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of Financial


7 (1979) 229-263.

0 North-Holland









of Technology, Cambridge, MA 02139, USA

Stanford University, Stanford, CA 94305, USA

Yale University,


New Haven, CT06520,



University ofCaliforma, Berkeley, CA 94720, USA Received March 1979, revised version received July 1979

This paper presents a simple discrete-time model for valumg optlons. The fundamental econonuc principles of option pricing by arbitrage methods are particularly clear In this setting. Its development requires only elementary mathematics, yet it contains as a special limiting case the celebrated Black-&holesmodel, which has previously been derived only by much more difficult methods. The basic model readily lends itself to generalization in many ways. Moreover, by its very constructlon, it gives rise to a simple and efficient numerical procedure for valumg optlons for which premature exercise may be optimal.

1. Introduction An option is a security which gives its owner the right to trade in a fixednumber of shares of a specified common stock at a fixed price at any time on or before a given date. The act of making this transaction is referred to as exercising the option. The fixed price is termed the striking price, and the given date, the expiration date. A call option gives the right to buy the shares; a put option gives the right to sell the shares. Options have been traded for centuries,but they remained relativelv obscure financial instruments until the introduction of a listed options exchange in 1973. Since then, options trading has enjoyed an expansion unprecedented in American securities markets. Option pricing theory has a long and illustrious history, but it also underwent a revolutionary change in 1973. At that time, Fischer Black and
*Our best thanks go to WilliamSharpe, who first suggested to us the advantages of the discrete-time approach to option prlcmg developed here. We are also grateful to our students over the past several years. Then favorable reactlons to this way of presenting things encouraged us to write this article. We have received support from the National Science Foundation under Grants Nos. SOC-77-18087 and SOC-77-22301.


J.C. Coxet al., Optwn prrcrng. A rrn~plr/wd ~~ppnw~ Ir

Myron &holes presented the first completely satisfactory equilibrium option pricing model. In the same year, Robert Merton extended their model in several important ways. These path-breaking articles have formed the basis for many subsequent academic studies. As these studies have shown, option pricing theory is relevant to almost every area offinance. For example, virtually all corporate securities can be interpreted as portfolios of puts and calls on the assets of the firm.’ Indeed, the theory applies to a very general class of economic problems - the valuation of contracts where the outcome to each party depends on a quantifiable uncertain future event. Unfortunately, the mathematical tools employed in the Black-Scholes and Mertonarticles are quite advanced and have tended to obscure the underlying economics. However, thanks to a suggestion by William Sharpe, it is possible to derive the same results using only elementary mathematics.2 In this article we will present a simple discrete-time option pricing formula. The fundamental economic principles of option valuation by arbitrage methods are particularly clear in this setting.Sections 2 and 3 illustrate and develop this model for a call option on a stock which pays no dividends. Section 4 shows exactly how the model can be used to lock in pure arbitrage profits if the market price of an option differs from the value given by the model. In section 5, we will show that our approach includes the BlackScholes model as a special limiting case. By taking the limits in a...