Distributional Transforms, Probability Distortions, and Their Applications

Abstract

In this paper we provide a general mathematical framework for distributional transforms, which allows for many examples that are used extensively in the literature of finance, economics and optimization. We put a special focus on the class of probability distortions, which is a fundamental tool in decision theory. As our main results, we characterize distributional transforms satisfying various properties and this includes an equivalent set of conditions which forces a distributional transform to be a probability distortion. As the first application, we construct new risk measures using distributional transforms. Sufficient and necessary conditions are given to ensure the convexity or coherence of the generated risk measures. In the second application, we introduce a new method for sensitivity analysis of risk measures based on composition groups of probability distortions. Finally, we construct probability distortions describing change of measures with an example in option pricing.

Keywords: distributional transforms, probability distortions, risk measures, option pricing, sensitivity analysis, change of measures, Value-at-Risk, Expected Shortfall, composition of groups