| Summary: | 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 <inline-formula><math display="inline"><semantics><mrow><mn>1</mn><mo>/</mo><mi>n</mi></mrow></semantics></math></inline-formula> and <inline-formula><math display="inline"><semantics><mrow><mn>1</mn><mo>/</mo><msup><mi>n</mi><mrow><mn>3</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow></semantics></math></inline-formula> 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.
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