Binomial approximations of shortfall risk for game options

Abstract

We show that the shortfall risk of binomial approximations of game (Israeli) options converges to the shortfall risk in the corresponding Black–Scholes market considering Lipschitz continuous path dependent payoffs for both discrete and continuous time cases. This results are new also for usual American style options. The paper continues and extends the study of [6] where estimates for binomial approximations of prices of game options were obtained. Our arguments rely, in particular, on strong invariance principle type approximations via the Skorokhod embedding, estimates from [6] and the existence of optimal shortfall hedging in the discrete time established in [2]. 1. Introduction. This paper deals with game (Israeli) options introduced in [5] sold in a standard securities market consisting of a nonrandom component bt representing the value of a savings account at time t with an interest rate r and of a random component St representing the stock price at time t. As usual, we view St, t > 0 as a stochastic process on a probability space (, F, P) and we assume that it generates a right continuous filtration {Ft}. The setup includes also two continuous stochastic payoff processes Xt ≥ Yt ≥ 0 adapted to the above filtration. Recall, that game contingent claim (GCC) or game option 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. If the buyer exercises the contract at time t then he receives the payment Yt, but if the seller exercises (cancels) the contract before the buyer then the latter receives Xt. The dif