Distributional Transforms, Probability Distortions, and Their Applications

Peng Liu (), Alexander Schied () and Ruodu Wang ()
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Peng Liu: Department of Mathematical Sciences, University of Essex, Colchester, CO4 3SQ, United Kingdom
Alexander Schied: Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
Ruodu Wang: Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada

Mathematics of Operations Research, 2021, vol. 46, issue 4, 1490-1512

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 a change of measures with an example in option pricing.

Keywords: Primary: 60E05; secondary: 60E15; Primary: Mathematics; secondary: distributions/probability; finance; distributional transforms; probability distortions; risk measures; option pricing; sensitivity analysis; change of measures; value-at-risk; expected shortfall; composition groups (search for similar items in EconPapers)
Date: 2021
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Citations: View citations in EconPapers (1)

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Persistent link: https://EconPapers.repec.org/RePEc:inm:ormoor:v:46:y:2021:i:4:p:1490-1512

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