Price Deviations of S&P 500 Index Options from the Black-Scholes Formula Follow a Simple Pattern
55 Pages Posted: 21 Mar 2005 Last revised: 6 Oct 2009
Date Written: November 21, 2004
It is well known that the actual prices of options deviate from values computed using the Black-Scholes formula or the binomial model with the same volatility for different strikes. For the S&P 500 index options, we find that these deviations from the Black-Scholes formula follow a simple pattern. Loosely, the slope and curvature of the differences between option prices and Black-Scholes values are described by a simple function of at-the-money-forward total volatility. Similarly, the slope and curvature of the volatility skew are described by a simple function of at-the-money-forward total volatility. This implies that the term structure of at-the-money-forward volatilities is sufficient to determine the entire volatility surface. Finally, we find that the implied risk-neutral probability density is bimodal. This finding has interesting implications for models of stochastic volatility.
Keywords: Implied volatility, volatility skew, index options
JEL Classification: G13
Suggested Citation: Suggested Citation