Binomial Models for Option Valuation - Examining and Improving Convergence

TLDR: In this article, the authors define new binomial models, where the calculated option prices converge smoothly to the Black-Scholes solution and remarkably, they even achieve order of convergence two with much smaller initial error.

Abstract: Binomial models, which rebuild the continuous setup in the limit, serve for approximative valuation of options, especially where formulas cannot be derived mathematically. Even with the valuation of European call options distorting irregularities occur. For this case, sources of convergence patterns are explained. Furthermore, it is proved order of convergence one for the Cox--Ross--Rubinstein[79] model as well as for the tree parameter selections of Jarrow and Rudd[83], and Tian[93]. Then, we define new binomial models, where the calculated option prices converge smoothly to the Black--Scholes solution and remarkably, we even achieve order of convergence two with much smaller initial error. Notably, solely the formulas to determine the constant up- and down- factors change. Finally, all tree approaches are compared with respect to speed and accuracy calculating relative root--mean--squared error of approximative option values for a sample of randomly selected parameters across a set of refinements. Approximation of American type options with the new models exhibits order of convergence one but smaller initial error than previously existing binomial models.