Content area
Full text
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.
Option pricing theory has become one of the most powerful tools in economics and finance. The celebrated Black-Scholes-Merton model not only led to a Nobel Prize but completely redefined the financial industry. Its sister model, the binomial or two-state model, has also attracted much attention and acclaim, both for its ability to illustrate the essential ideas behind option pricing theory with a minimum of mathematics and to value many complex options.
The origins of the binomial model are somewhat unclear. Options folklore has it that around 1975 William Sharpe, later to win a Nobel Prize for his seminal work on the Capital Asset Pricing Model, suggested to Mark Rubinstein that option valuation should be feasible under the assumption that the underlying stock price can change to one of only two possible outcomes.1 Sharpe subsequently developed the idea in the first edition of his textbook.2 Perhaps the bestknown and most widely cited original paper on the model is Cox, Ross, and Rubinstein (1979), but almost simultaneously, Rendleman and Bartter (1979) presented the same model in a slightly different manner.
Over the years, there has been an extensive body of research designed to improve the model.3 In the literature the model has appeared in a variety of forms. Anyone attempting to understand the model can become bewildered by the array of formulas that all purport to accomplish the desired result of showing how to value an option...