  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.
