Estimating Probability Distributions of Future Asset Prices: Empirical Transformations from Option-Implied Risk-Neutral to Real-World Density Functions

Abstract

The prices of derivatives contracts can be used to estimate ‘risk-neutral’ probability density functions that give an indication of the weight investors place on different future prices of their underlying assets, were they risk-neutral. In the likely case that investors are risk-averse, this leads to differences between the risk-neutral probability density and the actual distribution of prices. But if this difference displays a systematic pattern over time, it may be exploited to transform the risk-neutral density into a ‘real-world’ density that better reflect agents’ actual expectations. This work offers a methodology for performing this transformation. The resulting real-world densities may better represent market participants’ views of future prices, and so offer an enhanced means of quantifying the uncertainty around financial variables. Comparison with their risk-neutral equivalents may also reveal new and useful information as to how attitudes towards risk are affecting pricing.

Keywords: Asset prices, derivatives, expectations, options, option-implied density, risk premia, probability density forecasting, probability measure

JEL Classification: G10, G12, G13