Skewness and Kurtosis in S&p 500 Index Returns Implied by Option Prices

The Black‐Scholes (1973) model frequently misprices deep‐in‐the‐money and deep‐out‐of‐the‐money options. Practitioners popularly refer to these strike price biases as volatility smiles. In this paper we examine a method to extend the Black‐Scholes model to account for biases induced by nonnormal skewness and kurtosis in stock return distributions. The method adapts a Gram‐Charlier series expansion of the normal density function to provide skewness and kurtosis adjustment terms for the Black‐Scholes formula. Using this method, we estimate option‐implied coefficients of skewness and kurtosis in S&P 500 stock index returns. We find significant nonnormal skewness and kurtosis implied by option prices.