Reconstruction of Volatility: Pricing Index Options by the Steepest Descent Approximation

Marco Avellaneda, Dash Boyer-Olson, Jerome Busca and Peter Friz
Working Paper, April 2002, published in RISK, October 2002


We propose a  formula for calculating the implied  volatility of index options based on the volatility skews of the  options on the underlying stocks and  on a given correlation  matrix for the basket.  The derivation uses the steepest-descent approximation for the multivariate probability distribution function of forward prices. A simple financial justification is provided. We apply the formula to compute the implied volatilities of liquidly-traded options on exchanges-traded funds (ETF) across different strikes. Our theoretical results were found to be in good agreement with contemporaneous quotes on the Chicago Board of Options Exchange (CBOE) and the American Stock Exchange (AMEX).