Recovering Probability Distributions from Option Prices

TLDR

The study derives risk‑neutral probability distributions for S&P 500 European options and introduces a fast, smoothness‑maximizing optimization technique. Nonparametric methods minimize an objective function while enforcing consistency with observed option and underlying prices, and the new technique maximizes distribution smoothness. Different optimization specifications yield similar implied distributions, yet the risk‑neutral probability of a three‑ to four‑standard‑deviation index decline is roughly ten to one hundred times higher than under lognormal assumptions.

Abstract

ABSTRACT This article derives underlying asset risk‐neutral probability distributions of European options on the S&P 500 index. Nonparametric methods are used to choose probabilities that 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 percent (−46 percent) over a year) is about 10 (100) times more likely than under the assumption of lognormality.