Research Program in Finance Option Pricing: A Simplified Approach John c. cox Stephen ross Mark Rubinstein ptember 1978 Working Paper No 79 John Cox is at Stanford University, currently visiting at the Massachusetts Institute of Technology; Stephen Ross is at Yale University; and Mark rubinstein is at the University of Califomia at Berkeley. The authors acknowledge support for this research from the National Science Foundation, grant numbers SoC-77-18087 and SoC-77-22301
ABSTRACT The option pricing problem is examined in its simplest nontrivial setting--the uncertainty of the underlying stock price reduced to discrete binomal movements. Using only ele- mentary mathematics, this leads to a binomial option pricing formula,which contains the Black-Scholes and jump-process fo mulas as special Iimiting cases, The discrete binomial formula illuminates the economic intuition behind option pricing theory, thout any sacrifice of generall ty compared to the Black-Scholes analysis. Moreover, the binomial formulation, by its very con- struction, gives rise to a simple and efficent numerical pro- cedure for valuing options for which premature exercise may be desirable
I。 INTRODUCTI0N The original derivation of a Satisfactory exact option pricing formula was first published in the Journal of Policital economy, May 1973, in an article written by Fischer Black and Myron Scholes entitled The Pricing of Options and Corporate Liabilities. They specifically examined ordinary put and call options. A call is an option to buy a fixed number of shares of a specified common stock at a fixed price at any time until a fixed date. a put is similar, except it is an option to sell shares. The fixed price is termed the striking price, and the fixed date,the" expiration date."工t±s8 common to distingu⊥ sh options which can be exercised at any time prior to expiration (American options) from the hich can only be exercised at expiration (European options) Although the Black-Scholes paper dealt directly only with European op- tions, it has been possible to extend their approach to value American options as well. Unless otherwise indicated, this paper will likewise pertain to American options, of the type now traded on organized option markets in the United Sta If option pricing theory were confined to the valuation of ordi- nary pute and calle, despite recent instituonal developments, it would not have attracted widespread academic attention. As Black and Scholes them selves mentioned, virtually all corporate securities can be fruitfully A call should not be confused with a futures contract. The lat ter represents a commitment to buy or sell (1.e, on its expiration date, future must be e
interpreted as portfolios of puts and calls. Moreover, option pricing theory applies to a very general class of economc problems-the valuation of contracts where the outcome to each party depends on a quantifiable un- certain future event Unfortunately, the mathematical tools employed in the article are quite advanced, and have served to obscure the underlying economics. How- ever, thanks to a suggestion of William Sharpe, it is possible to derive the Black-Scholes formula using only elenentary Mathematics. This is done by treating the option pricing problem in its simplest nontrivial setting--the uncertainty of the underlying stock price reduced to discret binomial movements.- One of the conclusions of our paper is that the re sulting discrete binomial option pricing formula illuminates the economic intuition behind option pricing theory, without any sacrifice in general- ity compared to the Black-Scholes analysis Not only is the binomial formula of interest in itself, but the formula lies at the fork of two significant limiting cases--the Black- choles continuous stochastic process formula, and the Cox-Ross [1976] jump stochastic process formula. As we shall show, each can be derlved from the binomial formula by taking the appropriate limits To take an elementary case, consider a firm with a Bingle liabil- ity of a homogeneous class of pure discount bonds. The stockholders then have aca1ll"on the assets of the firm which they can choose to exercise at the maturity date of the debt by paying its principal to the bondhold ers. In turn, the bonds can be interpreted as a portfolio of a written put on the assets of the firm and a default-free loan with the same face value as the bonds lampe t⊥o pricing in l approach his recent book investments, Prentice-Hall, 1978. Rendleman and Bartter [1977] have also independently discovered the binomial formu lation of the option pricing problem
Other more general option pricing problems seem immune to reduction to a simple formula. Instead, numerical procedures must often be employ to price these more complex options. Following the approach of Michael Brennan and Edwardo Schwartz [1977], the Black-Scholes differential hedger ng equation is first reduced to a discrete-time difference equation and then the option price is obtained by somewhat elaborate numerical pro cedures. In contrast, the binomial formulation, by its very construction gives rise to an alternative numerical procedure which is both far simpler and, for many purposes, computationally more efficient
II. THE BASIC IDEA Suppose the current price of an underlying stock is S=$50, and at the end of a period a1 t ine, it: p: IIe mut t r'it irt 2:: 4-h.'t S:=$100. A call on the stock is available with a striking price of K=$50, expiring at the end of the period. It is also possible to bor row and lend at a 25 rate of interest. The one piece of information left unfurnished is the current Price c of the call. However, if riskless profitable arbitrage is not possible, we can deduce from the given information alone what the price of the call. must be! Consider forming the following levered hedge: (l)write 3 calls at c each (3)borrow $l0 at 25%, to be paid back at end of period Table 1 gives the return from this hedge, for each possible'level of he stock price at expiration. Regardless of the outcome, the hedge exactly breaks even or the expiration date. Therefore, to prevent profitable riskless arbitrage, its current value must be zero; chet is 3c+100-40=0 The current value of the call must then be C=$20 To keep matters sir against cash dividends mple, assume for now that the call is protecte le also ignore transactions costs, margin, and
Table I. Arbitrage Table Illustrating the Formation of a Riskless Hedge Present Expiration Date Date s大=25S大=100 Write 2 calls -150 Buy 2 shares 50 200 Borrow 50 Total 一一一 If the call were not priced at $20, a sure profit would be pos- sible. In particular, if C=$25, the above hedge would yield a current cash inflow of $15 and would experience no further gain or loss in the future. On the other hand, if C=$15, then the same thing could be ac- compIished by buying 3 calls, selling short 2 shares, and lending.$4 Table 1 can be interpreted as demonstrating that an, appropri- ateiy Levered position in stock wiiz replicate the future returns of a calL, That is, if we buy shares and borrow against them in the right proportion, we can, in effect, duplicate a pure position in calis. In view of this, it should seem less surprising that all we needed to de- termine the exact value of the call was its striking price, underlying stock price, ramge of movement in the under lying stock price, and the rate of interest. What may seem more incredible is what we don't need to know: among other things, we don't need to know the probability that
the stock price wiZz rise or fall. Bulls and bears tust agree on the value of the call, relative to its underlying stock price Clearly, our numerical example has been chosen for simplicity, not reals Among other things, it gives no consideration to the existing liquid secondary market, which permits closing transactions any tine priot to expiration, and it posiTs Very tmli't i
further by setting d=u and q =5, we choose to retain this greater leve】 of generality When the call expires, we know that its contract and a rational exercise policy imply that its value must either be c E max[O,uS-KI C,三如ax[0,uS一K] with probabi1ityq C, E max[O,ds -k] with probability 1-q Suppose we form a hedge at the beginning of the period by writing one call against a shares of stock, This would cost cS-C. The buyer of the call will either retain it until expiration or exercise it immed iately. This will depend on which is higher, the retention value or the exercise value, max[O, S-K]. To find out. we will first calculate the value of the call if he retains it. If the call is unexercised. then our hedge will return aus-c, with probability q ads-C d with probability q Now, since we can choose a any way we wish, suppose we select the"neu- tral hedge ratio, that is, the a that makes the hedge riskless. We ac complish this by selecting the a which equates the dollar returns in the two possibilities:
ads =auS〓 Solving this equation, the hedge ratio a which eliminates all risk is with this hedge ratio, since the return from the hedge, ads is riskless, to prevent riskless profitable arbitrage, it must have t same return as an investment of as-c dollars in riskless borrowing or lending. Therefore, ads-C, = r(as-c Rearranging this equality and substituting for a To state this more simply, observe that defining p==u, ther 1-p- Therefore (1) c=[ (1-p)c This is the exact formula for the value of a call one perlod prior to ex piration in terms of s, K, u, d, and r The formula gives the value of the call 1f, as we assumed, the buyer does not immediately exercise it. However, it easy to see that