Exchange rate volatility implied from option prices
-
Upload
srdjan-begovic -
Category
Documents
-
view
61 -
download
1
Transcript of 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
Exchange Rate Volatility Forecasting
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
Exchange Rate Volatility Forecasting
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
Exchange Rate Volatility Forecasting
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
Exchange Rate Volatility Forecasting
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
Exchange Rate Volatility Forecasting
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
Exchange Rate Volatility Forecasting
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
Exchange Rate Volatility Forecasting
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
Exchange Rate Volatility Forecasting
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
Exchange Rate Volatility Forecasting
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
Exchange Rate Volatility Forecasting
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
Exchange Rate Volatility Forecasting
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
Exchange Rate Volatility Forecasting
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
Exchange Rate Volatility Forecasting
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
Exchange Rate Volatility Forecasting
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
Exchange Rate Volatility Forecasting
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
Exchange Rate Volatility Forecasting
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
Exchange Rate Volatility Forecasting
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
Exchange Rate Volatility Forecasting
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
Exchange Rate Volatility Forecasting
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
Exchange Rate Volatility Forecasting
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
Exchange Rate Volatility Forecasting
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
Exchange Rate Volatility Forecasting
GBP/USD
JPY/USD
21
Exchange Rate Volatility Forecasting
EUR/USD
SEK/USD
22
Exchange Rate Volatility Forecasting
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
Exchange Rate Volatility Forecasting
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:
ଶ ଶ ଶߙ ൌ��Ͳand ߚൌ��ͳ
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
Exchange Rate Volatility Forecasting
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
Exchange Rate Volatility Forecasting
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
Exchange Rate Volatility Forecasting
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 Rate Volatility Forecasting
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
Exchange Rate Volatility Forecasting
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:
Exchange rates GARCH BSIV MFIV MinAUD/USD 0.00294267 0.00247095 0.002492099 0.002470948CAD/USD 0.00238481 0.00212963 0.002129653 0.002129631GBP/USD 0.00264525 0.00240595 0.002415455 0.002405951JPY/USD 0.00302915 0.00284617 0.002865150 0.002846174EUR/USD 0.00260017 0.00246102 0.00245972 0.002459721CHF/USD 0.00289666 0.00275689 0.00276422 0.002756891SEK/USD 0.00309225 0.00294346 0.00295830 0.002943460
MAE
Table 5.6a
Exchange rates GARCH BSIV MFIV MinAUD/USD 0.347914765 0.317775822 0.319235271 0.317775822CAD/USD 0.425957277 0.363727951 0.364552059 0.363727951GBP/USD 0.422777755 0.367825064 0.37077647 0.367825064JPY/USD 0.441987899 0.408391128 0.415198263 0.408391128EUR/USD 0.362851211 0.341692954 0.338094642 0.338094642CHF/USD 0.392015552 0.369628173 0.372778772 0.369628173SEK/USD 0.395502299 0.372726446 0.374573274 0.372726446
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
Exchange Rate Volatility Forecasting
Exchange rates GARCH BSIV MFIV MinAUD/USD 0.476278666 0.247643744 0.256183593 0.247643744CAD/USD 0.15086158 0.130385633 0.130017484 0.130017484GBP/USD 0.178744673 0.150061236 0.151327531 0.150061236JPY/USD 0.185279618 0.174449064 0.170125426 0.170125426EUR/USD 0.147991697 0.12644851 0.129351182 0.12644851CHF/USD 0.140468779 0.129136221 0.127491811 0.127491811SEK/USD 0.154088614 0.139143129 0.140083943 0.139143129
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
Exchange Rate Volatility Forecasting
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 Rate Volatility Forecasting
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
Exchange Rate Volatility Forecasting
Reference list
Alexander, C. (2001). Market Models: A Guide to Financial Data Analysis. John Wiley &
Sons.
Andrade, S. C. and Tabak, B. M. (2001). Is it worth tracking dollar/real implied volatility?
Working Paper, 1-25.
Benavides, G. (2009). Price volatility forecasts for agricultural commodities: an application of
volatility models, option implieds and composite approaches for futures prices of corn and
wheat. Journal of Management, Finance and Economics, 3, 40-59.
Beneder, R. and Elkenbracht-Huizing, M. (2003). Foreign Exchange Options and the
Volatility Smile. Medium Econometrische Toepassingen. 2, 30–36.
Blair, B. J., Poon, H. and Taylor, S. J. (2000). Forecasting S&P 100 Volatility : The
Incremental Information Content of Implied Volatilities and High Frequency Index Returns. Working
Paper.
Britten-Jones, M., and Neuberger, A. (2000). Option Prices, Implied Price Processes, and
Stochastic Volatility. Journal of Finance, 55, 839–66.
Busch, T., Christensen B. J. and Nielsen M. O. (2009). The Role of Implied Volatility in
Forecasting Future Realized Volatility and Jumps in Foreign Exchange, Stock, and Bond
Markets. Queen’s Economics Department Working Paper No. 1181.
Canina, L. and Figlewski, S. (1993). The Informational Content of Implied Volatility. The
Review of Financial Studies, 6, 659-681.
Cao, C., Yu, F. and Zhong, Z. (2009).The Information Content of Option-Implied Volatility
for Credit Default Swap Valuation. Working Paper, Penn State University.
Carr, P., and Madan, D. (1998), “Towards a Theory of Volatility Trading,” in R. Jarrow, ed.,
Volatility: New Estimation Techniques for Pricing Derivatives, London: Risk Books, 417–427.
Carr, P., and Wu, L. (2003), “What Type of Process Underlies Options? A Simple Robust
Test,” Journal of Finance, 2581–2610.
Reference list 33
Exchange Rate Volatility Forecasting
Day, E. T. and Lewis, C. M. (1992). Stock market volatility and the information content of
stock index options. Journal of Econometrics, 52, 267-287.
Day, E. T. and Lewis, C. M. (1993). Forecasting Futures Market Volatility. The Journal of
Derivatives.
Demeterfi, K. E., Derman, M. K., and Zou J. (1999). “More Than You Ever Wanted to Know
about Volatility Swaps,” Goldman Sachs Quantitative Strategies Research Notes.
Derman, E. and Kani, I. (1994). Riding on a smile. Risk, 7, 32-39.
Diebold, F. X. and Lopez, J. A. (1995). Forecast Evaluation and Combination. Handbook of
Statistics. University of Pennsylvania.
Dupire, B. (1994). Pricing with a Smile, Risk, 7, 18–20.
Ederington, L.H. and Guan, W. (2002). Measuring Implied Volatility: Is an Average Better?
Which Average? Journal of Futures Markets, 22, 811-837.
Egelkraut, T. M. and Garcia, P. (2006). Intermediate Volatility Forecasts Using Implied
Forward Volatility: The Performance of Selected Agricultural Commodity Options. Journal of
Agricultural and Resource Economics, 31, 508-528.
Figlewski, S. (2004). Forecasting volatility. Financial Markets, Institutions, and Instruments,
2-87.
Fleming, J. (1998). The quality of market volatility forecasts implied by S&P 100 index option
prices. Journal of Empirical Finance, 5, 317–345.
Giot, P. (2005). Relationships between implied volatility indexes and stock index returns. The
Journal of Portfolio Management, 92-100.
Giot, P. (2002). The Information Content of Implied Volatility in Agricultural Commodity
Markets. Journal of Futures Markets, 23, 441–454.
Hull, J. C.. (2012). Options, Futures and Other Derivatives. Pearson Education Limited.
Hung-Gay-Fung, Chin-Jen Lie, and Moreno, A. (1990). The Forecasting Performance of the
Implied Standard Deviation in Currency Options. Managerial Finance, 16, 24 - 29.
Jiang, G., and Y. Tian. (2005). Model-free Implied Volatility and Its Information Content.
Review of Financial Studies, 1305-42
Reference list 34
Exchange Rate Volatility Forecasting
Jorion, P. (1995). Predicting volatility in the foreign exchange market. Journal of Finance, 50,
507-528.
Kroner, F.K., Kneafsey, P.K. and Claessens S. (1995). Forecasting volatility in commodity
markets. Journal of Forecasting.
Lamoureux C. G. and Lastrapes, W. D. (1993). Forecasting stock-return variance: Toward and
understanding of stochastic implied volatilities. The Review of Financial Studies, 6, 293-326.
Lopez, J.A. (1995). Evaluating the Predictive Accuracy of Volatility Models. Manuscript,
Research and Market Analysis Group. Federal Reserve Bank of New York.
Lynch, D. P., and Panigirtzoglou, N. (2004). Option Implied and Realized Measures of
Variance, Working Paper.
Macbeth, J. D. and Merville, L. J. (1979). An Empirical Examination of the Black-Scholes
Call Option Pricing Model. Journal of Finance. 34, 1173-1186.
Malz, A. (2001). Do implied volatilities provide early warning of market stress? Risk Metrics
Journal, 41-60.
Malz, A. M., (1997). Option implied probability distributions and currency excess returns .
Federal Reserve Bank of New York Staff Reports No. 32.
Manfredo, M. R., Leuthold R. M. and Irwin, S. H. (2001). Forecasting Fed Cattle, Feeder
Cattle, and Corn Cash Price Volatility: The Accuracy of Time Series, Implied Volatility, and
Composite Approaches . Journal of Agricultural and Applied Economics, 3, 523-538.
Martens, M. and Zein, J. (2002). Predicting financial volatility: High-frequency time-series
forecasts vis-à-vis implied volatility. Working Paper, Erasmus University.
Pong, S., Shackleton, M. B., Taylor, S. J., and Xu, X. (2004). Forecasting currency volatility:
a comparison of implied volatilities and AR(FI)MA models. Journal of Banking and Finance, 28,
2541–2563.
Poon, H. and Granger, C., W., J. (2003). Forecasting Volatility in Financial Markets: A
Review. Journal of Economic Literature, 478–539.
Poon, S.H. (2005). A Practical Guide to Forecasting Financial Market Volatility. Chichester.
UK: John Wiley & Sons, Ltd.
Reference list 35
Exchange Rate Volatility Forecasting
Reiswich, D. and Wystup, U.. (2010). FX volatility smile construction. Working Paper No.
20. Frankfurt School of Finance & Management.
Ritter, J. R. (2003). Behavioral Finance. Pacific-Basin Finance Journal, 11, 429-437.
Rubinstein, M. E. (1994). Implied binomial trees, Journal of Finance, 69, 771-818.
Szakmary, A., Ors, E., Kim, K. J. and Davidson W. N. (2003). The predictive power of
implied volatility: Evidence from 35 futures markets. Journal of Banking & Finance, 27, 2151–2175.
Taylor, S. J. (2005). Asset Price Dynamics, Volatility and Prediction. New Jersey: Princeton
University Press.
Taylor, S. J., Yadav, P. K., Zhang, Y. (2007). The information content of implied volatilities
and model-free volatility expectations: evidence from options written on individual stocks. Journal of
Banking and Finance, 34, 871–881.
Taylor, S. J., Zhang, Y. and Wang, L. (2010). Investigating the Information Content of the
Model-free volatility Expectation by Monte Carlo Methods. Working Paper.
West, D. K. and Cho, D. (1995). The predictive ability of several models of exchange rate
volatility. Journal of Econometrics, 69, 367-391.
Xu, X. and Taylor, S.J. (1995). Conditional volatility and the informational efficiency of the
PHLX currency options markets. Journal of Banking and Finance, 19, 803-821.
Reference list 36
Exchange Rate Volatility Forecasting
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
Exchange Rate Volatility Forecasting
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
Exchange Rate Volatility Forecasting
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
Exchange Rate Volatility Forecasting
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
Exchange Rate Volatility Forecasting
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
Mean 0.007674 0.007665 0.011071 0.011522 Median 0.007535 0.007553 0.011247 0.011669 Maximum 0.017425 0.010589 0.016322 0.016912 Minimum 0.001346 0.006230 0.006975 0.007383 Std. Dev. 0.003036 0.000722 0.001878 0.001910 Skewness 0.346217 0.904350 0.072174 0.099777 Kurtosis 2.925562 4.110998 2.584093 2.596643
GBP/USD
RV GR BS MF
Mean 0.008102 0.012211 0.010975 0.011416 Median 0.007604 0.012002 0.010886 0.011294 Maximum 0.020499 0.017769 0.017834 0.018387 Minimum 0.001048 0.008454 0.006781 0.007284 Std. Dev. 0.003431 0.001550 0.001981 0.002005 Skewness 0.845081 0.784877 0.409644 0.454272 Kurtosis 3.990610 4.557422 3.485981 3.479948
Appendix 41
Exchange Rate Volatility Forecasting
EUR/USD
JPY/USD
CHF/USD
Appendix 42
RV GR BS MF
Mean 0.008826 0.013415 0.012564 0.013065 Median 0.008309 0.013255 0.012259 0.012678 Maximum 0.020203 0.016106 0.018180 0.018878 Minimum 0.001519 0.011510 0.006975 0.007296 Std. Dev. 0.003377 0.001030 0.002178 0.002250 Skewness 0.763459 0.406512 0.228675 0.226136 Kurtosis 3.685161 2.346633 2.599554 2.591573
RV GR BS MF
Mean 0.008993 0.014749 0.012732 0.013400 Median 0.008479 0.014535 0.012495 0.013148 Maximum 0.023408 0.018701 0.022327 0.023575 Minimum 0.001927 0.012509 0.007863 0.008180 Std. Dev. 0.003879 0.001358 0.002168 0.002288 Skewness 0.794526 0.596017 0.685294 0.681112 Kurtosis 3.780496 2.588794 3.774251 3.785253
RV GR BS MF
Mean 0.009836 0.015346 0.013503 0.014023 Median 0.009639 0.015314 0.013313 0.013774 Maximum 0.021539 0.016723 0.019082 0.019830 Minimum 0.001344 0.014030 0.008348 0.008681 Std. Dev. 0.003707 0.000571 0.002114 0.002188 Skewness 0.564556 0.139416 0.131295 0.136270 Kurtosis 3.377308 2.532101 2.453560 2.434748
Exchange Rate Volatility Forecasting
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