Hedging of game options with the presence of transaction costs

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 [Statist. Decisions 27 (2009) 357–369] and [Ann. Appl. Probab. 18 (2008) 491–520]) 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 [Weak convergence of stochastic processes for processes viewed in the strasbourg manner (1981) Preprint]. The second theory is essential in dealing with hedging which involves stopping times, like in the case of game options.