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Physica A 374 (2007) 773–782

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Does implied volatility of currency futures option implyvolatility of exchange rates?

Alan T. Wang�

Department of Accounting, National Cheng Kung University, No.1 Ta-Hsueh Rd., Tainan, 701, Taiwan

Received 16 March 2006; received in revised form 5 May 2006

Available online 11 September 2006

Abstract

By investigating currency futures options, this paper provides an alternative economic implication for the result reported

by Stein [Overreactions in the options market, Journal of Finance 44 (1989) 1011–1023] that long-maturity options tend to

overreact to changes in the implied volatility of short-maturity options. When a GARCH process is assumed for exchange

rates, a continuous-time relationship is developed. We provide evidence that implied volatilities may not be the simple

average of future expected volatilities. By comparing the term–structure relationship of implied volatilities with the process

of the underlying exchange rates, we find that long-maturity options are more consistent with the exchange rates process.

In sum, short-maturity options overreact to the dynamics of underlying assets rather than long-maturity options

overreacting to short-maturity options.

r 2006 Elsevier B.V. All rights reserved.

Keywords: Option; Implied volatility; GARCH

1. Introduction

Empirical evidence suggests that implied volatilities vary over moneyness (volatility smile) and maturities(term–structure effect). Explanations for these nonconstant volatility under the Black–Scholes [1] workinclude the fact that volatility is stochastic and the price of the underlying asset correlates with volatility.Term–structure effects can be attributed to stochastic volatilities, which have been examined by manyresearchers on finance.

For equity options, Stein [2] assumes the mean reversion process for volatility and concludes that long-maturity options overreact to changes in short-maturity options on the basis of S&P 100 index options. Engleand Mustafa [3] estimate and test the implied GARCH(1,1) models for several stock returns and S&P 500index returns using option prices. They conclude that significant IGARCH(1,1) processes use options prices.Heynen et al. [4] estimate the stock return processes in the European Option Exchange and the AmsterdamStock Exchange. They use the results from the stock return processes and perform ex ante efficiency tests onthe three alternative term–structure models of implied volatilities: the mean reversion process as in Stein [2],

e front matter r 2006 Elsevier B.V. All rights reserved.

ysa.2006.08.040

6 2757575x53439; fax: +886 6 2744104.

ess: wangt@mail.ncku.edu.tw.

ARTICLE IN PRESSA.T. Wang / Physica A 374 (2007) 773–782774

Bollerslev’s [5] GARCH process, and Nelson’s [6] Exponential GARCH (EGARCH) process. They claim thatthe joint hypothesis of a correct model specification and ex ante efficiency is rejected in the case of a meanreversion and a GARCH(1,1) stock return volatility specification, but is not rejected in the case of theEGARCH(1,1) model.

For foreign exchange rates, Taylor and Xu [7] investigate four currency options traded in the PHLX usingregression and Kalman filtering methods. They conclude significant term–structure effects, significantvariations in the long-term volatility expectations, and similar term–structures for pound, mark, Swiss franc,and yen at any moment in time. Campa and Chang [8] test the expectations hypothesis in the term–structure ofvolatilities in foreign exchange options. The main result is acceptance of the hypothesis that the long-datedvolatility quotes are consistent with expected future short-dated volatility quotes. They also document that thecurrent spread between long- and short-dated volatility rates is a significant predictor of future short-datedvolatility rates.1

Hull and White [11] develop an option pricing model on assets with stochastic volatilities. The main result isthat if the risk of volatility is not related with the aggregate consumption, or the volatility risk isnonsystematic; and if the stock returns and volatility changes are not correlated, the European option pricingmodel for assets with stochastic volatilities is the Black–Scholes [1] price integrated over the distribution of themean volatility. That is,

f ¼ E C V� �� �

,

where f is the H–W option price, E[ � ] is the expectation notation taken over V , C( � ) is the B–S model, and V

is the average of expected future volatilities over the remaining life of the option; that is, V over time interval[t, T] is

V�

t;T ¼1

T � t

Z T

t

E V s½ �ds.

Though the term–structure effects of implied volatilities of options have been investigated extensively, adirect comparison between the implied volatilities of options and the underlying exchange rates is lessexamined. The main purpose of this study is to answer the following question: if implied volatility can beregarded as the simple average of the expected volatilities over the remaining life of the option, are theseimplied volatilities consistent with the stochastic properties of the underlying assets? We recognize that thisstudy will provide important economic implications for the validity of Hull and White stochastic volatilityoption pricing model. In this paper, a continuous-time term–structure model for implied volatilities isdeveloped by assuming that the underlying asset returns follow a GARCH process. The model we develophere is similar to the discrete-time version developed by Heynen et al. [4].

If the volatility risk is nonsystematic and the Jensen’s inequality effect does not exist, we can directlycompare the coefficients in the GARCH process of the underlying assets with the coefficients in theterm–structure models of implied volatilities.2 These two prerequisites are met because first, the volatility riskis assumed not to correlate with the aggregate consumption. Second, the data used in this paper are at-the-money options, so the problem due to Jensen’s inequality is ruled out.

When implied volatilities are simple averages of expected future volatilities, we can investigate if thevolatility process of the underlying asset is consistent with the term–structure model. If this is the case, then weconclude that the implied volatility of the option denotes the volatility of the underlying asset.

In this study we obtain the following results. First, the mean reversion parameter in implied volatilities islarger than what the underlying exchange rates suggest. Second, mean reversion coefficients decrease inmaturity, suggesting that implied volatilities may not be the simple average of the volatilities over theremaining lives of the options. This yields an alternative explanation for the overreactions in options market

1Some researchers argue that jump-diffusion models can explain these volatility smiles and term–structure effects. See, for example,

Bates [9] and Das and Sundaram [10].2The Jensen’s inequality occurs because in Hull–White stochastic volatility model, the option price is the expected value of the

Black–Scholes model with the average of expected volatilities being the volatility input.

ARTICLE IN PRESSA.T. Wang / Physica A 374 (2007) 773–782 775

reported by Stein [2]. Only implied volatilities of longer-maturity options are more consistent with thevolatilities of underlying assets.

2. Analytical framework

In this study, there are two stochastic processes for the futures price and its corresponding volatility:

dFt=F t ¼ mdtþffiffiffiffiffiffiVt

pdwt; and (1a)

dV t ¼ aðy� V tÞdtþ bV t dzt, (1b)

where Ft represents the currency futures price at time t, Vt represents the volatility at time t, m is the driftcoefficient for futures price, a represents the mean reversion parameter for Vt, y represents the long-termunconditional mean variance for Vt, and b represents the volatility coefficient for Vt. dwt and dzt represent thechanges in Gaussian–Wiener processes for Ft and Vt, respectively, with correlation coefficient r. That is,dwtdzt ¼ rdt.

Eq. (1a) describes the stochastic process followed by the underlying asset, in which the diffusion coefficientis governed by (1b). Eq. (1a) is a standard assumption for the process of the underlying asset, which assumesthat the process of the underlying asset follows a random walk with drift. (1b) is a mean reversion process, andthe parameter a measures the degree of mean reversion. If the parameter a is larger, then the volatility shouldtend to revert faster to the long-run unconditional mean value. Nelson [12] shows that, if the process ofexchange rate returns is governed by an ARCH type process, the volatility process will converge to (1b) in thelimit.

Though the underlying asset is currency futures instead of exchange rate, under fairly general assumption, itcan be shown that the volatility of futures price is the same as the volatility of the underlying exchange rate.3

Because this study compares the stochastic properties of exchange rate volatility with the implied volatilityimplicit in the option prices, it is necessary to know the different specifications in the real world and the risk-adjusted world. Consider the stochastic processes of the exchange rate and the volatility in the risk-adjustedworld:

dFt

F t

¼ffiffiffiffiffiffiVt

pd ~wt; and (2a)

dV t ¼ aðy� VtÞ � l½ �dtþ bV t d~zt, (2b)

where l represents the risk premium associated with the volatility risk. d ~wt and d~zt represent the changesin Gaussian–Wiener processes for Ft and Vt, respectively in the risk-adjusted world, with correlationcoefficient r.

Suppose that the price of the underlying asset and volatility change at only n equally spaced times in theinterval from 0 to T. Each time interval denoted by Dt is given by Dt ¼ T=n. Then (2a) and (2b) together implythe mean of lnðF tþDt=F tÞ conditional on V tþDt in the risk-adjusted world as

�Vt

2Dtþ r

ffiffiffiffiffiffiV t

p

bln V tþDt=V t

� ��

ay�

Vt

� a�b2

2

� �Dt

, (3a)

and the variance of lnðFtþDt=FtÞ conditional on VtþDt as

VtDtð1� r2Þ, (3b)

where y� ¼ y� l=a.

3Because futures price is a function of the underlying asset, convenience yield and opportunity cost, if the futures price and the

underlying asset are driven by the same state variable(s), the volatility of futures price and the volatility of underlying exchange rate are the

same.

ARTICLE IN PRESSA.T. Wang / Physica A 374 (2007) 773–782776

If Vs is the variance of the asset returns under consideration for time s at time t, where sXt, then the averageof the expected volatilities at time t over the time interval [t, T], denoted by V

�

t;T , is

V�

t;T ¼1

T � t

Z T

t

E V s½ �ds. (4)

If the parameters in Eq. (2) satisfy the Lipschitz and growth conditions, there exists a unique solution

VT ¼ Vt þ

Z T

t

aðy� � VsÞds

Z T

t

þ

Z T

t

bV s dzs. (5)

We can then obtain the expected variance in the future time, t, tXt:

Et V t½ � ¼ Vt þ

Z t

t

aðy� � V sÞds ¼ y� þ ðV t � y�Þ e�aðt�tÞ. (6)

It can be shown that the average of the expected future volatilities (i.e., the implied variance) over the timeinterval [t, T] can be represented as

V�

t;T ¼1

T � t

Z T

s¼t

Et Vs½ �ds ¼ y� þ1

aðT � tÞðy� � V tÞ e

�aðT�tÞ � 1� �

. (7)

Eq. (7) describes the relationship between the implied variance and the maturity, given the current level ofvariance, Vt, the mean reversion parameter, a, and the unconditional mean of variance, y*. It is important torecognize from this equation that when a is very large, the implied volatility is very close to the unconditionalmean, y*.

Given (3b), it is straightforward to show that the variance of lnðFT=FtÞ is given by V�

t;T ðT � tÞð1� r2Þ.If Eq. (7) represents the implied volatility, then we can develop the term–structure relationship for any two

implied volatilities of the same underlying asset with different maturities observed at any time:

V�

t;T ð1� r2Þ ¼ y�ð1� r2Þ þðt� tÞð1� e�aðT�tÞÞ

ðT � tÞð1� e�aðt�tÞÞðV�

t;t � y�Þð1� r2Þ; or

V�

t;T ¼ y� þðt� tÞð1� e�aðT�tÞÞ

ðT � tÞð1� e�aðt�tÞÞðV�

t;t � y�Þ, ð8Þ

where maturity tXt, and maturity TXt. This term–structure (8) describes the relative behavior of any twoimplied volatilities with different maturities of the same asset. The discrete-time version of this equation can befound in Heynen et al. [4]. There are two important implications when we apply (8) in the empirical works.First, the correlation coefficient between the future price and its volatility plays no role in this volatilityterm–structure relation.4 Second, the risk premium for volatility risk is implicit in y*, and this premium is notdirectly observable unless there is a volatility-based security traded in the market. Alternatively, if we assumethat the volatility risk is nonsystematic, then l is equal to zero. In this case, y* is equal to y. Because l is notdirectly observed, we assume that volatility risk is not systematic for empirical purposes.

3. Times series model and continuous-time model

Financial researchers such as Bollerslev et al. [13] suggest that foreign exchange rate returns follow GARCHprocesses. Let yt denote the foreign exchange rate in log, if there exists a GARCH(1,1) process for the returnon yt, or (1�L)yt, where L is the lag operator, then the dynamics of yt can be described by the following meanequation and volatility equation:

ð1� LÞyt ¼ lþ ut; and

ht ¼ kþ d1ht�1 þ g1u2t�1 þ �t, ð9Þ

4Although Stein [2] assumes stochastic volatility, the effect of correlation between volatility and the underlying asset is not incorporated

in his model.

ARTICLE IN PRESSA.T. Wang / Physica A 374 (2007) 773–782 777

where l,k,d1 and g1 are constants, ut and et are error terms, and ut ¼ffiffiffiffiht

pvt, where n is i.i.d. with zero mean and

unit variance.5

If the term–structure effect of implied volatilities is generated by the stochastic properties of the underlyingforeign exchange rates, then the properties of the term–structure should be reconciled with the properties ofthe volatility process of those underlying exchange rates. If the foreign exchange rates are governed by theGARCH(1,1) process as specified by (9), then the mean reversion parameter in (8) can be approximated by thefollowing relationship (see, for example, Heynen et al. [4]): d1 þ g1! e�a, or equivalently, a ¼ � lnðd1 þ g1Þ inthe limit.

4. Data and empirical results

4.1. Data

The implied volatility data used in this study are the implied volatilities of currency futures options traded inthe International Monetary Market (IMM) of the Chicago Mercantile Exchange (CME), which are providedby the Bridge-CRB. The implied volatility data are calculated using the Black-Scholes model from the dailyclosing price of at-the-money options. The exchange rate data are obtained from the Federal Reserve Bank ofChicago. The exchange rates used here are defined as the prices of foreign currencies in US dollar. Thecurrencies under study include the Australian dollar (AD), British pound (BP), Canadian dollar (CD),Deutsche mark (DM), and Japanese yen (JY). We use the sample of daily exchange rates and options withmaturities from the beginning of January 1998 to the beginning of September 2001.

The implied volatility data are merged by date. For example, each date has several options on the samecurrency futures being traded and these options have different maturities. These implied volatilities are sortedby maturity and then given an ordinal label beginning with ‘‘1’’ for the shortest-maturity implied volatility andso on. For example, if there are five implied volatility observations for each date, then option 1 has the shortestmaturity and option 5 has the longest maturity. The number of options for each date and for each currencyranges from three to five observations, depending on the availability of the data.

4.2. Empirical results

The hypothesis that term–structure effects exist in the implied volatilities of currency options is tested, andthen the term–structure model of Eq. (6) is estimated. As described in the Data subsection, observed impliedvolatilities are classified into groups by their maturity. For example, for each trading day, there are severaloptions on the same currency being traded that differ in their maturity. We classify these options traded on thesame date into several groups. Group 1 has the shortest maturity (however, no less than 10 days), and Group 2has the second-shortest maturity, and so forth. The number of groups is determined by the number ofobservations available during the sample period for each currency. For example, we can classify the impliedvolatilities of BP into more than five groups, say six groups, but we will have less than 400 observationsavailable for Group 6.

To keep the number of observations above 400, the numbers of groups available for each currency is asfollows. For AD we have three groups, for BP, CD and JY we have five groups, and we have four groups forDM. Table 1 shows the number of observations, average maturity, average implied standard deviation, andstandard deviation of the implied standard deviations for each group and each currency. The number ofobservations available ranges from 406 to 1055. During our sample period, it can be verified that the averageof the maturities for each group increased from Group 1 to Group 3, 4 or 5. The average of the impliedvolatilities is the largest for Group 1 and there is a trend that as the average maturity of the implied volatilitiesfor each group increases, the average of the implied volatilities decreases for each currency.

5GRACH(1,1) assumption for exchange rate may be subject to empirical justifications. However, GARCH(1,1) is parsimonious and

quite general for empirical comparison with other models. For example, in comparison with implied volatility in predicting future realized

volatilities of the underlying asset, Lamoureux and Lastrapes [14] use GARCH(1,1) model as a representative time series model.

ARTICLE IN PRESS

Table 1

Descriptive statistics

Group by maturity Number of observations Maturity (in year) Implied volatility

(Mean) (Mean) (Standard deviation)

(A) Australian dollar

1 972 0.08277 0.13040 0.03564

2 806 0.23577 0.11655 0.01887

3 406 0.37700 0.11129 0.01851

(B) British pound

1 1049 0.08126 0.09118 0.02097

2 975 0.14657 0.08554 0.01232

3 928 0.24794 0.08800 0.05548

4 843 0.42030 0.08569 0.00985

5 638 0.64246 0.08439 0.01025

(C) Canadian dollar

1 1032 0.07895 0.06965 0.02364

2 967 0.16310 0.06537 0.02633

3 911 0.29676 0.06025 0.01175

4 774 0.48798 0.05950 0.01021

5 455 0.69524 0.06150 0.00706

(D) Deutsche mark

1 1055 0.10591 0.14872 0.11243

2 931 0.19715 0.11460 0.03340

3 772 0.32212 0.10893 0.02237

4 493 0.44947 0.10601 0.01579

(E) Japanese yen

1 1002 0.06421 0.148076 0.05450

2 954 0.14518 0.136991 0.03496

3 895 0.24244 0.137054 0.02881

4 832 0.43200 0.135793 0.02370

5 564 0.64493 0.136058 0.01882

A.T. Wang / Physica A 374 (2007) 773–782778

Table 2 reports the estimates of the mean-reversion coefficient in the term–structure relationship. Eq. (8) isestimated by using two time series of implied volatilities with different maturities over time. For example, BPhas five implied volatilities for each date, so there are 10 pairs of implied volatilities for each estimate. Becauseimplied volatilities are autocorrelated, a is estimated with the nonlinear least square method with instrumentalvariables (two lags of implied volatilities). Several findings are concluded. First, the mean-reversion coefficient,a, tends to decrease in maturity. For example, for BP, the estimate of a is 39.42 when Groups 1 and 2 are used,and 0.71 when Groups 4 and 5 are used. Second, for DM, no convergent results were found for any pair exceptfor Groups 2 and 3, with which the estimate of a is 31.11. Third, the adjusted R-squares are very unstablewhen the different implied volatilities are used, which ranges from 0% to 85%. Finally, the Durbin–Watson(DW) tests for residuals show positive and significant first-order autocorrelations in (8), which suggest thatsuch model may not capture all the dynamic properties of implied volatilities. Because DW tests have alreadyshown significant results, we do not report Ljung–Box or Box–Pierce Q statistics.

Table 3 shows the GARCH(1,1) estimates for daily exchange rates of AD, BP, CD, DM and JY during thesample period from 01/02/98 to 08/31/01. The results suggest significant ARCH(1) effect for AD, CD and JY,significant GARCH(1,1) for BP and no ARCH effect for DM during this period. These estimates are thencompared with the estimates of a from (8).

Because (d1+g1) (from (8)) converges to e�a within the limit, we can obtain a related value interval from theestimates of (d1+g1). If the estimates of a fall into such interval, then we conclude that the term–structure ofthe implied volatilities indicates the volatility process of the underlying exchange rates. Comparing the

ARTICLE IN PRESS

Table 2

Nonlinear least square ARCH term structure estimation

Maturity pair a T-stat R Bar square(%) D.W.

(A) Australian dollar

1, 2 243 0.70 28.86 0.42

1, 3 NA

2, 3 2.1066 5.09 56.30 0.91

(B) British pound

1, 2 39.42 7.53 28.24 0.69

1, 3 16.62 0.17 0.00 2.00

1, 4 9.36 10.74 7.43 0.66

1, 5 12.82 8.84 8.13 0.52

2, 3 NA

2, 4 2.53 13.32 73.30 0.71

2, 5 1.73 11.98 62.80 0.63

3, 4 101.98 0.00 72.00 0.77

3, 5 NA

4, 5 0.71 4.28 82.19 0.94

1, 2 17.79 3.38 4.97 0.81

1, 3 29.77 8.96 26.68 0.33

1, 4 48.84 2.83 2.36 0.29

1, 5 25.66 4.47 0.00 0.26

2, 3 NA

2, 4 NA

2, 5 NA

3, 4 3.68 7.98 53.25 0.63

3, 5 4.42 9.80 42.61 0.56

4, 5 14.39 0.36 40.05 0.81

(D) Deutsche mark

1, 2 NA

1, 3 NA

1, 4 NA

2, 3 31.107 8.10 30.45 1.34

2, 4 NA

3, 4 NA

(E) Japanese yen

1, 2 187.42 0.97 78.63 0.22

1, 3 20.37 9.59 55.89 0.63

1, 4 12.32 8.94 30.45 0.67

1, 5 8.49 13.26 19.08 0.66

2, 3 9.02 14.15 85.53 0.82

2, 4 6.33 17.14 64.63 0.60

2, 5 5.32 13.55 39.29 0.71

3, 4 4.00 12.94 80.93 0.89

3, 5 5.36 12.71 51.98 0.67

4, 5 2.86 9.07 74.32 0.63

Nonlinear term structure estimation when the implied volatility on currency options follows the ARCH process. The parameter a is

estimated from the following term structure relationship:

V�

t;T ¼ V�

þðt� tÞð1� e�aðT�tÞÞ

ðT � tÞð1� e�aðt�tÞÞðV�

t;t �V�

Þ,

where Vt;T

�

is the implied variance observed at time t with maturity T, and V�

is the unconditional mean for the implied variance. NA: not

applicable.

A.T. Wang / Physica A 374 (2007) 773–782 779

ARTICLE IN PRESS

Table 3

GARCH(1,1) estimation for exchange rate returns

Currency k d1 g1 Log-likelihood function value

AD 0.0000 0.0934 0.0118 4040.19

(18.72)*** (2.60)*** (0.65)

BP 0.0000 0.0802 0.1260 4439.78

(17.17)*** (2.76)*** (5.26)***

CD 0.0000 0.0945 �0.0000 4689.68

(21.31)*** (3.01)*** (�0.01)

DM 0.0196 0.5645 �0.0020 1354.39

(8.99)*** (1.13) (�0.06)

JY 0.0000 0.2542 0.0000 3952.92

(20.95)*** (8.05)*** (0.00)

The GARCH(1,1) process for daily exchange rates is based on the following specifications:

Model : ð1� LÞyt ¼ lþ ut; ht ¼ kþ d1u2t�1 þ g1ht�1 þ �t,

and ut ¼ffiffiffiffiht

pvt; v�iidð0; 1Þ,

where (1�L)yt is the change in exchange rate, and the exchange rate is defined as the USD value of one foreign currency. l,k,d1 and g1 areconstants, ut and et are error terms. The numbers in the parentheses are t-statistics. *** indicates significant at 1% significance level.

Table 4

Comparison of the confidence interval for mean-reverting parameters and the coefficients in the term structure models

Currency d1+g1 95% Confidence intervals for d1+g1 Corresponding intervals for a a � � ln d1 þ g1� �� �

AD 0.0934 [0.0228, 0.1640] [1.8079, 3.7810]

BP 0.2062 [0.1022, 0.3102] [1.1705, 2.2808]

CD 0.0945 [0.0330, 0.1560] [1.8579, 3.4112]

DM NA

JY 0.2542 [0.1923, 0.3161] [1.1517, 1.6487]

The confidence interval is obtained from the GARCH(1,1) parameters for each currency. We examine if the parameters in the continuous-

time term–structure models from Tables 4 and 5 fall into the confidence intervals of (d1+g1) from the discrete-time GARCH(1,1) models:

Model : ð1� LÞyt ¼ lþ ut,

ht ¼ kþ d1u2t�1 þ g1ht�1 þ �t; and

ut ¼ffiffiffiffiht

pvt; v�iidð0; 1Þ,

where (1�L)yt is the change in exchange rate, and the exchange rate is defined as the USD value of one foreign currency. l,k,d1 and g1 areparameters. ut and et are error terms. NA: not applicable.

A.T. Wang / Physica A 374 (2007) 773–782780

estimates of a from (8) with the value intervals reported in Table 4, it is found that, except for long-maturityimplied volatilities, most of the estimates are much larger than the values implied by the GARCH processes ofexchange rates. In other words, the overall mean reversion coefficients implicit in implied volatilities are muchlarger than what the underlying exchange rates suggest.

5. Conclusion

Market practitioners use relative magnitudes of implied volatilities for both long- and short-maturityoptions to predict future movement in the underlying assets. If short-term option volatilities are significantly

ARTICLE IN PRESSA.T. Wang / Physica A 374 (2007) 773–782 781

lower than long-term ones, we expect a potential breakout; while if short-term option volatilities aresignificantly higher than long-term ones, then we expect reversion to range trading.6 The intuition behind suchstrategies suggests mean reversion of the volatility processes, which are consistent with what this study andprevious literature on finance suggest. In this paper, we use the framework of stochastic volatility optionpricing model of Hull and White [11] with nonsystematic volatility risk to provide alternative explanation forthe observations documented by Stein [2].

Stein [2] suggests that long-maturity implied volatility is fully determined by short-maturity impliedvolatility and the mean reversion parameter, if agents in the options market have rational expectations. Steinfinds that long-maturity implied volatilities tend to overreact to short-maturity implied volatilities. In thispaper, we have two major findings. First, the mean reversion parameter in implied volatilities is too largecompared with what the underlying exchange rates suggest for most cases. Second, mean reversion coefficientsdecrease in maturity, which suggests that implied volatilities may not be the simple average of the expectedvolatilities over the remaining lives of options.

A larger mean reversion parameter suggests a flatter term–structure in implied volatilities. It is very likelythat agents in an options market tend to weigh short-term expected volatilities of exchange rates more whenformulating implied volatilities. The idea is described by the following example. Consider an option with amaturity of four days. The expected volatilities of the coming four trading days are 12%, 13%, 14% and 15%.If the implied volatility is the simple average of the four, then the implied volatility would be 13.5%. If theimplied volatility is a weighted average of the four with larger weights for nearer volatilities, such as12%*0.5+13%*0.3+14%*0.1+15%*0.1, or 12.8%, a flatter term–structure is obtained. The reasons whyimplied volatility weighs nearer expected volatilities more could be that more distant expected volatilities aremore uncertain while nearer volatilities are more ‘‘reliable’’. As a result, options are priced closer to thecurrent volatility level. Another finding is that longer-maturity options are more consistent with what theunderlying exchange rates suggest with respect to the mean reversion parameter. This suggests that short-termoptions overreact to underlying assets.

The argument of this paper that implied volatility may weigh more short-term expected volatility than long-term expected volatility may provide following implication for option traders: though the short-term impliedvolatility is more volatile than long-term implied volatility, the difference may not be as large as what theimplied volatility as the average of future expected volatility suggests. Finally, it is argued that bid-ask spreadswill generate a bias in asset mean returns calculated with transaction prices. This will yield a biased constantterm in the autoregressive process, whereas our mean reversion parameter will not be affected by such biases.7

References

[1] F. Black, M. Scholes, The pricing of options and corporate liabilities, J. Pol. Econ. 81 (1973) 637–654.

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data are used. In this study, bid-ask spreads will affect the constant term in the term-structure model instead of the mean reversion

parameter a.

ARTICLE IN PRESSA.T. Wang / Physica A 374 (2007) 773–782782

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