  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 for i.i.d. random vectors which were obtained
[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)
