Risk measures beyond frictionless markets

TL;DR

This paper develops a comprehensive framework for risk measures in markets with transaction costs and portfolio constraints, extending classical theories by removing translation invariance and establishing dual representations.

Contribution

It introduces a general theory of risk measures that incorporates market frictions and constraints, providing new properties and dual representations beyond classical models.

Findings

Characterizes properties like star shapedness, convexity, and subadditivity in the new setting.

Establishes dual representations for convex and quasiconvex risk measures.

Shows how market frictions influence risk measurement and duality theory.

Abstract

We develop a general theory of risk measures that determines the optimal amount of capital to raise and invest in a portfolio of reference traded securities in order to meet a pre-specified regulatory requirement. The distinguishing feature of our approach is that we embed portfolio constraints and transaction costs into the securities market. As a consequence, we have to dispense with the property of translation invariance, which plays a key role in the classical theory. We provide a comprehensive analysis of relevant properties such as star shapedness, positive homogeneity, convexity, quasiconvexity, subadditivity, and lower semicontinuity. In addition, we establish dual representations for convex and quasiconvex risk measures. In the convex case, the absence of a special kind of arbitrage opportunities allows to obtain dual representations in terms of pricing rules that respect market bid-ask spreads and assign a strictly positive price to each nonzero position in the regulator's acceptance set.