Good deal bounds induced by shortfall risk

TL;DR

This paper develops good deal pricing bounds for contingent claims based on shortfall risk, using convex risk measures and minimal penalty functions, with explicit representations and optimal strategies in simple cases.

Contribution

It introduces a novel approach to good deal bounds induced by shortfall risk, with mild assumptions and explicit representations involving convex risk measures.

Findings

Bounds expressed by convex risk measures on Orlicz hearts

Representation with minimal penalty function

Optimal strategies for simple cases

Abstract

We shall provide in this paper good deal pricing bounds for contingent claims induced by the shortfall risk with some loss function. Assumptions we impose on loss functions and contingent claims are very mild. We prove that the upper and lower bounds of good deal pricing bounds are expressed by convex risk measures on Orlicz hearts. In addition, we obtain its representation with the minimal penalty function. Moreover, we give a representation, for two simple cases, of good deal bounds and calculate the optimal strategies when a claim is traded at the upper or lower bounds of its good deal pricing bound.