Recovering Probability Distributions from Option Prices

TLDR: In this paper, the authors derived underlying asset risk-neutral probability distributions of European options on the S&P 500 index and used nonparametric methods to choose probabilities which minimize an objective function subject to requiring that the probabilities are consistent with observed option and underlying asset prices.

Abstract: This paper derives underlying asset risk-neutral probability distributions of European options on the S&P 500 index. Nonparametric methods are used to choose probabilities which minimize an objective function subject to requiring that the probabilities are consistent with observed option and underlying asset prices. Alternative optimization specifications produce approximately the same implied distributions. A new and fast optimization technique for estimating probability distributions based on maximizing the smoothness of the resulting distribution is proposed. Since the crash, the risk-neutral probability of a three (four) standard deviation decline in the index (about-36% (-46%) over a year) is about 10 (100) times more likely than under the assumption of lognormality.