Distributional Transforms, Probability Distortions, and Their Applications

33 Pages Posted: 15 Jul 2019 Last revised: 13 Apr 2020

See all articles by Peng Liu

Peng Liu

University of Waterloo - Department of Statistics and Actuarial Science

Alexander Schied

University of Waterloo

Ruodu Wang

University of Waterloo - Department of Statistics and Actuarial Science

Date Written: July 13, 2019

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

Suggested Citation

Liu, Peng and Schied, Alexander and Wang, Ruodu, Distributional Transforms, Probability Distortions, and Their Applications (July 13, 2019). Available at SSRN: https://ssrn.com/abstract=3419388 or http://dx.doi.org/10.2139/ssrn.3419388

Peng Liu (Contact Author)

University of Waterloo - Department of Statistics and Actuarial Science ( email )

200 University Avenue West
Waterloo, Ontario N2L 3G1
Canada

Alexander Schied

University of Waterloo ( email )

200 University Ave W
Waterloo, Ontario
Canada

Ruodu Wang

University of Waterloo - Department of Statistics and Actuarial Science ( email )

Waterloo, Ontario N2L 3G1
Canada

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