Using Illiquid Option Prices to Recover Probability Distributions

Abstract

This paper describes a method for recovering the risk neutral market’s perceived probability distribution (RND) of European options on the FTSE100 Index on an hourly time basis. A nonparametric procedure is used to choose probabilities that minimise an objective function subject to requiring that the obtained probabilities comply with observed option prices. The procedure is based on the idea that probability distributions of asset returns can be expressed as a mixture of lognormal variables. The use of a lognormal mixture provides a natural and robust way to model distributions with fat tails, a key feature of financial returns. The optimisation technique for estimating probability distributions incorporates a “smoothness” and a “variability” factor in the objective function to account for situations where little smoothness and high variability in the posterior distributions are plausible due to problems in the data, such as illiquidity. With this method we are able to resolve some of the numerical difficulties presented by earlier procedures, plot consistent pictures in a timely fashion of the implied distribution and measure the first four implied moments of the future probability distribution implied by the FTSE 100 option market.