Pricing Currency Options Under Stochastic Volatility

Pricing Currency Options Under Stochastic Volatility

Ming-Hsien Chen Department of Finance National Cheng Chi University

Yin-Feng Gau*

Department of International Business Studies National Chi Nan University

August 2004

Keywords: Stochastic volatility; Currency options; Implied risk premium of volatility.

JEL codes: F31, G13.

* _{Corresponding Author. Department of International Business Studies, National Chi Nan University, 1 University Rd., Puli,}

Abstract

This paper investigates the relative pricing performance between constant volatility and stochastic volatility pricing models, based on a comprehensive sample of options on four currencies, including the British pound, Deutsche mark, Japanese yen and Swiss franc, traded frequently in the Philadelphia Stock Exchange (PHLX) from 1994 to 2001. The results show that the model of Heston (1993) outperforms the model of Garman and Kohlhagen (1983) in terms of sum of squared pricing errors for all currency options. Furthermore, the adjustment speed toward the long-run mean volatility in the currency market is faster than that in the stock market. It may be attributed to the larger momentum effect in the stock. We also find that the stock market exhibits larger implied skewness than the currency market. This may be due to stronger ‘trend effect’ in the stock market, especially involved in the bear market.

Keywords: Stochastic volatility; Currency options; Implied risk premium of volatility.

1. Introduction

It has been shown that the Black-Scholes (1973) option pricing model is subject to systematic biases originated from the violation of the normal distribution assumption on the underlying returns (e.g., Melino and Turnbull (1990), Day and Lewis (1992), Rubinstein (1994), Bakshi, Cao, and Chen (1997), Nandi (1998), Bates (1996, 2000, 2003), and Lin, Strong and Xu (2001)). Empirically, the negative implicit skewness causes the out-of-the-money option price bias, whereas the implicit leptokurtosis increases the prices of deeply in-the-money and out-of-money options and lower prices of near-the-money options.

To enhance the pricing accuracy, the stochastic volatility (SV) option pricing models, such as Hull and While (1987), Scott (1987), Wiggins (1987), Melino and Turnbull (1990), and Heston (1993), incorporate the leptokurtosis or excess kurtosis of the underlying asset returns by allowing the volatility process to behave randomly.1 Black and Scholes (1973) specified that the source of volatility risk comes from stochastic returns of the underlying, however, the SV option pricing models allow the pricing risk to emerge from both the stochastic processes of price and the volatility. To model the stochastic process of volatility in the SV option pricing model, one has to specify the market price of volatility risk, the volatility of variance, and the correlation between underlying price (return) and its volatility. Assuming zero correlation of price and volatility, Hull and White (1987) used the Taylor expansion technique and proposed a power series approximation for the European stock call option; Scott (1987) specified the volatility process as a mean-reverting Ornstein-Uhlenbeck process; Wiggins (1987) included investors’ utility function into the model to incorporate the market price of volatility risk. The major limitation of Hull and White (1987), Scott (1987), and Wiggins (1987) models is the assumption of zero correlation between the

underlying return and its volatility, although they do consider the phenomenon of leptokurtosis of the return process. Fleming (1998) and Guo (1996) argued that the implied variance extracted from the model of Hull and White (1987) is dominant but still a biased estimator in terms of out-of-sample forecasting performance in the stock-index and foreign currency options markets.

Stein and Stein (1991) offered the stock price distribution evaluated by an inverse Fourier series transformation based on the no correlation between the underlying return and its volatility, and obtained a closed-form formula for the European options. Specifically, Hull and White (1987), Scott (1987), Wiggins (1987) and Stein and Stein (1991) contributed the consideration of the kurtosis by modeling the volatility to a stochastic process, but left the skewness of the underlying returns to be not incorporated in the models.

Heston (1993) argued that the correlation between the underlying asset return and its volatility affected both the skewness and the leptokurtosis of the distribution of underlying returns. This correlation is important in explaining for the skewness that affects the pricing of in-the-money options. With the zero correlation, the stochastic volatility is only related to the kurtosis that influences the pricing of near-the-money and far-from-the-money options, thus it can induce the pricing errors in options. Bates (1996) offered an empirical study based on the closed form solution of Heston (1993), with a diffusion-jump process. By estimating the parameters in the volatility process on the Deutsche mark options, Bates (1996) found that the stochastic volatility could explain the implicit leptokurtosis only by considering the jump risk. Guo (1998) empirically studied the parameters on the risk-neutral variance process and the market price of variance risk implied in the PHLX currency options, based on the model of Heston (1993). Guo found that the market price of variance risk was nonzero, time

varying, and of sufficient magnitude to imply that the compensation for variance risk was a sufficient component of the risk premium in the currency market. On the stock options, Bakshi, Cao and Chen (1997) evaluated the relative in-sample fitness, out-of-sample pricing and hedging performances for S&P 500 stocks call options among various options pricing models, including SV, SV with jump, and stochastic interest option pricing models. However, their results based on stock index options suggest that adding jumps or stochastic interest rates to the SV model does not significantly improve the hedging performance, although the SV model with jumps performs best in out-of-sample pricing performance. Nandi (1998) also pointed out the importance of the correlation between the underlying return and its volatility. Based on the options on the S&P 500 index, Nandi (1998) observed a significant improvement on the pricing performance when considering a nonzero correlation in the empirical investigation. Bates (2000) obtained similar results based on the S&P 500 futures options and demonstrated a negative correlation between index returns and volatility innovation. Lin, Strong and Xu (2001) empirically studied how the Heston (1993) model reduced the pricing errors of FTSE 100 index options.

Although Bakshi, Cao, and Chen (1997) have evaluated the empirical performances of constant volatility and SV option pricing models, their results based on index options may not hold for currency options. In this paper, we use a 8-year sample of historical prices of currency options on the Philadelphia Stock Exchange (PHLX), including options on the British pound, Deutsche mark, Japanese yen and Swiss franc, from 1994 to 2001, to compare the pricing performances of constant volatility and SV options pricing models. We also extract the parameters implicit in the model of Heston (1993) by allowing an nonzero correlation between the volatility and the underlying asset return. With different estimation periods, such as weekly, monthly or 6-months

interval, we compare the different interval-estimated dynamic processes of asset price and volatility based on these parameters. In this paper, we investigate the exchange rates process implicited in currency options and compare the relative pricing performance between the model of Garman and Kohlhagen (1983), a modified version of the Black-Scholes model for the currency option, and the model of Heston (1993) which specifies both the stochastic processes of the underlying asset return and volatility.

There are three major findings in this study. First, we find that the model of Heston (1993) outperforms the model of Garman and Kohlhagen (1983), in terms of the sum of squared pricing errors for all currency options from 1994 to 2001. The percentage difference of the sum of squared pricing errors between the Garman and Kohlhagen and the Heston models are from nearly 30% to 70%, suggesting that the specification of stochastic volatility does improve the pricing accuracy on currency options. Furthermore, we estimate the implicit parameters of the risk-neutral stochastic variance process in the model of Heston (1993) by using a nonlinear-least-square-error approach. We find that, in the currency market, the adjustment speed toward the long-run mean volatility implicit in currency options is faster than that in the stock market. It may be attributed to the fact that there is a larger momentum effect in the stock market than that in the currency market, resulting in the slow mean-reverting speed in the stock market. Finally, we find that the stock market exhibits a more negative skewness than that in the currency market. The phenomenon could be explained based on the fact that the stock market exhibits a stronger ‘trend effect,’ especially involved in the bear market. Another explanation could be supported by the exchange-rate target zones hypothesis that states that there exist no extreme values on currency returns.

The remainder of this paper is organized as follows. Section 2 introduces models of currency option pricing, including Garman and Kohlhagen (1983) and Heston (1993). Section 3 describes the data of the PHLX currency option prices. Section 4 reports the estimation results of implied parameters and the pricing performance of different options pricing models. Section 5 concludes the paper.

2. Models of Currency Option Prices

Black and Scholes (1973) assumed that the volatility of the returns is constant and used the concept of hedging portfolios of options and their underlying stocks to derive the non-dividends European option theoretical valuation formula. Garman and Kohlhagen (1983) is one of the versions of the Black-Scholes options pricing model on the currency option. Similar to the Black-Scholes model, based on the arbitrage-free condition, Garman and Kohlhagen (1983) compared the advantages of holding a foreign exchange option with those of holding its underlying currency. Under these conditions, spot and options markets are frictionless and the domestic ( rD ), and foreign ( rF) interest rates are constant, Garman and Kohlhagen (1983) applied the interest-rate parity to derive the European currency option valuation formula:

1 2 ( , ) rF ( ) rD ( ) C S τ =Se− τN d −Ke− τN d

, (1)

where 2 1 ln( /S K) [(rD rF) 0.5 ] d σ τ σ τ + − + = ; d₂ =d₁−σ τ . Symbols S, K, σ , and T t

τ ≡ − denote the spot rate, exercise price, the constant volatility of the underlying asset return, and the time to maturity, respectively. By the put-call parity, the European style put option can be obtained:

2 1

( , ) r TD ( ) r TF ( )

P S T =Ke− N −d −Se− N −d . (2) Many studies, such as Bollerslev, Chou, and Kroner (1992), Heieh (1989), and

Poon and Granger (2003), have argued that the exchange rate volatility follows a stochastic process. Specifically, tails of the distribution computed with intraday or daily market prices are fatter than those of the lognormal distribution, exhibiting leptokurtosis (Gesser and Poncet, 1997). Thus, the estimated implied volatility from the constant volatility assumption from the Black-Shcoles or Garman–Kohlhage model will be a biased estimate. Various stochastic volatility option pricing models, such as Hull and White (1987), Scott (1987), Wiggins (1987), Melino and Turnbull (1991), Heston (1993), and Bates (1996), were developed to solve the unrealistic assumption, the constant return volatility. In this paper, we adopt Heston’s (1993) SV model to investigate its relative performance against the model of Garman-Kohlhagen (1983), and explore the exchange rate process implicit in the options based on the model of Heston (1993).

Releasing the assumption that correlation between volatility and the spot return is zero, Heston (1993) obtained a closed-form solution for the European option on an asset, including stock, currency and bond, with mean-reverting square-root stochastic volatility. Heston (1993) asserted that the correlation between the underlying asset returns and its volatility affected the skewness and leptokurtosis of the distribution of the underlying asset returns. The volatility is specified to follow an Ornstein-Uhlenbeck mean reverting process given below:

( ) ( ) ( ) ( ) S( )

dS t =µS t dt+ V t S t dz t , (3)

( ) ( ( )) _V ( ) _V( )

dV t =κ θ−V t dt+σ V t dz t , (4) where ( )dz t_S and dz t_V( ) are Wiener processes with instantaneous correlation ρ, i.e.,

(

_S( ), _V( )

)

corr dz t dz t =ρdt. The major difference between the model of Hull and White (1987) and that of Heston (1993) is the specification of the correlation. The

instantaneous variance V t( ) follows a square-root process proposed by Cox, Ingersoll and Ross (1985). The dynamic process, dV t( ), follows a mean-reverting process with long term mean, θ, mean reversion speed, κ, and volatility of volatility, σV.

Following the risk-neutral pricing probabilities derived by Cox, Ingersoll, and Ross (1985), Equation (4) can be rewritten as:

* * ( ) ( ( )) _V ( ) _V( ) dV t =k θ −V t dt+σ V t dz t , (5) where κ*= +κ λ, * k k θ θ λ =

+ , and λ is the risk premium parameter compensated for

the volatility risk from σ_V .2 The instantaneous risk-neutral variance process ( )V t

drifts toward a long-run mean of θ*, with the mean-reversion speed dominated by κ*. In general, if the risk-neutral probability density function of a future security’s price is

( T)

f S , the exercise price is K, the time to maturity is τ = −T t, and the price of a European call option can be written as,

( ) ( )

r K

C =e−τ

∫

∞ S_τ −K f S dS_{τ τ}. (6) Heston (1993) derived the density function of S satisfying Equations (3) and (5) by inverting the Fourier transform for the convolution of the lognormal density of S_T

conditional on the average variance and the density of average variance. The derivation relies on properties of the conditional distribution and involves numerical integration of the characteristic function of the probability process. The European call option on currency is as follows:

1 2

f d

r r

C=Se− τP−Ke− τP . (7)

C, S, and K have the same meaning in the Balck-Scholes model. In order to get a closed-form solution, Heston used the Fourier transformation and derived the probability density function Pj :3

ln( ) 0 ( , , ; ) 1 1 ( , , ; ln[ ]) Re 2 i K j j e f x v t P x v t K d i ϕ _ϕ ϕ π ϕ − ∞ _{ } = + _{ }    

∫

, (8)

where j=1, 2, Re means the real part of the square bracket, f x v t_j( , , ; )ϕ represents the characteristic functions of the conditional probability P_j.

(

)

0 exp ln( ) 1 1 Re 2 1, 2 j j i K f P d i j ϕ ϕ π ϕ ∞  − ⋅  = + _{ }   =

∫

_{, (9)}

where f_j, j=1, 2 is the characteristic function of the parameters set,

* *

{ ,k θ σ ρ, _v, }

Φ = .4 Given no-arbitrage condition and the price of a call option, C, we can use the put-call parity to obtain the European put option price as,

f d

r r

P= −C Se− τ +Ke− τ. (10) Since the American style currency options traded in the PHLX are used in this study, we employ the quadratic approximation method of Barone-Adesi and Whaley (1987) to adjust the early exercise premium to derive a more accurate empirical study.5

3. Data Description and Sample Properties

We use the “Foreign Currency Options Pricing History (FCOPH)” database of the PHLX.6 The FCOPH data of the PHLX contain options on the Australian dollar, British pound, Canadian dollar, Deutsche mark, Euro, Japanese yen, Swiss franc and other related customized option contracts. The trade-by-trade option prices cover the period from January 1994 to December 2001. The major advantage of using PHLX database is that the daily closing option prices and the simultaneous spot exchange rate quotes are recorded on the same transaction report. The trading hours is from 2:30 a.m. to 2:30 p.m., Eastern Time, Monday to Friday. We use the mid-month option, which expires in March, June, September, December and the two near-term months and

month-end option, which expires at three nearest months call and put options. The precise expiration date is the Friday preceding the third Wednesday of the month for mid-month options and the last Friday of the month for month-end options.

Daily closing prices of options and the simultaneous spot exchange rate quotes on British pound, Deutsche mark, Japanese yen and Swiss franc against US dollar are selected as our samples. Two reasons for choosing these four currencies are that they are highly volatile and are most frequently traded in the PHLX in terms of the trading volumes. Since the European Union started using EURO in 1999, the currency options of British pound and Deutsche mark were traded less over the period from 1999 to 2001 than that from 1994 to 1998. Thus, the selected sample periods are from January, 1994 to December, 1998 for the British pound and Deutsche mark and those are from January, 1994 to December, 2001 for Japanese yen and Swiss franc. Since the database of PHLX contains the null values of simultaneous spot rates on Japanese yen over the period from October, 1994 to March, 1996, we fill the blank with daily spot rate from the DataStream.

In this study we use the American-style call and put options and discard the uninformative options records from samples in the following categories. First, since for the implied volatility of options, the maturity day is less than 5 calendar days that is erratic, we use options with the period from 5 to 90 calendar days to expiration. Second, options violate the European-style boundary conditions, for example, the condition is _C<_Se−rFτ −_Ke−rDτ _{for currency call options. And finally, the daily options}

with the strike/spot price ratio (moneyness, denoted as S/K for call options, K/S for put options) are from 0.8 to 1.2.

We use the daily closing quotes of London Euro-Dollars rates collected from DataStream, including the U.S. dollar against the British pound, Deutsche mark,

Japanese yen and Swiss franc as likely riskless interest rates to derive the implied volatility from two alternative option models. To match the maturity of currency options, the 30-, 60- and 90-day interest rates are used to be the rates whose maturity are very close to the option expiration. Table 1 summarizes the sample sizes of four currency options. There are a total of 394,940 records in the PHLX database including the British pound, Deutsche mark, Japanese yen, Swiss franc and other currencies (such as, Australian dollar, Canadian dollar and EURO) over the period from 1994 to 2001. From Table 1, we can find that most options traded in the PHLX are American-style, especially on the British pound and Deutsche mark.

To investigate the existence of volatility smile of implied volatility, we first divide the option data into several categories in terms of either moneyness or time to maturity. We classify the money of a call (put) option to three categories, at-the-money (ATM) option, in which its S/K (K/S) is from 0.99 to 1.01, out-of-the-money option (OTM), in which its S/K (K/S) is below 0.99, and in-the-money option (ITM), in which its S/K (K/S) is above 1.01. A finer partition resulted in six moneyness categories. According to the term of expiration, options are classified into (i) short term maturity (5~30 days); (ii) medium term maturity (30~60 days); and (iii) long term maturity (60~90 days). Table 2.1 to 2.4 offers the sample properties of PHLX currency options in our studies. The reported numbers are the average quoted prices, the standard deviations shown in the parentheses, and the total observed options listed in braces, for each moneyness-maturity category. The sample period extends from January, 1994 to December, 1998 including a total of 30410 options for British pound and 80779 options for Deutsche mark, respectively; and from January, 1994 to December, 2001 including a total of 63016 for Japanese yen, and a total of 50990 for Swiss franc, respectively. Note that for British pound, OTM, ITM and ATM options take up 45.3 percent, 10.8

percentage and 43.9 percent of the total sample, and the average call price ranges from 0.1995 cents for short-term, deep OTM options to 8.0601 cents for long-term, and deep ITM calls. Sample properties across moneyness and time to maturity for Deutsch mark, Japanese yen and Swiss franc are similar to that of British pound as shown in Tables 2.2 - 2.4.

4. Implied Parameters Estimation and In-Sample Pricing

Performance

Our analysis intend to present pricing biases of the Black-Scholes model caused by the stochastic volatility of underlying returns and to investigate this volatility process implicit in currency options. First of all, we back up the implied volatility from Garman and Kohlhagen (1983) model on four currency options. If the volatility pattern showed the effect of smile, the constant-volatility assumption could not be applied for currency options and the pricing biases did exist. Furthermore, we could apply a more complicit model, the Heston model (1993), to investigate the relative pricing performance between Garman and Kohlhagen (1983) model, and Heston (1993) stochastic volatility model. Finally, we investigate the volatility process implicit in currency options.

4.1 Implied volatility of Garman and Kohlhagen (1983) model

There are many currency options traded on the same underlying currency with different maturities and strike prices. To estimate the implied volatility of one currency on certain period, we follow the approach of Whaley (1982) by using an equally-weighted nonlinear least squares method to estimate the daily implied volatility.

, 2 , , , 1, min ( ) [ ] GK i t N GK GK i t i t i t i t SSE C C σ σ =

∑

₌ − , (11)

where C_{i t,} is the currency i option’s market price, N is the trading call/put options in sample period t in each moneyness-maturity category, and ,

GK i t

C is Garman and Kohlhagen’s theory call/put price derived from equation (1).

Table 3 reports the average Garman and Kohlhagen implied volatility estimated by minimizing the sum of squared errors across six moneyness and three maturity categories. It is obvious that the Garman and Kohlhagen’s implied volatility exhibits a smile pattern as shown in Figure 1, especially in the deep OTM options. The same pattern can also be found across the maturity, but in some cases the pattern exhibits a slight smirk effect, as the term to expiration increases. For example, the implied volatility of deep-OTM and short maturity category on Japanese yen options, is from 0.141, 0.128 to 0.105. A non-flat implied volatility curve across moneyness is an indication of model misspecification. These findings of moneyness-related and maturity-related biases resulted from the constant volatility assumption are consistent with the prior empirical studies (e.g., on stock options, Bakshi, Cao and Chen (1997); on currency options, Bates (1996)). Therefore any improvement offered by an alternative model must be able to show the ability to price OTM and ITM options more accurately. As the smile effect is indicative of negative-skewed implicit return distributions with leptokurtosis, a better model, such as Heston (1993), must take the phenomenon of the negative-skewed and leptokurtic distributions of underlying returns into account.

4.2 Implied risk-neutral parameters from Heston (1993) model

of the risk-neutralized stochastic variance process in Equation (4). Similar to the process of backing up the implied volatility from Garman and Kohlhagen in Equation (11), we follow Whaley (1982) to use an equally weighted nonlinear least square to estimate the implied parameters. The principle is that the estimated parameters are chosen to minimize the sum of square pricing errors between the quoted options prices and theoretical option prices from the Heston (1993) model. Specifically, for each sample period in the sample, we collect all the call and put option i and trading on period t, then solve the following optimization function which minimizes the sum of squared errors between model prices and observed premiums,

2 , , , 1 1 min [ ( , )] t T N H i t i t t v t i C C v Φ

∑ ∑

_{= =} − Φ , (12)

and Φ ={κ θ σ ρ*, *, _v, } is the parameters set in Heston’s return and volatility processes.

T is the number of trading days in an estimating period, N is the number of options on period t; C_{i t,} and C_{i t}H_, ( ,v_t Φ) are the traded options prices and Heston’s option price, respectively. The computational intensity makes the direct estimation of the entire parameter vector Φ ={κ θ σ ρ*, *, _v, } and v_t. Therefore, the parameters in Φ and v_t

are estimated simultaneously until the parameter estimates converge within a desired tolerance level.7

Summary statistics for volatilities and risk-neutral parameter estimates from the Heston model implicit in options prices are shown in Tables 4 and 5, respectively. Implied volatility smile graphs show the cross-sectional pricing ability of an option pricing model. The information on implied volatility taken from an option model is directly related to the option trader’s expectations of the underlying volatility over the remaining life of the option. In other words, implied volatilities over substantially different horizons indicate different market perceptions of the underlying volatility.

Therefore, alternative volatility smiles should be compared for a given trading horizon. Figure 2 presents the implied volatility curves associated with the Garman and Kohlhagen as well as Heston models, backed out from the four PHLX currency options across moneyness. It shows that the implied volatilities of Heston model display a flat curve across moneyness than that of Garman and Kohlhagen model. Figure 2 illustrates a rough idea that for currency options, the stochastic volatility model provides a superior pricing fitness as the constant volatility model does, though the sharps of Heston’s implied volatilities across moneyness still exhibit a slight smirk-pattern, especially for short-maturity out-of-money options.

The risk-neutral mean-reverting volatility process is controlled by three parameters: the long-term mean θ* to which volatility reverts, the speed of reversion

*

κ , and the variation coefficient σ_v. For example, the parameters of British pound in

1994 are estimated to the values of {κ*, θ* _, σ_v} = {10.469, 0.2243, 0.1491}, and volatility takes about 24 days to revert halfway towards a long-term mean of 22.43%. Based on the whole sample period, average reverting speed of the volatility on four currencies are 10.376%, 8.1663%, 13.330%, and 7.8516% for British pound, Deutsche mark, Japanese yen, and Swiss franc, respectively. The range of average half-life of shocks is from 19 to 33 days. Compared with Guo’s (1998) findings on Deutsche mark traded in the PHLX from 1987 to 1992, our estimates of the parameter on the speed of reversion κ* (an average value 8.1663) is similar. However, we find that the speed of reversion on the volatility shock in the currency market is faster than that in stock markets. Bakshi, Cao and Chen (1997) found that the value of speed of reversion for S&P 500 index options during the period from 1988 to 1991 is equal to 1.15, similar to the finding of 1.49 for the period from 1988 to 1993 in Bates (2000), while Nandi (1998) found it on stock futures options as equal to 3.29. This phenomenon could be

contributed to the more resourceful information and more frequent trading session in the currency market than those in the stock markets. Thus, the adjusting speed toward the long-run mean volatility in the currency market is quicker than that in the stock market. Another story is that the momentum effect could be large in the stock market resulting in a slow mean-reverting speed.

The average variation coefficients, σ_v which catch leptokurtosis effect of the return process, are 0.1494, 0.1090, 0.1008, and 0.2012 for British pound, Deutsche mark, Japanese yen, and Swiss franc, respectively, during each sample period. While the correlation coefficients, which responde to the negative skewness of the return distribution, between the two Brownian motions in return and volatility process are estimated to be -0.0090, -0.0143, -0.0097, and -0.0071 (average) on the four currencies in Table 5. These empirical findings show that the stochastic volatility model does provide a flexible structure to consider both skewness and leptokurtosis effects on the distribution on the underlying returns, which is oversimplifiedly assumed to follow the log-normal distribution in the Black-Scholes type option models. Previous studies, e.g. Bakshi, Cao and Chen (1997), Nandi (1998) and Bates (2000), show that in the stock market, coefficients range from -0.54 to -0.79 which suffer more series skewness effects than that in the currency market. A potential explanation for this difference between the stock and currency markets may be attributed to the former that owns the stronger ‘trend effect’ especially involved in the bear market. It also could be supported by the exchange-rate target zones hypothesis, causing that there does not exist extreme values on currency returns, in foreign exchange markets.

4.3 In-sample pricing fitness

Kohlhagen model as well as the Heston model are estimated by minimizing the sum of squared errors between the market price and the theoretical price. The daily average sum of squared errors, as shown in Table 6, are calculated by collecting each option pricing squared errors and then are divided by the 365 calendar days for each model. These reported statistics are quite informative about the internal working of the models. As such, several observations are in order. First, from Table 6, it can be easily found that the Heston model performs better than the model of Garman and Kohlhagen in terms of sum of squared pricing errors for all currency options. The percentage difference of the sum of squared pricing errors between Garman and Kohlhagen model and Heston model are 38.64%, 28.80%, 41.05%, and 68.11% for the British pound, Deutsche mark, Japanese yen, and Swiss franc, respectively.

The improved pricing accuracy could be attributed to two major modifications, the skewness and the high kurtosis in the Heston model. Comparing Tables 5 and 6, we can find that the largest improved pricing performance occurred in 1996 for the British pound, 1996 for the Deutsche mark, and 2000 for Japanese yen, respectively. From Table 5, these years also show the largest negative skewness effects among the three currencies. (The values of ρ are -0.0099, -0.0149, and -0.107.) Among the four currency options, Heston SV model performs better than the Garman-Kohlhagen model, especially in Swiss franc. This result shows another source, the estimated σ_v, which catches the leptokurtosis effect, that could improve the pricing performance. The value of the largest improved pricing performance in Swiss franc is almost twice as that of Japanese yen (84% larger than that of Deutsche mark), and is near 35% larger than that of British pound. Similar to the studies of Guo (1998) in currency options, and of Bakshi, Cao and Chen (1997) in the stock index options, our results support that including the stochastic volatility process in the option pricing model can catch the

actual return distribution better than the log-normal distribution specification in Black-Scholes type models, in terms of in sample pricing fitness.

5. Conclusion

Using a comprehensive sample of the PHLX currency options, including the British pound, Deutsche mark, Japanese yen, and Swiss franc, we find that considering the skewness and kurtosis of the underlying asset returns significantly improve the accuracy of option pricing, especially for the out-of-money options.

There are three principal contributions in this paper. First, we employ currency options on the British pound, Deutsche mark, Japanese yen, and Swiss franc to examine the pricing performances of the constant volatility option pricing model of Garman and Kohlhagen (1983) and the stochastic volatility option pricing model of Heston (1993). Second, we back out the volatility implied in the foreign-currency option market via the models of Garman and Kohlhagen as well as Heston (1993) and identify whether the implied volatility is characterized by a smile. The results show that the volatility smile effects are significant especially in the deep OTM options across moneyness and time to maturity in the PHLX currency option market. Third, utilizing the nonlinear-least-square method we estimate the parameters of Garman and Kohlhagen (1983) model and Heston (1993) model. Similar to the related studies in stock markets, we find that Heston model does provide a superior pricing fitness than the constant volatility model, though the sharps of Heston’s implied volatilities across moneyness still exhibit a slight smirk-pattern, especially for short-maturity out-of-money options.

Moreover, we estimate the parameters of the underlying process implicit in Heston model, based on four foreign-currency options. We find that the speed of reversion on the volatility shock in the currency market is faster than that in stock markets proposed

by Bakshi, Cao and Chen (1997) and Bates (2000) for S&P 500 index options, and Nandi (1998) for stock futures options. We think that the divergence could be caused by the degree of efficiency differing from stock market to the currency market. We presume the different reverting speed between the stock and currency markets could be attributed to the more resourceful information and more frequent trading sessions in the currency market than those in the stock market. Another reason may be due to the momentum effect. It could be larger in the stock market than in the currency market resulting in the slow mean-reverting speed in the later market. Regarding the negative correlation parameter, we find that prior studies on stock markets suffer more series skewness effects than our results in the currency market. A potential reason to explain this difference between stock and currency market may be attributed to the former that owns the stronger ‘trend effect’ especially involved in the bear market. It could also be supported by the exchange-rate target zones hypothesis, which states that there does not exist extreme values on currency returns, in foreign exchange markets. Finding out factors that result in the different behaviors between the stock market and currency market is an interesting direction of future research. A further exanimation by out-of-sample pricing performance and hedging efficiency will be considered in our future investigation.

Footnotes

1. Although the SV option pricing models with jumps could price options better, Bakshi et al. (1997) found that adding jumps or stochastic interest rates does not improve the performance of the SV model. Therefore, the focus of this paper is on the improvement of considering stochastic volatility in pricing foreign-currency options.

2. Since there is no traded security that can be used to hedge the risk of exchange rate volatility, the volatility of variance, the valuation of the option is no longer preference free, and the market prices of volatility risks need to be determined. Similar to the argument of Breeden (1979), the risk premium can be set as proportional to the

volatility V, that is, ( , , )λ S V t =λV.

3. It assumes that the moment-generating function M(t) of a random variable X is finite on an interval –a < t < a. If M(t) is finite on such an interval, then k

EX is finite for all k; in particular, if EX does not exist or is infinite, then M(t) does not exist on such an interval. To handle such a distribution, the characteristic function is often used. The characteristic function of a random variable X is defined by

)] (sin( ) [cos( ) ( ) (t E e E Xt iE Xt

c = iXt = + . In the applied mathematics, the characteristic function is often called the Fourier transform of the density function. See Arnold (1990), Mathematical Statistics, Prentice-Hall Englewood Cliffs, New Jersey, pp. 117-122, for details.

4. See Heston (1993) for more details on the derivation of the characteristic functions. 5. Barone-Adesi and Whaley (1987) used the quadratic approximation method to price American call and put options on an underlying asset with cost-of-carry rate b. When

b≥r (r is the domestic risky-free interest rate, and the cost-of-carry rate b is equal to the difference between domestic risky-free rate and foreign risky-free rate, i.e.,

d f

found by using the generalized Black-Scholes formula.

6. PHLX provide two kinds of currency options traded in exchange, the standardized and customized options. In the present paper, we just use the standardized contracts. 7. We use the Nelder-Meadsimplexalgorithm, which is the most widely used direct search method for nonlinear unconstrained optimization problems to solve the problem in Equation (12).

References

Arnold (1990), Mathematical Statistics, Prentice-Hall Englewood Cliffs, New Jersey. Bakshi, G., Cao, C. and Chen, Z. (1997), “Empirical Performance of Alternative Option

Pricing Models,” Journal of Finance, 52, 2003-2049.

Barone-Adesi, G., and Whaley, R. (1987), “Efficient Analytic Approximation of American Option Values,” Journal of Finance, 42, 301-320.

Bates, D. (1996), “Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in Deutsche Market Options,” Reviews of Financial Studies, 9, 69-107.

Bates, D. (2000), "Post-'87 Crash Fears in the S&P 500 Futures Option Market,"

Journal of Econometrics, 94, 181-238.

Bates, D. (2003), "Empirical Option Pricing: A Retrospection," Journal of Econometrics, 116, 387-404.

Black, F. and Scholes, M., (1973), “The Pricing of Options and Corporate Liabilities,”

Journal of Political Economy, 81, 637-659.

Bollerslev, T., R. Y. Chou, and K. F. Kroner (1992), “ARCH Modeling in Finance: A Review of the Theory and Empirical Evidence,” Journal of Econometrics, 52, 5-59.

Cox, J. C., Ingersoll, J. E, and Ross, S. A. (1985), “An Intertemporal General Equilibrium Model of Asset Prices,” Econometrica, 53, 363-384.

Day, T. E., and Lewis, C. M. (1992), “Stock Market Volatility and the Information Content of Stock Index Options,” Journal of Econometrics, 52, 267-287.

Fleming, J. (1998), “The Quality of Market Volatility Forecasts Implied by S&P 100 Index Option Prices,” Journal of Empirical Finance, 5, 317-345.

Garman, M. B., and Kohlhagen, S. W., (1983), “Foreign Currency Option Values,”

Gesser, V. and P. Poncet (1997), “Volatility Patterns: Theory and Some Evidence from the Dollar-Mark Option Market,” Journal of Derivatives, 5, 46-65.

Guo, D. (1996), “The Predictive Power of Implied Stochastic Variance From Currency Options,” Journal of Futures Markets, 16, 915-942.

Guo, D. (1998), “The Risk Premium of Volatility Implicit in Currency Options,”

Journal of Business & Economic Statistics, 16, 498-507.

Heston, S. (1993), “A Closed Form Solution for Options With Stochastic Volatility,”

Review of Financial Studies, 6, 327-343.

Hsieh, D. A. (1989), “Modeling Heteroscedasticity in Daily Foreign Exchange Rates,”

Journal of Business and Economic Statistics, 7, 307-317.

Hull, J., and White, A. (1987), “The Pricing of Options on Assets With Stochastic Volatility,” Journal of Finance, 42,281-300.

Lin, Y. N., Strong, N., and Xu, X. (2001), “Pricing FTSE 100 Index Options Under Stochastic Volatility,” Journal of Futures Markets, 21, 197-211.

Melino, A., and Turnbull, S. (1990), “Pricing Foreign Currency Options With Stochastic Volatility,” Journal of Econometrics, 45, 7-39.

Nandi, S. (1998), “How Important Is the Correlation Between Returns and Volatility in A Stochastic Volatility Model? Empirical Evidence from Pricing and Hedging in the S&P 500 Index Options Market,” Journal of Banking and Finance, 22, 589-610.

Rubinstein, M. (1994), “Implied Binomial Trees”, Journal of Finance 49, 771-818. Scott, L. O. (1987), “Option Pricing When the Variance Changes Randomly: Theory,

Estimation, and an Application,” Journal of Financial and Quantitative Analysis, 22, 419-439.

Volatility: An Analytic Approach,” Reviews of Financial Studies, 4, 727-752. Whaley, R. E. (1982), “Valuation of American Call Options on Dividend Paying Stocks:

Empirical Tests,” Journal of Financial Economics, 10, 29-58

Wiggins, J. (1987), “Option Values Under Stochastic Volatility: Theory and Empirical Estimates,” Journal of Financial Economics, 19, 351-372.

Table 1. Samples of Currency Options

British Pound Deutsche Mark Japanese Yen Swiss Franc European Styles 8,580 18,401 12,929 46,733 American Styles 32,810 81,724 26,714 51,416 Total 41,390 100,125 39,643 98,149

Note: The sample periods begin from January, 1994 through December, 1998 for the British pound and the Deutsche mark, and from January, 1994 through December, 2001 for the Japanese yen and the Swiss franc. The American-style options include mid-month and month-end options for the four currencies.

Table 2.1 Sample Properties of PHLX Currency Options (British pound)

Calendar Days to Maturity Moneyness <30 30~60 60~90 Subtotal Percentage <0.97 ₡0.1995 (0.4485) {348} ₡0.2861 (0.2202) {1281} ₡0.7989 (0.6363) {6728} {8357} Out of the money 0.97~0.99 ₡0.4281 (0.3039) {1634} ₡0.6720 (0.3203) { 757} ₡1.6782 (0.8760) {3024} {5415} 45.3 0.99~1 ₡0.7302 (0.3863) {2700} ₡1.2549 (0.4997) {3451} ₡1.9213 (1.3311) {2097} {8248} At the money 1~1.01 ₡1.2957 (0.5046) {1814} ₡2.0751 (0.4794) {1488} ₡2.1914 (1.5357) {1795} {5097} 43.9 1.01~1.03 ₡3.0442 (0.8688) {806} ₡3.4778 (0.8445) {801} ₡3.8673 (1.3288) {713} {2320} In the money >1.03 ₡6.8037 (2.3737) {287} ₡6.8455 (2.3103) {275} ₡8.0601 (3.5928) {411} {973} 10.8 Subtotal {7589} {8053} {14768} {30410} Percentage 24.9 26.5 48.6 100%

Note: The reported numbers are the average quoted prices; the standard deviations are shown in the parentheses; the numbers of options are listed in braces for each category of moneyness - maturity. The sample period is from January, 1994 to December, 1998, covering a total of 30,410 observations of options. Moneyness is defined as S/K (K/S) for call (put) options, where S denotes the spot exchange rate recorded synchronously by the PHLX and K is the exercise price.

Table 2.2 Sample Properties of PHLX Currency Options (Deutsche mark)

Calendar Days to Maturity Moneyness (S/K) <30 30~60 60~90 Subtotal Percentage <0.97 ₡0.1395 (0.1350) {1610} ₡0.2449 (0.1835) {2312} ₡0.6267 (0.4190) {6443} {10365} Out of the money 0.97~0.99 ₡0.2572 (0.1945) {8533} ₡0.4276 (0.2537) { 2295} ₡0.6924 (0.5837) {8514} {19342} 36.8 0.99~1 ₡0.3478 (0.1981) {7218} ₡0.6399 (0.2792) {6331} ₡0.9314 (0.5941) {6174} {19723} At the money 1~1.01 ₡0.6036 (0.2912) {6795} ₡1.0102 (0.3002) {3683} ₡1.0342 (0.7250) {7104} {17582} 46.2 1.01~1.03 ₡1.1829 (0.4366) {3163} ₡1.5011 (0.3893) {2215} ₡1.7062 (0.7685) {4326} {9704} In the money >1.03 ₡3.0234 (1.6238) {1025} ₡3.1684 (1.5187) {900} ₡3.7515 (1.6883) {2138} {4063} 17.0 Subtotal {28344} {17736} {34699} {80779} Percentage 35.1 22.0 42.9 100%

Note: The reported numbers are the average quoted prices; the standard deviations are shown in the parentheses; the total observed options are listed in braces for each moneyness - maturity category. The sample period is from January, 1994 to December, 1998, covering a total of 80,779 observations of options. Moneyness is defined as S/K

(K/S) for call (put) options, where S denotes the spot exchange rate recorded synchronously by the PHLX and K is the exercise price.

Table 2.3 Sample Properties of PHLX Currency Options (Japanese yen)

Calendar Days to Maturity Moneyness <30 30~60 60~90 Subtotal Percentage <0.97 ₡0.2522 (0.4065) {1576} ₡0.4548 (0.3627) {2758} ₡1.0970 (0.8323) {9934} {14268} Out of the money 0.97~0.99 ₡0.4643 (0.3047) {4392} ₡0.7548 (0.3714) { 1898} ₡1.4172 (0.9486) {7822} {14112} 45.0 0.99~1 ₡0.6874 (0.3918) {3921} ₡1.0858 (0.4876) {5214} ₡1.7203 (1.1810) {5609} {14744} At the money 1~1.01 ₡1.0774 (0.4589) {2981} ₡1.5753 (0.5631) {1912} ₡1.9878 (1.2430) {4037} {8930} 37.6 1.01~1.03 ₡1.9680 (0.6237) {1597} ₡2.4165 (0.7367) {1438} ₡2.7946 (1.2395) {3116} {6151} In the money >1.03 ₡5.4757 (3.2989) {991} ₡5.6654 (3.6888) {1033} ₡6.7650 (4.3197) {2787} {4811} 17.4 Subtotal {15458} {14253} {33305} {63016} Percentage 24.5 22.6 52.9 100%

Note: The reported numbers are the average quoted prices (×10-2); the standard deviations are shown in the parentheses; the total observed options listed in braces, for each moneyness - maturity category. The sample period is from January, 1994 through December, 2001, covering a total of 63,016 observations of options. Moneyness is defined as S/K (K/S) for call (put) options, where S denotes the spot exchange rate recorded synchronously by PHLX, and K is the exercise price.

Table 2.4 Sample Properties of PHLX Currency Options (Swiss franc) Days to Maturity Moneyness <30 30~60 60~90 Subtotal Percentage <0.97 ₡0.1504 (0.1680) {928} ₡0.3506 (0.2812) {1876} ₡0.8003 (0.5237) {6412} {9216} Out of the money 0.97~0.99 ₡0.3618 (0.2920) {3241} ₡0.6073 (0.3023) {1417} ₡1.2849 (0.6622) {6910} {11568} 40.8 0.99~1 ₡0.5378 (0.2841) {2938} ₡0.8689 (0.3277) {4696} ₡1.3979 (0.7140) {2857} {10491} At the money 1~1.01 ₡0.8370 (0.3531) {3349} ₡1.2189 (0.3695) {2152} ₡1.4705 (0.8844) {3951} {9452} 39.1 1.01~1.03 ₡1.5075 (0.5331) {1993} ₡1.8450 (0.4883) {1728} ₡2.1072 (0.9425) {3145} {6866} In the money >1.03 ₡3.6442 (1.9680) {761} ₡3.9773 (1.7972) {825} ₡4.5946 (2.1941) {1811} {3397} 20.1 Subtotal {13210} {12694} {25086} {50990} Percentage 26.0 24.9 49.1 100%

Note: The reported numbers are the average quoted prices; the standard deviations are shown in the parentheses; the total observed options listed in braces, for each moneyness - maturity category. The sample period begins from January, 1994 through December, 2001, covering a total of 63,016 observations of options. Moneyness is defined as S/K (K/S) for call (put) options, where S denotes the spot exchange rate recorded synchronously by PHLX, and K is the exercise price.

Table 3. Implied Volatility from the Garman and Kohlhagen (1983) Model

Time to Maturity

British pound Deutsche mark Japanese yen Swiss franc

Moneyness <30 30~60 60~90 <30 30~60 60~90 <30 30~60 60~90 <30 30~60 60~90 <0.97 0.10427 0.09549 0.10427 0.13419 0.11895 0.11649 0.14141 0.12828 0.10563 0.14059 0.13089 0.11497 Out of the money 0.97~0.99 0.08182 0.08725 0.08182 0.11858 0.11616 0.11441 0.12953 0.11917 0.07320 0.12270 0.12485 0.10772 0.99~1 0.07728 0.08438 0.07728 0.10231 0.11021 0.08778 0.11045 0.10897 0.04365 0.11231 0.11515 0.08581 At the money 1~1.01 0.07842 0.08499 0.07842 0.09846 0.10609 0.10159 0.10243 0.09899 0.09266 0.11091 0.10435 0.09923 1.01~1.03 0.07563 0.06242 0.07563 0.08820 0.09612 0.08423 0.10329 0.10523 0.06261 0.09200 0.10019 0.07451 In the money >1.03 0.09395 0.09420 0.09395 0.10560 0.10329 0.09427 0.12714 0.11779 0.08996 0.10768 0.09993 0.10631

Note: We calculate the implied volatility separately by inverting the Garman and Kohlhagen model from each trading day for the call and put

option contracts. The daily implied volatilities are then averaged within each category across the sample period. Moneyness is defined as S/K (K/S)

Table 4. Implied Volatility from the Heston (1993) Model

Time to Maturity

British pound Deutsche mark Japanese yen Swiss franc

Moneyness <30 30~60 60~90 <30 30~60 60~90 <30 30~60 60~90 <30 30~60 60~90 <0.97 0.07656 0.08471 0.07125 0.10251 0.11033 0.10205 0.12555 0.13125 0.13737 0.11829 0.12919 0.14396 Out of the money 0.97~0.99 0.06649 0.05505 0.07307 0.07813 0.10863 0.10233 0.11845 0.12653 0.12092 0.12919 0.11242 0.10549 0.99~1 0.07175 0.06564 0.07455 0.10154 0.10075 0.10137 0.12360 0.13605 0.12287 0.10846 0.10549 0.10279 At the money 1~1.01 0.07262 0.06365 0.07297 0.09863 0.10091 0.10741 0.13519 0.13149 0.12186 0.10442 0.11550 0.10937 1.01~1.03 0.07384 0.07105 0.07308 0.09527 0.11953 0.10438 0.12183 0.12691 0.12058 0.13016 0.13696 0.10966 In the money >1.03 0.07381 0.07194 0.08439 0.10996 0.11925 0.10645 0.11385 0.12965 0.12359 0.12838 0.12268 0.12017

Note: We calculate the implied volatility separately by inverting the Garman and Kohlhagen model from each trading day for the call and put

option contracts. The daily implied volatilities are then averaged within each category across the sample period. Moneyness is defined as S/K (K/S)

Table 5. Return and Volatility Processes Parameters Implicit in Heston Model

British pound Deutsche mark Japanese yen Swiss franc

Year κ* θ* v σ ρ Half-Life * κ θ* v σ ρ Half-Life * κ θ* v σ ρ Half-Life * κ θ* v σ ρ Half-Life 1994 10.469 0.0503 0.1491 -0.0090 24.167 8.8247 0.0110 0.1109 -0.0145 28.669 13.335 0.0092 0.1041 -0.0102 18.972 8.1639 0.0204 0.2041 -0.0071 30.990 1995 9.817 0.0202 0.1483 -0.0088 25.771 8.0808 0.0111 0.1131 -0.0146 31.309 13.304 0.0099 0.1008 -0.0100 19.016 8.3475 0.0103 0.1999 -0.0069 30.308 1996 10.528 0.0201 0.1497 -0.0099 24.031 8.3108 0.0112 0.1119 -0.0149 30.442 12.723 0.0091 0.1012 -0.0071 19.885 8.3808 0.0202 0.2090 -0.0073 30.188 1997 10.835 0.0216 0.1482 -0.0085 23.349 8.2401 0.0092 0.1031 -0.0132 30.704 12.445 0.0103 0.1027 -0.0096 20.329 9.3496 0.0184 0.2046 -0.0060 27.060 1998 10.232 0.0203 0.1518 -0.0091 24.727 7.3751 0.0104 0.1062 -0.0146 34.304 13.858 0.0101 0.1037 -0.0102 18.256 8.4215 0.0191 0.2017 -0.0068 30.042 1999 13.330 0.0096 0.1007 -0.0097 18.979 4.8961 0.0202 0.1992 -0.0091 51.673 2000 12.760 0.0101 0.1003 -0.0107 19.827 7.4487 0.0195 0.1936 -0.0068 33.965 2001 14.886 0.0088 0.0927 -0.0104 16.996 7.8047 0.0200 0.1974 -0.0071 32.416 Average 10.376 0.0265 0.1494-0.0090 24.409 8.1663 0.0106 0.1090 -0.0143 31.086 13.330 0.0096 0.1008 -0.0097 19.032 7.8516 0.0185 0.2012 -0.0071 33.330

Table 6. In-Sample Pricing Fitness

Sum of Square Pricing Errors

British pound Deutsche mark Japanese yen Swiss franc

Sample period

G&K Heston Difference % G&K Heston Difference % G&K Heston Difference % G&K Heston Difference %

1994 17.626 11.698 5.928 33.63 18.592 16.923 1.669 8.97 38.715 28.355 10.360 26.76 15.592 13.401 2.191 14.05 1995 18.086 13.239 4.847 26.99 19.056 14.776 4.280 22.46 25.608 19.865 5.743 22.43 15.624 14.953 0.671 4.29 1996 32.309 14.013 18.296 56.63 34.067 21.841 12.226 35.89 21.896 10.768 11.128 50.82 17.639 14.076 3.563 20.19 1997 26.377 18.847 7.530 28.55 35.521 24.559 10.962 30.86 15.639 5.035 10.604 67.80 29.593 13.838 15.755 53.24 1998 19.078 11.837 7.241 37.95 33.278 21.944 11.334 34.06 15.572 8.036 7.536 48.39 16.403 12.784 3.619 22.06 1999 16.083 11.184 4.899 30.46 19.007 11.641 7.366 38.75 2000 30.732 15.066 15.666 50.97 17.449 15.636 1.813 10.39 2001 15.702 7.774 7.928 50.49 13.999 13.625 0.374 2.67 Total 113.48 69.63 113.48 69.63 140.51 100.04 43.85 38.64 179.94 106.08 73.86 41.05 125.30 39.96 85.34 68.11 Average 22.695 13.926 28.103 20.009 22.493 13.260 15.663 4.994

Note: For each year in the sample, the structural parameters of models of Garman and Kohlhagen (G&K) as well as Heston (1987) are estimated by minimizing the sum of squared errors between the market price and theoretical price. The daily average sum of squared errors is calculated by collecting each option’s squared pricing errors and then is divided by the 365 calendar days for each model.

Figure 1. Implied Volatility of Garman-Kohlhagen (G&K) Model Across Moneyness and Maturity

British pound Deutsche mark

Figure 2. Implied Volatility Pattern: Garman-Kohlhagen (G&K) Model versus Heston Model Across Maturity

Short-term Maturity < 30 days Middle-term Maturity 30 ~ 60 days British pound Deutsche mark

Figure 2. (continued) Implied Volatility Pattern: Garman-Kohlhagen (G&K) Model versus Heston Model Across Maturity

Short-term Maturity < 30 days Middle-term Maturity 30 ~ 60 days Japanese yen Swiss franc