A Synthesis of Binomial Option Pricing Models for Lognormally Distributed Assets

Abstract

The finance literature has revealed no fewer than 11 alternative versions of the binomial option pricing model for options on lognormally distributed assets. These models are derived under a variety of assumptions and in some cases require information that is ordinarily unnecessary to value options. This paper provides a review and synthesis of these models, showing their commonalities and differences and demonstrating how 11 diverse models all produce the same result in the limit. Some of the models admit arbitrage with a finite number of time steps and some fail to capture the correct volatility. This paper also examines the convergence properties of each model and finds that none exhibit consistently superior performance over the others. Finally, it demonstrates how a general model that accepts any arbitrage-free risk neutral probability will reproduce the Black-Scholes-Merton model in the limit.