will be used as a

comparison criteria. The rule of thumb suggests selecting the model that has the highest R2 criterion as

the most accurate. On the other hand, encompassing regression will serve as a tool for comparing

impacts of BS and MF implied volatility models on explanatory power of regression when they are

separately added to GARCH (1, 1). Such comparison will actually show which of the two implied

volatility models has incremental information over the other one. The model that subsumes the

information contained in the other one will have significantly higher coefficient of determination when

added to GARCH (1, 1).

17


4. Data processing and description

Before getting onto calculations and the empirical part of work, it is necessary first to describe the way

that provided data was used and processed, and also to give an attention to adjustments and

assumptions applied. To begin with, this peace of research examines three different volatility

forecasting models for seven exchange rates in the out-of-sample period between 2002 and 2006 using

weekly data on options and exchange rates. The data are obtained from the Bank of England and

British Bankers Association online databases. However, these are not available in a weekly format but

rather in daily, thus obtaining weekly returns and options data is the first adjustment conducted.

The approach used to generate weekly returns considers filtering out only Monday prices for each

exchange rate and then computing logarithmic returns out of these. Similarly, options data, which

includes at-the-money implied volatility and 25-delta risk reversals and strangles, are also filtered out

on the “Monday basis” to match the returns data. As it is known, if the sample size is increased, the

final results will be more accurate, thus another adjustments involves enlarging the sample size. Since

the provided data only considers trading days, the sample size is now improved by including all the

other days that were left out. Each added day was assigned a value from the day before. For example,

if Friday and Monday were left out and then were added back, they are assigned values from the

Thursday. Another reason of why these excluded days were added back is the fact that there was a lot

of problems in computation of realized volatility. Namely, the proxy for realized volatility in this

study is computed by taking an average of daily squared returns within a week (Monday to Friday).

Since a lot of days were excluded from online database, some weeks had only 3 to 4 days instead of 5,

so computed realized volatility would be inaccurate. Once this adjustment has been conducted and

Mondays were filtered out, the new sample size counts 260 observations.

Due to liquidity reasons there was no options data for some Mondays’ that were already included in

the examined sample. Therefore, the adjustment applied in this case assumed that the Monday for

which the options data is not available will be assigned the same values as that of previous Monday.

Since there are only a few such data gaps, this adjustment won’t significantly affect the research. The

same approach was used when dealing with extremely large values on some Mondays, so called

outliers, but this was a rear situation and it occurred only twice. In order to match the selected forecast

horizon of one week to options data, the paper uses implied volatility data on options which maturity

is one week. Since there was no weekly risk-reversals and strangles to be associated with these weekly

ATM implied volatilities in the smile equation, this has called for a new assumption. Thus, assuming a

flat yield curve allows the use of monthly risk-reversal and strangle prices as weekly in calculation of

volatility smile. Although an alternative approach would be a linear interpolation, it won’t provide

significantly different results.

18


Another important rule worth mentioning considers transformation of all forecasts and realized

variance into standard deviations since this makes comparison more convenient to some extent.

Moreover, Poon (2005) suggests that errors computed from variances may complicate model

comparisons since confidence interval of error statistics can widen drastically making it difficult to

determine significant differences between examined models. In addition, measures of accuracy that are

based on errors between realized and forecasted volatility may be wrongly computed if variance is

used instead of standard deviation. For example, using variance in derivation of MSE will result in

huge errors since the squared variance error is actually the error computed from standard deviation

raised to the fourth power. The practice of simply taking the square root of variance to get standard

deviation is not the case with the BS implied volatility, which needs a bit different treatment. BS

implied volatility is scaled as follows:

BSIV scaled=√ HN

∗BSIV

Where H presents the forecast horizon, which is 5 in this case, while N presents the sample size.

Final adjustment relates to LIBOR rates, obtained from Global rates website (www.global-rates.com),

which were used in this case as domestic and foreign interest rates. As this research uses seven

different currencies against US dollar, then US dollar LIBOR is considered as domestic interest rate,

while foreign interest rates are LIBOR rates of the seven chosen currencies. Since only monthly

LIBOR rates are available, the adjustment suggests allocating each month’s LIBOR rate to each

Monday within that month. However, the extreme case worth mentioning is that of the Swedish Krone

LIBOR rates which are available only from the beginning of 2006 till to date. Due to the fact that the

examined out-of-sample period goes from the beginning of 2002 till the end of 2006, such lack of data

presents a potential problem and the expected bias in derivation of model-free implied volatility. The

only solution here considers assigning zero values to LIBOR rates for the part of the out-of-sample

period before 2006.

19


5. Empirical analysis

5.1. Forecasted volatility for the out-of-sample period (Jan. 2002 – Dec. 2006)

The following graphs present volatility forecasts for the period between January 2002 and December

2006 based on GARCH (1, 1), BS and MF models. In the bottom of graphs, there is description

showing which line on the graph corresponds to what model, thus RV stands for realized volatility,

GR for GARCH (1, 1), BS is for Black-Scholes implied, and MF is Model-Free implied volatility.

Moreover, the dates in the horizontal axis are in the American style, thus the first number corresponds

to month, second is day and third presents a year.

AUD/USD

CAD/USD

20


GBP/USD

JPY/USD

21


EUR/USD

SEK/USD

22


CHF/USD

5.2. Examining incremental information in volatility forecasting models

Following research of Jiang and Tian (2005), this study paper will apply univariate and encompassing

regressions in order to examine the informational content and efficiency of the volatility forecasting

models. As it is known, univariate regression involves regressing realized volatility against each

forecasting model individually in order to check whether there is any relationship between forecasted

values and the actual ones. Therefore, univariate regression examines forecasting ability and

informational content of each model. On the other hand, the encompassing regression analysis

involves regressing realized volatility against two or more forecasting models for comparison

purposes, determining whether one model subsumes the information contained in the other having on

such way more stronger informational content.

To begin with the analysis, the univariate regressions are formulated as:

RV t+H=α+β GARCH t+H+et+H

RV t+H=α+β BSIV t+H+et+H

RV t+H=α+β MFIV t+H +e t+H

Whether the model contains specific information that is related to realized volatility will be decided

based on a hypothesis with the null stating that the beta coefficient of the model is zero(β=0).

Namely, beta coefficient is also known as a coefficient of correlation between the dependent and

independent variables showing the strength of their relationship, thus if it is zero, that means that

dependent variable cannot be explained by the independent. The results of regressions are presented in

Table 5.1 below:

23


Models GARCH BSIV MFIVExchange ratesAUD/USD 0.8785 (0.000) 0.7944 (0.000) 0.706629 (0.000)CAD/USD 0.961 (0.000) 0.8012 (0.000) 0.7883 (0.000)GBP/USD 0.3728 (0.000) 0.7905 (0.000) 0.7697 (0.000)JPY/USD 0.1907 (0.000) 0.57 (0.000) 0.5329 (0.000)EUR/USD 0.2964 (0.000) 0.5469 (0.000) 0.5267 (0.000)CHF/USD 0.0638 (0.000) 0.5123 (0.000) 0.4908 (0.000)SEK/USD 0.1894 (0.000) 0.5992 (0.000) 0.5704 (0.000)

Univariate regression results

ߚ ߚ ߚ

Table 5.1

As it is shown, the beta values for all three models across seven different exchange rates are

significantly different from zero. By looking at the p-values in brackets, corresponding to each beta, it

can be concluded that the null hypothesis is rejected at 1% level of significance and that there is a

statistically significant correlation between the forecasted volatilities and the realized one. In addition

to univariate regression, most studies have a usual practice to check the biasness of forecasts. Namely,

a bias in statistics presents the deviation of the forecast from the actual results, thus if the bias is zero,

that means that forecasted values are 100% equal to actual values which is not realistic in practice.

Therefore, this study paper applies a joint hypothesis with the null stating that a forecast is an unbiased

estimator of the future realized volatility if its intercept equals zero (α=0) while slope coefficient

equals 1(β=1). The results are presented in the following Table 5.2:

Models GARCH BSIV MFIVExchange rates

AUD/USD 953.2668 (0.0000) 1067.273 (0.0000) 1283.668 (0.0000)CAD/USD 420.3015 (0.0000) 434.955 (0.0000) 557.9634 (0.0000)GBP/USD 403.6365 (0.0000) 234.2421 (0.0000) 308.7745 (0.0000)JPY/USD 593.4499 (0.0000) 284.2075 (0.0000) 392.4892 (0.0000)EUR/USD 494.1774 (0.0000) 387.6324 (0.0000) 494.6371 (0.0000)CHF/USD 577.3267 (0.0000) 298.7824 (0.0000) 385.9079 (0.0000)SEK/USD 491.5695 (0.0000) 352.4148 (0.0000) 443.222 (0.0000)

Testing unbiasness with Ho:

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Table 5.2

The decision rule in this case is based on the Chi-square values generated by Wald’s test of

coefficients restrictions. As it can be seen, all Chi-square values are very high and their p-values

suggest that the null hypothesis of unbiased estimators can be rejected at 1% level of significance.

This was expected, due to the fact that in practice, forecasts are not perfect estimators of realized

volatility. Moreover, descriptive statistics analysis, presented in appendix A2.1, shows that implied

volatility forecasts of BS and MF on average have much greater mean values than RV, which also

contributes to biasness. Therefore, all three forecasting models produce biased forecasts of realized

volatility.

24


In order to fully conduct the analysis of univariate regression, the comparison between models based

on the coefficients of determination obtained from these regressions is also included. As it was

mentioned previously in the methodology, the coefficient of determination presents what is known as

“goodness of fit” or how well the examined forecasted volatility explains the realized volatility (Pong

et al, 2003). Therefore, the higher the R2 value is the better the forecast, or in other words, the closer

the forecasted volatility is to the realized one.

Taking a bigger picture of the Table 5.3 below, it can be easily noticed that percentage values of the

coefficient of determination are less than 40%. In accordance with some general rules of statistics, this

would mean that none of independent variables are statistically significant determinants of the realized

volatility. However, such low values for coefficient of determination were expected since the realized

volatility is computed from daily squared returns, which are considered as a too noisy approximation

of actual volatility relative to examined models that provide much smoother forecasts. Ignoring this

magnitude mismatch between realized volatility and its forecasts, R2 results are still considered as

valid evaluation criteria, due to the fact that previous tests have found significant correlation between

three volatility models and realized volatility. The R2 values obtained from univariate regressions are

following:

Exchange rates GARCH BSIV MFIV

AUD/USD 0.2681 0.3476 0.3453CAD/USD 0.3153 0.3417 0.3421GBP/USD 0.2685 0.3089 0.3070JPY/USD 0.1725 0.1832 0.1831EUR/USD 0.1746 0.2152 0.2150CHF/USD 0.1506 0.1751 0.1749SEK/USD 0.2184 0.2433 0.2419

Coeffi cient of determination analysis based on univariate regression

�ଶ �ଶ�ଶ

Table 5.3

In a more general view the coefficient of determination analysis shows that both BS implied and MF

implied models dominate GARCH (1, 1) as it was expected. On the other hand, when comparing all

three models individually it can be noticed that BS implied volatility has the highest coefficient of

determination except for the CAD/USD case where it is dominated by MF implied. However, it is not

that easy to conclude that BS implied forecasting model is the most accurate one due to the fact that

the difference between MF implied and BS implied R2 values is not that significant. Moreover, there

are other tests to be performed which will hopefully strengthen the analysis and make the comparison

process easier. Another thing to spot in the table is a discrepancy between the levels of R2 values. For

example, the highest values are that of AUD/USD, GBP/USD, CAD/USD, which coefficient of

determination reaches the level of 30% to 35%, while in the case of other four exchange rates, values

25


oscillate between 17% and 24%. The major reason for such differences could be liquidity/illiquidity of

some of these currencies.

Another important test that has to be conducted considers checking superiority of one forecasting

model over another in explaining realized volatility. If one model is “information richer” than the

other, it is expected that once plugged into the regression it significantly improves the coefficient of

determination. Therefore, in order to check the informational efficiency of the models and to prove the

expectations that there is an incremental information in implied volatilities relative to GARCH (1, 1),

the study will separately regress realized volatility against each of the two implied volatility models in

combination with the time-series one. Once this is done, the two implied models BS and MF will be

evaluated based on improvements in the coefficient of determination. In other words, the one that

improves coefficient of determination more when added to GARCH (1, 1), will be considered as

“informationally superior”. However, before getting on the encompassing regression analysis, it is

necessary to perform specific data adjustment in order to solve problems of heteroscedasticity and

autocorrelation that are present in data. Namely, avoidance of these two can lead to wrong decision-

making when it comes to accuracy evaluation as well as wrong results in other measures of accuracy

that will be applied. Including the lagged value of dependent variable and applying the adjustment for

heteroscedasticity, so called White’s test in E-views statistic software, present satisfying solutions.

The regressions are as follows:

RV t+H=α+β1GARCH t+H+β2 LRV +β3 BSIV t+H+et+H

RV t+H=α+β1GARCH t+H+β2 LRV +β3 MFIV t+H +e t+H

In the two regressions LRE stands for Lagged Realized Volatility. Observing results from the Table

5.4 presented below for the two encompassing regressions, it can be argued that once either BS

implied or MF implied volatility are added to GARCH (1, 1) the explanatory power of the regression

significantly increases for all exchange rates. Taking GBP/USD exchange rate as an example,

coefficient of determination for GARCH (1, 1) alone is around 26.85%, while once BS or MF

forecasts are plugged in the regression this jumps to 33.83% or 33.38% respectively.

Exchange rates GARCH+BSIV GARCH+MFIV

AUD/USD 0.350672 0.347688CAD/USD 0.345912 0.346483GBP/USD 0.338305 0.333878JPY/USD 0.195477 0.19552EUR/USD 0.235374 0.236382CHF/USD 0.20461 0.205588SEK/USD 0.256187 0.254285

Incremental information based on encompassing regression

�ଶ �ଶ

26


Table 5.4

However, what makes things a bit difficult is deciding upon the informational domination between BS

and MF models. Namely, differences between coefficients of determination when each of the two are

separately plugged into regression are very small, and as such insignificant in the analysis. For

example, in the case of EUR/USD the difference between the computed R2 is 0.001008 which can be

to some extent even ignored. Moreover, when these small differences are taken into consideration, the

thing that contributes to such difficulty in deciding between BS and MF models is the fact that for four

out of seven exchange rates the MF implied volatility dominates BS implied, while in other cases BS

is the most accurate model. For example, in the case of AUD/USD BS model dominates MF since its

coefficient of determination is by 0.0029 higher than that MF, while in the case of JPY/USD MF beats

BS by 0.000043.

This is probably a result of variation in data availability between currencies, or from a mathematical

aspect, it can be even viewed as a consequence of less losses in data when it comes to computation of

the MF implied volatility for exchange rates where MF model is the most accurate one. On the other

hand, lack of input data as well as various adjustments conducted in data processing can also

contribute to results that favor BS model. For example, there is no data on SEK LIBOR rates for the

period between 2002 and 2005 which can present a potential problem in estimating the MF implied

volatility and not very accurate results, thus weakening MF model relative to BS. Unfortunately, the

goal of this work is to come up with the most accurate forecasting model and thus comparisons will

have to be conducted more strictly including other measures of accuracy.

In order to strengthen the previous regression analysis, this study paper applies few measures of

accuracy that are based on the errors between the forecast and the realized volatility. These measures

are grouped according to their loss functions, for example, MSE and RMSE are based on squared

errors, and are thus explained in parallel. Similarly, MAE and MAPE are based on absolute errors,

while HMSE stands alone. General rule in this analysis suggests that the best forecast will show the

lowest values in these tests, since this will mean that the error between the forecast and realized

volatility for that specific model is the smallest.

To begin with the analysis the first two tables below show MSE and RMSE results respectively. As

previously explained in methodology, both Mean Squared Error (MSE) and Root Mean Squared Error

(RMSE), are two widely used measures of forecast accuracy which computation is a very

straightforward procedure. The first one, MSE, involves a calculation of squared errors between the

forecast and realized volatility across the whole sample and then estimation of the average of these

squared errors, while the second one, RMSE, is just a squared root of this average. Calculations are

performed in Excel and the results are as follows:

27


Exchange rates GARCH BSIV MFIV MinAUD/USD 1.5039E-05 1.02819E-05 1.04727E-05 1.02819E-05CAD/USD 8.66077E-06 6.92561E-06 6.92088E-06 6.92088E-06GBP/USD 1.13957E-05 9.35286E-06 9.28425E-06 9.28425E-06JPY/USD 1.49255E-05 1.3472E-05 1.35131E-05 1.3472E-05EUR/USD 1.12696E-05 9.94901E-06 9.96276E-06 9.94901E-06CHF/USD 1.36926E-05 1.25242E-05 1.25456E-05 1.25242E-05SEK/USD 1.47907E-05 1.33588E-05 1.3427E-05 1.33588E-05

Mean Squared Error (MSE)

Table 5.5a

Exchange rates GARCH BSIV MFIV MinAUD/USD 0.0038780 0.0032065 0.003236154 0.00320654CAD/USD 0.0029429 0.0026317 0.002630757 0.00263076GBP/USD 0.0033758 0.0030582 0.003047007 0.00304701JPY/USD 0.0038633 0.0036704 0.003676016 0.00367043EUR/USD 0.0033570 0.0031542 0.003156384 0.00315421CHF/USD 0.0037003 0.0035390 0.003541975 0.00353896SEK/USD 0.0038459 0.0036550 0.003664282 0.00365497

RMSE

Table 5.5b

The last column in both tables (5.5a and b) presents the smallest error value between three models for

each exchange rate. This is done in order to make comparisons easier and clearer. Numbers colored in

green are the values computed for the BS implied model, while the red ones refer to MF model.

In general, GARCH (1, 1) model has the largest MSE and RMSE when compared to both, BS and MF,

implied volatilities for all examined exchange rates which is consistent with previous regression

analysis. For example, in the case of EUR/USD, GARCH (1, 1) MSE is around1.127∗10−5, while BS

and MF are of the order of10−6. Similarly, for the same exchange rate, GARCH (1, 1) has RMSE of

0.0033570 which is again the largest RMSE value when compared to BSIV and MFIV. However, the

same difficulty in decision-making, regarding greater accuracy between BS and MF model, arises in

this analysis as well. Like in the coefficient of determination analysis, differences between the errors

of BS and MF are too small to be considered as statistically significant. A good example of how small

this difference is, could be the case of CHF/USD where MSE values of the two models differ at the

ninth decimal (BS: 0.00001252, MF: 0.00001254), while in the case of RMSE similar example is with

JPY/USD exchange rate where the difference between the two occurs at the sixth decimal. Taking into

consideration even these small differences, the BS model is the most accurate one, according to MSE

and RMSE, for all the exchange rates except CAD/USD and GBP/USD where MF is dominant.

28


Besides MSE and RMSE which provide their results by using squared errors as a loss function,

additional accuracy measurement parameters described in parallel are MAE and MAPE or Mean

Absolute Error and Mean Absolute Percentage Error respectively. These two compare forecasting

models based on the loss function that considers absolute errors between forecasts and realized

volatility. In the case of MAE, absolute errors are averaged, while MAPE firstly divides these absolute

errors by realized volatility and then estimates the average. The results are presented in the Tables 5.6a

and b:


MAE

Table 5.6a


MAPE

Table 5.6b

As it can be noticed, GARCH (1, 1) is again the weakest forecasting model according to both MAE

and MAPE. However, when it comes to comparing BS and MF models, results differ much from the

MSE and RMSE analysis. Namely, the only case here in which BS model is dominated by MF is the

case of EUR/USD, while in previous analysis these were CAD/USD and GBP/USD. Such a difference

in results is probably a consequence of change in loss function since there is a shift from squared to

absolute errors.

Finally, the last comparison between the three models in this work is based on a version of MSE that is

adjusted for heteroscedasticity also known as HMSE. Its loss function significantly differs from those

of the previous accuracy measures since it is expressed as a square of the relative deviation of the

forecasted from the realized volatility. Due to the fact that it involves an adjustment for

heteroscedasticity it can be considered as an improved MSE and as such is expected to give more

accurate results. The HMSE table (Table 5.7) is provided below:

29



HMSE

Table 5.7

Results now significantly differ from those provided by previous measures of forecasting accuracy.

The BS implied volatility is the best forecast of realized volatility in the four out of seven exchange

rates (AUD/USD, GBP/USD, EUR/USD, and SEK/USD), while in the remaining cases MF is the

dominant one (CAD/USD, JPY/USD, CHF/USD). As with the previous measures of accuracy,

GARCH (1, 1) still remains the weakest model.

30


6. Conclusion

Practice has shown that forecasting volatility of financial assets plays an immense role in investment

decision-making, portfolio management and the construction of hedging strategies. However, none of

these would be successful if inappropriate forecasts are used. Therefore, the more important and

challenging step is deciding upon the forecasting model to rely on. Considering analysis and results of

this study, it seems that evaluation of the forecasts derived by different models cannot always give a

direct answer, since various parameters may negatively impact it. For the purpose of obtaining clearer

picture and strengthening the evaluation procedure, several accuracy measures were applied in this

work together with the extensive regression analysis.

In general, the most straightforward conclusion common for all the tests applied in the empirical

analysis part of the work suggests that time-series forecasting model GARCH (1, 1) really is

dominated by the other two implied volatility models, which is consistent with prior literature and

studies. What was shown is that not only there is a significant incremental information contained in

implied volatility models, but the errors between implied forecasts and realized volatility are lower

relative to that between GARCH (1, 1) and realized volatility. However, the difficult part throughout

the empirical analysis was deciding about dominance between BS and MF implied volatility forecasts.

Namely, results of the encompassing regression analysis, which compares BSIV to MFIV by

separately adding them to GARCH (1, 1) in a regression, favors MFIV in four out of seven exchange

rates (CAD/USD, JPY/USD, EUR/USD, and CHF/USD) while BSIV proves to be the prevailing one

for the remaining three cases. On the contrary, MSE and RMSE favor MFIV just in case of CAD/USD

and GBP/USD, while MAE and MAPE show that the BSIV is the most accurate model for all

exchange rates except EUR/USD. Finally, heteroscedasticity-adjusted HMSE stresses out the

dominance of MFIV in three out of seven cases (CAD/USD, JPY/USD and CHF/USD).

However, another important fact spotted in all the tests applied is a significantly small difference

between results on BSIV and MFIV. For some measures of accuracy the two models start to differ at

ninth decimal, while in case of other tests this difference appears at sixth or fifth decimal place. Due to

the fact that such strict comparisons, that consider all the decimal places, won’t help much in

concluding this work, final test applied here involves checking the significance of the difference

between the BS and MF forecasts. Diebold-Mariano’s test has shown the following:

31


Exchange rates GARCH-BS GARCH-MF BS-MFAUD/USD 0.000 0.000 0.774CAD/USD 0.000 0.000 0.8459GBP/USD 0.000 0.000 0.6133JPY/USD 0.000 0.000 0.5723EUR/USD 0.000 0.000 0.3988CHF/USD 0.000 0.000 0.418SEK/USD 0.000 0.000 0.3459

DM test (p-values) applying 1% Level of significanceHo: Difference between the models is insignificant

Table 6.1

As it can be seen, the null hypothesis can be rejected only when BS or MF forecasts are compared to

GARCH (1, 1), while this is not the case when two implied volatility models are compared with one

another. Therefore, as their forecasts do not differ drastically it can be concluded that this work is

indifferent between using BSIV or MFIV in exchange rate volatility forecasting. Although it was

expected that the MFIV will be the dominant one, it seems that in the case of exchange rates results

are not always stabile, either due to problems regarding data availability or even due to mathematical

losses that emerge from the various adjustments conducted in estimation of the MFIV.

32


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Appendix

A1.1 Volatility smile plot for CAD/USD exchange rate:

0.010.05

0.090.13

0.170.21

0.250.29

0.330.37

0.410.45

0.490.53

0.570.61

0.650.69

0.73

0.770000000000001

0.810000000000001

0.850000000000001

0.890000000000001

0.930000000000001

0.9700000000000010.0540.0560.058

0.060.0620.064

0.0660.068

0.07

Volatility Smile

OTM OTM NTM and ATM zone

This volatility smile plot is an example from CAD/USD exchange rate taken at 11 th of March 2002,

where horizontal axis presents delta values ranging from 0.01 till 0.99, while vertical axis refers to

implied volatilities estimated by second-degree polynomial approach. As it can be noticed, the lowest

implied volatility, which is approximately 0.06, corresponds to near-the-money and at-the-money

options which cover a narrow range of delta values between 0.45 and 0.65. Moving outside these

barriers, implied volatility increases as options are slowly becoming out-of-money.

Appendix 37


A1.2 Plot of strike prices across delta values for CAD/USD exchange rate:

0.010.05

0.090.13

0.170.21

0.250.29

0.330.37

0.410.45

0.490.53

0.570.61

0.650.69

0.73

0.770000000000001

0.810000000000001

0.850000000000001

0.890000000000001

0.930000000000001

0.9700000000000011.52

1.54

1.56

1.58

1.6

1.62

1.64

Strike price against Delta

As diagram presents, the horizontal axis corresponds to delta values, while strike prices are plotted on

the vertical one. The fall in the strike price as the delta increases is normal due to the nature of delta

and its positive relationship to call price. Since the delta presents the percentage change in price of

option when there is a change in the price of underlying asset. For example, if the options delta is 0.80,

that means that if the price of underlying asset changes by small amount, the price of option will

change by 80% of that amount. As the price of the underlying increases, delta tends to increase.

Moreover, increase in price of underlying further increases the price of a call option since it is more

likely that the option will be exercised by the holder. Therefore, getting the call price and delta into

linkage, it can be concluded that they are positively related. On the other hand, call price falls as the

strike price increases, thus delta of the option and strike price can be also considered as negatively

related.

Appendix 38


A1.3 Diagrammatical plot of call prices against options delta values:

0.010.05

0.090.13

0.170.21

0.250.29

0.330.37

0.410.45

0.490.53

0.570.61

0.650.69

0.73

0.770000000000001

0.810000000000001

0.850000000000001

0.890000000000001

0.930000000000001

0.9700000000000010

0.005

0.01

0.015

0.02

0.025

0.03

0.035

Plot of Call prices against Delta

As explained previously in Appendix A1.2, call price and delta are positively related, thus as delta

increases, call price should increase as well. The diagram above shows exactly this relationship,

having delta values plotted on the horizontal and call prices on the vertical axis. Thus, this prove that

calculations were correctly conducted, taking as an example the case of CAD/USD.

Appendix 39


A1.4 Calculation of the integral

The formula for the MF implied volatility presented in methodology has a following form:

2 ∫Kmin

Kmax CF (T , K )−Max (0 , F0−K)K2 dK

This integral is solved in MS Excel as follows:

2∑i=1

98 f ( K i )+ f ( K i+1 )2

∆ K i

Where: K i is a strike price that corresponds to δ i

δ i is options delta, which ranges from 0.01 till 0.99

∆ K i= K i+1−K i

f ( K i )=CF (T , K i )−Max (0 , F0−K i)

K i2

Appendix 40


A2.1 Descriptive statistics for forecasts

AUD/USD

RV GR BS MF

Mean 0.007893 0.013674 0.013551 0.014114 Median 0.007402 0.013167 0.013216 0.013846 Maximum 0.019522 0.019796 0.023006 0.023760 Minimum 0.001894 0.010189 0.008140 0.008466 Std. Dev. 0.003416 0.002052 0.002652 0.002733 Skewness 0.739959 0.796022 0.699450 0.668494 Kurtosis 3.311410 3.002486 3.219745 3.178978

CAD/USD

RV GR BS MF


GBP/USD

RV GR BS MF


Appendix 41


EUR/USD

JPY/USD

CHF/USD

Appendix 42

RV GR BS MF


RV GR BS MF


RV GR BS MF



SEK/USD

Appendix 43

RV GR BS MF

Mean 0.009814 0.014948 0.014005 0.014530 Median 0.009411 0.014965 0.013701 0.014283 Maximum 0.021320 0.018798 0.020607 0.021307 Minimum 0.002547 0.010729 0.009097 0.009437 Std. Dev. 0.003866 0.001670 0.002070 0.002125 Skewness 0.556733 -0.102421 0.390518 0.410658 Kurtosis 2.920336 2.564752 2.855770 2.846146

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