The inverse problem of option pricing

Abstract

Valuation of options and other financial derivatives critically depends on the underlying stochastic process specified for a particular market. An inverse problem of option pricing is to determine the nature of this stochastic process, namely, the distribution of expected asset returns implied by current market prices of options with different strikes. We give a rigorous mathematical formulation of this inverse problem, establish uniqueness, and suggest an efficient numerical solution. We apply the method to the S&P 500 Index and conclude that the index is negatively skewed with a higher probability of the sudden decline of the US stock market.