Joshi's split tree for option pricing

In a thorough study of binomial trees, Joshi introduced the split tree as a two-phase binomial tree designed to minimize oscillations, and demonstrated empirically its outstanding performance when applied to pricing American put options. Here we introduce a "flexible" version of Joshi's tree, and develop the corresponding convergence theory in the European case: we find a closed form formula for the coefficients of 1/n and 1/n3/2 in the expansion of the error. Then we define several optimized versions of the tree, and find closed form formulae for the parameters of these optimal variants. In a numerical study, we found that in the American case, an optimized variant of the tree significantly improved the performance of Joshi's original split tree.