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BINOMIAL APPROXIMATIONS OF SHORTFALL RISK FOR GAME OPTIONS
, 2008
"... 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 pathdependent payoffs for both discrete and continuoustime cases. These results are new also for usual Am ..."
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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 pathdependent payoffs for both discrete and continuoustime 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
Binomial approximations for barrier options of Israeli style
, 2009
"... 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 ..."
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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.
Dynkin games and Israeli options
 ISRN Probability and Statistics, volume 2013 (2013), Id.856458
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Applications of Weak Convergence for Hedging of American and Game Options
, 2009
"... This paper studies stability of Dynkin’s games value under weak convergence. We use these results to approximate game options prices with path dependent payoffs in continuous time models by sequence of game options prices in discrete time models which can be calculated by dynamical programming algor ..."
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This paper studies stability of Dynkin’s games value under weak convergence. We use these results to approximate game options prices with path dependent payoffs in continuous time models by sequence of game options prices in discrete time models which can be calculated by dynamical programming algorithms. We also show that shortfall risks of American options in a sequence of multinomial approximations of the multidimensional Black–Scholes (BS) market converge to the corresponding quantities for similar American options in the multidimensional BS market with path dependent payoffs. In comparison to previous papers we work under more general convergence of underlying processes, as well, as weaker condition on the payoffs.
ERROR ESTIMATES FOR MULTINOMIAL APPROXIMATIONS OF AMERICAN OPTIONS IN MERTON’S MODEL
, 2009
"... We derive error estimates for multinomial approximations of American options in a multidimensional jump–diffusion Merton’s model. We assume that the payoffs are Markovian and satisfy Lipschitz type conditions. Error estimates for such type of approximations were not obtained before. Our main tool ..."
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We derive error estimates for multinomial approximations of American options in a multidimensional jump–diffusion Merton’s model. We assume that the payoffs are Markovian and satisfy Lipschitz type conditions. Error estimates for such type of approximations were not obtained before. Our main tool is the strong approximations theorems fori.i.d. random vectors which were obtained in [14]. For the multidimensional Black–Scholes model our results can be extended also to a general path dependent payoffs which satisfy Lipschitz type conditions. For the case of multinomial approximations of American options for the Black–Scholes model our estimates are a significant improvement of those which were obtained in [8] (for game options in a more general setup).