Binomial approximations of shortfall risk for game options

TL;DR

This paper proves that the shortfall risk of binomial approximations for game options converges to the continuous market risk, extending results to American options and using advanced probabilistic techniques.

Contribution

It establishes convergence of shortfall risk in binomial models to the Black--Scholes market for path-dependent payoffs, including American options, using Skorokhod embedding.

Findings

Convergence of shortfall risk in binomial approximations to continuous models.

Extension of results to American style options.

Application of strong invariance principles and existing hedging estimates.

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. These results are new also for usual American style options. The paper continues and extends the study of Kifer [Ann. Appl. Probab. 16 (2006) 984--1033] 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 Kifer [Ann. Appl. Probab. 16 (2006) 984--1033] and the existence of optimal shortfall hedging in the discrete time established by Dolinsky and Kifer [Stochastics 79 (2007) 169--195].