The Black-Scholes European Call Option Formula Corrected Using the Gram-Charlier Expansion

The first systematic empirical study of the mispricing of options by the Black–Scholes model appears to be that of F. Black [1] (of Black–Scholes fame), who observed that the Black–Scholes formula overprices deep in-the-money options and underprices deep out-of-the-money options. A later study by Macbeth and Merville [2] reached the exact opposite conclusion. The reason for this was explained in [3]. The authors used the Edgeworth expansion (see [5]), which enabled them to obtain a formula for the value of a call option that accounts for the possibility of nonzero skewness and excess kurtosis in the distribution of stock returns. They showed that the sign of skewness determines the overpricing-underpricing behavior of Black–Scholes for in-the-money and out-of-the-money options, while the kurtosis has the dominant effect for at-the-money options. This Demonstration is based on the formula for the call option obtained in [4]. The difference between this work and [3] is that in [4] the Gram–Charlier expansion is applied to the distribution of log-returns, while in [3] the Edgeworth expansion is applied to the distribution of stock prices. The conclusions of both studies are thus essentially equivalent.

It should be noted that the "probability density function" obtained by truncating an infinite series is not a true PDF and can assume negative values. This can cause the formula for option price to return negative values for deep out-of-the-money options.

[1] F. Black, "Fact and Fantasy in the Use of Options," Financial Analysts Journal, 1975 pp. 55–72.

[2] J. Macbeth and L. Merville, "An Empirical Examination of the Black–Scholes Call Option Pricing Model," Journal of Finance, 34, pp. 1173–86.

[3] R. Jarrow and A. Rudd, "Approximate Option Valuation for Arbitrary Stochastic Processes," Journal of Financial Economics, 10, 1982 pp. 347–69.

[4] C. J. Corrado and T. Su, "Skewness and Kurtosis in S&P500 Index Returns Implied by Option Prices," Journal of Financial Research, 19(2), 1996 pp. 175–192.

[5] M. Kendall and A. Stuart, The Advanced Theory of Statistics, Vol. 1: Distribution Theory, 4th ed., New York: Macmillan, 1977.