The Black-Scholes European Call Option Formula Corrected Using the Gram-Charlier Expansion
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It has long been well-known that the Black–Scholes model frequently misprices deep in-the-money and out-of-the-money options. A large part of the problem seems to lie in the normality assumptions of the Black–Scholes model. Empirical evidence shows that actual stock prices and stock returns have a distribution that is usually skewed and has a larger kurtosis than the log-normal distribution. There are a number of approaches that attempt to correct this problem. Here we illustrate an approach based on using the Edgeworth (or Gram–Chalier) series, which allows one to expand a given probability density function in terms of the probability density function of the normal distribution and cumulants of the given PDF. Using a finite truncation of this series instead of the original PDF we obtain a formula for option prices with correction terms for nonzero values of skewness and excess kurtosis (kurtosis -3).[more]
The plot shows the Black–Scholes and the corrected Black–Scholes values of the European call option on a stock with initial price of 100 that pays no dividend against the "percentage moneyness" of the option defined as , where is the initial price of the stock, is the strike price, is the time to expiry, and is the interest rate (which in this Demonstration is taken to be 0).[less]
The first systematic empirical study of the mispricing of options by the Black–Scholes model appears to be that of F. Black  (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  reached the exact opposite conclusion. The reason for this was explained in . The authors used the Edgeworth expansion (see ), 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 . The difference between this work and  is that in  the Gram–Charlier expansion is applied to the distribution of log-returns, while in  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.
 F. Black, "Fact and Fantasy in the Use of Options," Financial Analysts Journal, 1975 pp. 55–72.
 J. Macbeth and L. Merville, "An Empirical Examination of the Black–Scholes Call Option Pricing Model," Journal of Finance, 34, pp. 1173–86.
 R. Jarrow and A. Rudd, "Approximate Option Valuation for Arbitrary Stochastic Processes," Journal of Financial Economics, 10, 1982 pp. 347–69.
 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.
 M. Kendall and A. Stuart, The Advanced Theory of Statistics, Vol. 1: Distribution Theory, 4th ed., New York: Macmillan, 1977.