Price Deviations of S&P 500 Index Options from the Black-Scholes Formula Follow a Simple Pattern
Neil D. Pearson
University of Illinois at Urbana-Champaign - Department of Finance
November 21, 2004
AFA 2006 Boston Meetings Paper
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: G13working papers series
Date posted: March 21, 2005 ; Last revised: October 6, 2009
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