Pricing, Hedging and Optimally Designing Derivatives Via Minimization of Risk Measures

TL;DR

This paper explores pricing and hedging of non-tradable claims through risk measure minimization, combining utility maximization, risk transfer, and dynamic approaches with BSDEs for tractability.

Contribution

It introduces a unified framework linking utility-based pricing, risk transfer, and dynamic risk measures via BSDEs, emphasizing practical tractability.

Findings

Dynamic risk measures relate to BSDEs.

Static and dynamic approaches are integrated.

Framework addresses incomplete markets.

Abstract

The question of pricing and hedging a given contingent claim has a unique solution in a complete market framework. When some incompleteness is introduced, the problem becomes however more difficult. Several approaches have been adopted in the literature to provide a satisfactory answer to this problem, for a particular choice criterion. In this paper, in order to price and hedge a non-tradable contingent claim, we first start with a (standard) utility maximization problem and end up with an equivalent risk measure minimization. This hedging problem can be seen as a particular case of a more general situation of risk transfer between different agents, one of them consisting of the financial market. In order to provide constructive answers to this general optimal risk transfer problem, both static and dynamic approaches are considered. When considering a dynamic framework, our main purpose is to find a trade-off between static and very abstract risk measures as we are more interested in tractability issues and interpretations of the dynamic risk measures we obtain rather than the ultimate general results. Therefore, after introducing a general axiomatic approach to dynamic risk measures, we relate the dynamic version of convex risk measures to BSDEs.