Recovering an Asset's Implied PDF from Option Prices: An Application to Crude Oil during the Gulf Crisis

Abstract

We develop a general method for estimating the implied, martingale equivalent, probability density function (PDF) for futures prices from American options prices. The early exercise feature of American options precludes expressing the price of the option in terms of the PDF. There exist tight bounds for the price of American options in terms of the PDF. We demonstrate how these bounds, together with observed option prices, can be used to estimate the parameters of the PDF. We estimate the distribution for crude oil during the Persian Gulf crisis and find the distribution differs significantly from that recovered using standard techniques.