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

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

Black-Scholes

Corrected Black-Scholes

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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).

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 100%, 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).

S-Kexp(-rt)

Kexp(-rt)

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