Pricing Currency Options Under Stochastic Volatility

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AbstractThis paper investigates the relative pricing performance between constant volatility andstochastic volatility pricing models, based on a comprehensive sample of options onfour currencies, including the British pound, Deutsche mark, Japanese yen and Swissfranc, 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 Garmanand 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 currencymarket is faster than that in the stock market. It may be attributed to the largermomentum effect in the stock. We also find that the stock market exhibits largerimplied skewness than the currency market. This may be due to stronger ‘trend effect’in the stock market, especially involved in the bear market.Keywords: <strong>Stochastic</strong> volatility; <strong>Currency</strong> options; Implied risk premium of volatility.JEL Codes: F31, G13.2
1. IntroductionIt has been shown that the Black-Scholes (1973) option pricing model is subject tosystematic biases originated from the violation of the normal distribution assumptionon 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 skewnesscauses the out-of-the-money option price bias, whereas the implicit leptokurtosisincreases the prices of deeply in-the-money and out-of-money options and lower pricesof near-the-money options.To enhance the pricing accuracy, the stochastic volatility (SV) option pricingmodels, such as Hull and While (1987), Scott (1987), Wiggins (1987), Melino andTurnbull (1990), and Heston (1993), incorporate the leptokurtosis or excess kurtosis ofthe underlying asset returns by allowing the volatility process to behave randomly. 1Black and Scholes (1973) specified that the source of volatility risk comes fromstochastic returns of the underlying, however, the SV option pricing models allow thepricing risk to emerge from both the stochastic processes of price and the volatility. Tomodel the stochastic process of volatility in the SV option pricing model, one has tospecify the market price of volatility risk, the volatility of variance, and the correlationbetween underlying price (return) and its volatility. Assuming zero correlation of priceand volatility, Hull and White (1987) used the Taylor expansion technique andproposed 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 themarket 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 the3
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Page 5 and 6: varying, and of sufficient magnitud
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Page 11 and 12: month-end option, which expires at
Page 13 and 14: percentage and 43.9 percent of the
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Page 21 and 22: Footnotes1. Although the SV option
Page 23 and 24: ReferencesArnold (1990), Mathematic
Page 25 and 26: Volatility: An Analytic Approach,
Page 27 and 28: Table 2.1 Sample Properties of PHLX
Page 29 and 30: Table 2.3 Sample Properties of PHLX
Page 31 and 32: Table 3. Implied Volatility from th
Page 33 and 34: Table 5. Return and Volatility Proc
Page 35 and 36: Figure 1. Implied Volatility of Gar
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Pricing Currency Options Under Stochastic Volatility

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