A recursive algorithm for multivariate risk measures and a set-valued Bellman's principle

TL;DR

This paper introduces a recursive, set-valued Bellman's principle for calculating multivariate risk measures in discrete-time market models with transaction costs, illiquidity, and trading constraints, using backward recursion and optimization techniques.

Contribution

It develops a novel recursive algorithm for multivariate risk measures that incorporates set optimization and extends Bellman's principle to a set-valued context.

Findings

The method computes capital requirements recursively in complex market models.

Conditions are provided for reducing the problem to linear or convex vector optimization.

Numerical examples demonstrate applications to superhedging, entropic risk, and average value at risk.

Abstract

A method for calculating multi-portfolio time consistent multivariate risk measures in discrete time is presented. Market models for $d$ assets with transaction costs or illiquidity and possible trading constraints are considered on a finite probability space. The set of capital requirements at each time and state is calculated recursively backwards in time along the event tree. We motivate why the proposed procedure can be seen as a set-valued Bellman's principle, that might be of independent interest within the growing field of set optimization. We give conditions under which the backwards calculation of the sets reduces to solving a sequence of linear, respectively convex vector optimization problems. Numerical examples are given and include superhedging under illiquidity, the set-valued entropic risk measure, and the multi-portfolio time consistent version of the relaxed worst case risk measure and of the set-valued average value at risk.