Binomial models for option valuation - examining and improving convergence

Abstract

Binomial models, which describe the asset price dynamics of the continuous-time model in the limit, serve for approximate valuation of options, especially where formulas cannot be derived analytically due to properties of the considered option type. To evaluate results, one inevitably must understand the convergence properties. In the literature we find various contributions proving convergence of option prices. We examine convergence behaviour and convergence speed. Unfortunately, even in the case of European call options, distorted results occur when calculating prices along the iteration of tree refinements. These convergence patterns are examined and order of convergence one is proven for the Cox-Ross-Rubinstein model as well as for two alternative tree parameter selections from the literature. Furthermore, we define new binomial models, where the calculated option prices converge smoothly to the Black-Scholes solution, and we achieve order of convergence two with much smaller initial error. Notably, ...