A Synthesis of Binomial Option Pricing Models for Lognormally Distributed Assets

Article excerpt

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 and hedge an option position. These formulas have many similarities but notable differences. Another source of confusion is that some presentations use opposite notation.4 But more fundamentally, the obvious question is how so many different candidates for the inputs of the binomial model can exist and how each can technically be correct.

The objective of this paper is to synthesize the different approaches within a body of uniform notation and provide a coherent treatment of each model. Each model is presented with its distinct assumptions. Detailed derivations are omitted but are available in a supplemental document on the journal website or from the author.

Some would contend that it is wasteful to study a model that, for European options, in the limit equals the Black-Scholes-Merton model. Use of the binomial model, they would argue, serves only a pedagogical purpose. But it is difficult to consider the binomial model as a method for deriving the values of more complex options without knowing how well it works for the one scenario in which the true continuous limit is known. An unequivocal benchmark is rare in finance.

For options on lognormally distributed assets, the literature contains no less than 11 distinct versions of the binomial model. Some of the models are improperly specified and can lead to arbitrage profits for a finite number of time steps, while some do not capture the exogenous volatility. Several models focus first on fitting the binomial model to the physical process, rather than the risk neutral process, thereby requiring that the expected return on the stock be known, an unnecessary requirement in arbitrage-free pricing. …