Convex hedging of non-superreplicable claims in discrete-time market models

Tomasz J Tkalinski

doi:10.1007/s00186-014-0461-1

Abstract

All of the papers written so far deal with efficient hedging of contingent claims for which superhedging exists. The goal of this paper is to investigate the convex hedging of contingent claims for which superhedging does not exist. Without superhedging assumption it is still possible to prove the existence of a solution, but one cannot obtain structure of the solution using techniques known so far. Therefore, we develop a new approximative approach to deduce structure of the solution in case of non-superreplicable claims.

Highlights

We consider a hedging problem of nonnegative European contingent claims in discrete time arbitrage-free financial market models
In complete markets every contingent claim is attainable, i.e. it can be replicated by a self-financing trading strategy and its price is uniquely determined
All of the papers mentioned above deal with the efficient hedging of nonnegative contingent claims for which superhedging exists, which is equivalent to assumption that supremum of expectations of the claim with respect to all martingale measures is finite

Summary

Introduction

We consider a hedging problem of nonnegative European contingent claims in discrete time arbitrage-free financial market models. All of the papers mentioned above deal with the efficient hedging of nonnegative contingent claims for which superhedging exists, which is equivalent to assumption that supremum of expectations of the claim with respect to all martingale measures is finite. We show that this assumption may be violated even for a standard plain vanilla call with payoff based on a basket of non-traded securities in a one-period model. 3 we provide examples which motivate consideration of the convex hedging problem of contingent claims for which superhedging does not exist This problem is solved, where a new approximative technique is presented

Formulation of the problem

Motivating examples

Convex hedging of non-superreplicable claims

Approximating problems and their solution