Hedging of game options under model uncertainty in discrete time

Yan Dolinsky

doi:10.1214/ecp.v19-2714

Abstract

We introduce a setup of model uncertaintyin discrete time. In this setup wederive dual expressions for the super-replication prices of game options with upper semicontinuous payoffs. We show that the super-replication price is equal to the supremum over a special (non dominated) set of martingale measures, of the corresponding Dynkin games values. This type of results is also new for American options.

Highlights

We introduce a setup of model uncertainty in discrete time
We show that the super–replication price is equal to the supremum over a special set of martingale measures, of the corresponding Dynkin games values
A game contingent claim (GCC) or game option, which was introduced in [11], is defined as a contract between the seller and the buyer of the option such that both have the right to exercise it at any time up to a maturity date T

Summary

Introduction

A game contingent claim (GCC) or game option, which was introduced in [11], is defined as a contract between the seller and the buyer of the option such that both have the right to exercise it at any time up to a maturity date (horizon) T. For the case of volatility uncertainty, there are only few papers which deal with American options and game options (see [14] and [15]). Game options under model uncertainty [19] and [20]), and super–replication under volatility uncertainty in continuous time models. This proof is quite elementary and does not use advanced tools. This extension is technically involved and requires the establishment of some stability results for Dynkin games under weak convergence.

Preliminaries and main results

Proof of the main result

Extension for upper semicontinuous payoffs

Findings

Auxiliary Lemma