Hedging of Game Options With the Presence of Transaction Costs

TL;DR

This paper extends super-replication pricing to game options under transaction costs in a multidimensional continuous-time model, introducing a game variant of the concave envelope and utilizing advanced convergence and pricing theories.

Contribution

It introduces a novel super-replication price characterization for game options with transaction costs, extending previous European option results to more complex game options.

Findings

Super-replication price equals the cheapest trivial super-replication strategy.

Super-replication price for game options is given by a game variant of the concave envelope.

The approach combines consistent price systems and extended weak convergence theories.

Abstract

We study the problem of super-replication for game options under proportional transaction costs. We consider a multidimensional continuous time model, in which the discounted stock price process satisfies the conditional full support property. We show that the super-replication price is the cheapest cost of a trivial super-replication strategy. This result is an extension of previous papers (see [3] and [7]) which considered only European options. In these papers the authors showed that with the presence of proportional transaction costs the super--replication price of a European option is given in terms of the concave envelope of the payoff function. In the present work we prove that for game options the super-replication price is given by a game variant analog of the standard concave envelope term. The treatment of game options is more complicated and requires additional tools. We combine the theory of consistent price systems together with the theory of extended weak convergence which was developed in [1]. The second theory is essential in dealing with hedging which involves stopping times, like in the case of game options.