Three-Benchmarked Risk Minimization for Jump Diffusion Markets

Abstract

The paper discusses the problem of hedging not perfectly replicable contingent claims by using a benchmark, the numerraire portfolio, as reference unit. The proposed concept of benchmarked risk minimization generalizes classical risk minimization, pioneered by Follmer, Sondermann and Schweizer. The latter relies on a quadratic criterion, requesting the square integrability of contingent claims and the existence of an equivalent risk neutral probability measure. The proposed concept of benchmarked risk minimization avoids these restrictive assumptions. It employs the real world probability measure as pricing measure and identifies the minimal possible price for the hedgable part of a contingent claim. Furthermore, the resulting benchmarked profit and loss is only driven by nontraded uncertainty and forms a martingale that starts at zero. Benchmarked profit and losses, when pooled and sufficiently independent, become in total negligible. This property is highly desirable from a risk management point of view. It is making a symptotically benchmarked risk minimization the least expensive method for pricing and hedging for an increasing number of not fully replicable benchmarked contingent claims.