Option Prices and the Underlying Asset's Return Distribution

Abstract

This work examines the relation between option prices and the true, as opposed to risk-neutral, distribution of the underlying asset. If the underlying asset follows a diffusion with an instantaneous expected return at least as large as the instantaneous risk-free rate, observed option prices can be used to place bounds on the moments of the true distribution. An illustration of the paper's results is provided by the analysis of the information concerning the mean and standard deviation of market returns contained in the prices of S&P 100 Index Options. ALTHOUGH IT SEEMS NATURAL that the prices of claims to various parts of the underlying asset's distribution should contain information about the shape of that distribution, Cox and Ross (1976) established that an option's price equals its expected payoff discounted at the risk-free rate where the expectation is taken over the 'risk-neutral', rather than the true, distribution of the underlying asset. Linking the risk-neutral distribution implicit in option prices to the true distribution remains a comparative mystery. A necessary condition for the risk-neutral pricing methodology to be applicable is that the true and the risk-neutral distribution share a common support. The only information about the true distribution that can be obtained from observed option prices alone is information about that support. This paper demonstrates though that observed option prices, when used in conjunction with simple assume,d restrictions on the true distribution, do contain information about the noncentral moments of the true distribution not directly implied by those assumed restrictions alone.' The intuition for the result that option prices can be useful in placing bounds on the moments of the true distribution is straightforward. First, we generalize the results in Lo (1987) to show how the expected payoff to a call can be bounded above in terms of any chosen set of the noncentral moments of the return on the underlying asset. Second, we establish restrictions on the