Skewness and Kurtosis Adjusted Black-Scholes Model: A Note on Hedging Performance

Abstract This article investigates the delta hedging performance of the skewness and kurtosis adjusted Black-Scholes model of Corrado and Su (1996) and Brown and Robinson (2002). The empirical tests in the FTSE 100 index option market show that the more sophisticated skewness and kurtosis adjusted model performs worse than the simplistic Black-Scholes model in terms of delta hedging. The hedging errors produced by the skewness and kurtosis adjusted model are consistently larger than the Black-Scholes hedging errors, regardless of the moneyness and maturity of the options and the length of the hedging horizon. Keywords: Option Hedging, Delta Hedging, Skewness, Kurtosis JEL classification: G10, G13 1. INTRODUCTION The Black-Scholes (1973) model assumes the asset price dynamics to be described by a geometric Brownian motion, and thus, implies constant volatility and Gaussian log-returns. However, both the assumptions of constant volatility and Gaussian returns are obviously violated in financial markets. It has been recognized for a long time that asset return distributions tend to be leptokurtic. More recently, a vast literature has documented volatility to be time-varying. In addition, there seems to be a tendency for changes in stock prices to be negatively correlated with changes in volatility, due to which stock return distributions tend to be negatively skewed. Given the possible misspecifications of the Black-Scholes model, a substantial literature has devoted to the development of option pricing models which account for the observed empirical violations, such as the volatility smile. The Black-Scholes constant volatility assumption is relaxed in stochastic volatility models such as Hull and White (1987) and Heston (1993a), in deterministic volatility models by Dupire (1994), Derman and Kani (1994), and Rubinstein (1994), and in ARCH models of Engle and Mustafa (1992), Duan (1995), and Heston and Nandi (2000). The assumption of lognormal terminal price distribution is relaxed, e.g., in skewness and kurtosis adjusted models of Jarrow and Rudd (1982), and Corrado and Su (1996), log-gamma model of Heston (1993b), lognormal mixture model by Melick and Thomas (1997), and hyberbolic model of Eberlein et al. (1998). This article focuses on the delta hedging performance of the skewness and kurtosis adjusted Black-Scholes model of Corrado and Su (1996) and Brown and Robinson (2002). Despite the intensive empirical research on different option pricing models, surprisingly little is known about the hedging performance beyond the Black-Scholes model. Previous studies on hedging performance have focused on different time-varying volatility option pricing models. The hedging performance of stochastic volatility option pricing models is investigated, e.g., in Bakshi et al. (1997, 2000), Nandi (1998), and Lim and Guo (2000) whereas Dumas et al. (1998), Engle and Rosenberg (2000), Coleman et al. (2001), and Lim and Zhi (2002) examine option hedging under deterministic volatility models. A bit surprisingly, previous studies indicate that although time-varying volatility option pricing models clearly outperform the Black-Scholes model in terms of pricing [see e.g., Bakshi et al.