Binomial Approximations for Barrier Options of Israeli Style

Abstract

We show that prices and shortfall risks of game (Israeli) barrier options in a sequence of binomial approximations of the Black-Scholes (BS) market converge to the corresponding quantities for similar game barrier options in the BS market with path dependent payoffs and the speed of convergence is estimated, as well. The results are new also for usual American style options and they are interesting from the computational point of view, as well, since in binomial markets these quantities can be obtained via dynamical programming algorithms. The paper continues the study of [11] and [7] but requires substantial additional arguments in view of pecularities of barrier options which, in particular, destroy the regularity of payoffs needed in the above papers.

Limit Theorems for Partial Hedging Under Transaction Costs

“…Similar results were obtained in Dolinsky and Kifer (, ) for game options without the presence of transaction costs. The proof of the results there relied heavily on the completeness of the markets, which is no longer the case with the presence of transaction costs.…”

“…For any λ > 0, 0 < μ < 1 and there exists a portfolio strategy such that Next, we introduce the binomial models. Similar binomial models were used to approximate option prices and shortfall risks in the complete setup (see Kifer ; Dolinsky and Kifer , ), i.e., in the absence of transaction costs. For any n consider the n ‐step binomial market, which consists of a savings account B ( n ) ( t ) given by and of a risky stock S ( n ) given by the formulas S ( n ) ( t ) = S 0 for t ∈ [0, T / n ) and where ξ 1 , ξ 2 , … are i.i.d.…”

Dynkin's Games and Israeli Options

“…In [33] similar approximation results as above were extended to barrier game options. Namely, [33] deals with double knock-out barrier option with two constant barriers L, R such that 0 ≤ L < S 0 < R ≤ ∞ which means that the option becomes worthless to its holder (buyer) at the first time τ I the stock price S t exits the open interval I = (L, R). Thus for t ≥ τ (L,R) the payoff is X t = Y t = 0.…”

Error estimates for multinomial approximations of American options in a class of jump diffusion models

“…Strong approximation theorem allows to construct a probability space which contain all the markets models such that the risky assets in the discrete time models will be "close" with respect to the sup norm to the risky assets in the continuous model. Several authors applied strong invariance principles in order to obtain error estimates for American and game options in the one dimensional BS model (see, [5], [7] and [3]). In all of these papers the authors used the Skorokhod embedding tool of i.i.d.…”