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
Abstract
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).