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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

Minqiang Li

Bloomberg LP

Neil D. Pearson

University of Illinois at Urbana-Champaign - Department of Finance

Date Written: November 21, 2004

Abstract

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

Li, Minqiang and Pearson, Neil D., Price Deviations of S&P 500 Index Options from the Black-Scholes Formula Follow a Simple Pattern (November 21, 2004). AFA 2006 Boston Meetings Paper. Available at SSRN: https://ssrn.com/abstract=686803 or http://dx.doi.org/10.2139/ssrn.686803

Minqiang Li

Bloomberg LP ( email )

731 Lexington Avenue
New York, NY 10022
United States

Neil Pearson (Contact Author)

University of Illinois at Urbana-Champaign - Department of Finance ( email )

1206 South Sixth Street
Champaign, IL 61820
United States
217-244-0490 (Phone)
217-244-9867 (Fax)

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