Error bounds and convergence for American put option pricing based on translation-invariant Markov chains

Consider a discrete finite-dimensional, Markovian market model. In this setting, discretely sampled American options can be priced using the so-called ``non-recombining'' tree algorithm. By successively increasing the number of exercise times, the American option price itself can be computed; for combinatorial reasons, we shall consider a recursive algorithm that doubles the number of exercise times at each recursion step. First we prove, by elementary arguments, error bounds for the first order differences in this recursive algorithm. From this, bounds on the higher order differences can be obtained using combinatorial arguments that are motivated by the theory of rough paths. We shall obtain an explicit $L^1(C)$ convergence estimate for the recursive algorithm that prices a discretely sampled American $\max$-put option (on a basket of size $d$) at each recursion step, $C$ belonging to a certain class of compact subset of $\RR^d$, in under the assumption of sufficiently small volatilities. In case $d=1$, $L^1(C)$-bounds for an even more natural choice of $C$ will be derived.