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Option Pricing A Simplified Approach

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Option Pricing A Simplified Approach

    Option Pricing: A Simplified Approach

    John C. Cox

    Massachusetts Institute of Technology and Stanford University

    Stephen A. Ross

    Yale University

    Mark Rubinstein

    University of California, Berkeley

    March 1979 (revised July 1979)

    (published under the same title in Journal of Financial Economics (September 1979))

    [1978 winner of the Pomeranze Prize of the Chicago Board Options Exchange]

     [reprinted in Dynamic Hedging: A Guide to Portfolio Insurance, edited by Don Luskin (John Wiley and

    Sons 1988)]

    [reprinted in The Handbook of Financial Engineering, edited by Cliff Smith and Charles Smithson

    (Harper and Row 1990)]

     [reprinted in Readings in Futures Markets published by the Chicago Board of Trade, Vol. VI (1991)]

    [reprinted in Vasicek and Beyond: Approaches to Building and Applying Interest Rate Models, edited by

    Risk Publications, Alan Brace (1996)]

     [reprinted in The Debt Market, edited by Stephen Ross and Franco Modigliani (Edward Lear Publishing

    2000)]

     [reprinted in The International Library of Critical Writings in Financial Economics: Options Markets

    edited by G.M. Constantinides and A..G. Malliaris (Edward Lear Publishing 2000)]

    Abstract

    This paper presents a simple discrete-time model for valuing options. The fundamental economic 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-Scholes model, 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 construction, it gives rise to a simple and efficient numerical procedure for valuing options for which premature exercise may be optimal.

____________________

     Our best thanks go to William Sharpe, who first suggested to us the advantages of the discrete-time approach to option pricing developed here. We are also grateful to our students over the past several years. Their favorable reactions 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.

1. Introduction

    An option is a security that gives its owner the right to trade in a fixed number 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 strike 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 relatively 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 Myron Scholes 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 of finance. For example, virtually all corporate securities can be interpreted as portfolios of puts and calls on 1the 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 Merton articles 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 2mathematics.

    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 that 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 Black-Scholes model as a special limiting case. By taking the limits in a different way, we will also obtain the Cox-Ross (1975) jump process model as another special case.

     1 To take an elementary case, consider a firm with a single liability of a homogeneous class of pure discount bonds. The stockholders then have a ―call‖ 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 bondholders. In turn, the bonds can be interpreted as a portfolio containing a default-free loan with the same face value as the bonds and a short position in a put on the assets of the firm.

     2 Sharpe (1978) has partially developed this approach to option pricing in his excellent new book, Investments.

    Rendleman and Bartter (1978) have recently independently discovered a similar formulation of the option pricing problem.

     2

    Other more general option pricing problems often seem immune to reduction to a simple formula. Instead, numerical procedures must be employed to value these more complex options. Michael Brennan and Eduardo Schwartz (1977) have provided many interesting results along these lines. However, their techniques are rather complicated and are not directly related to the economic structure of the problem. Our formulation, by its very construction, leads to an alternative numerical procedure that is both simpler, and for many purposes, computationally more efficient.

    Section 6 introduces these numerical procedures and extends the model to include puts and calls on stocks that pay dividends. Section 7 concludes the paper by showing how the model can be generalized in other important ways and discussing its essential role in valuation by arbitrage methods.

2. The Basic Idea

Suppose the current price of a stock is S = $50, and at the end of a period of time, its price must

    be either S* = $25 or S* = $100. A call on the stock is available with a strike price of K = $50, 3expiring at the end of the period. It is also possible to borrow and lend at a 25% rate of interest. The one piece of information left unfurnished is the current value of the call, C. However, if

    riskless profitable arbitrage is not possible, we can deduce from the given information alone

    what the value of the call must be!

Consider the following levered hedge:

    (1) write 3 calls at C each,

    (2) buy 2 shares at $50 each, and

    (3) borrow $40 at 25%, to be paid back at

     the end of the period.

    Table 1 gives the return from this hedge for each possible level of the stock price at expiration. Regardless of the outcome, the hedge exactly breaks even on the expiration date. Therefore, to prevent profitable riskless arbitrage, its current cost must be zero; that is,

    3C 100 + 40 = 0

The current value of the call must then be C = $20.

     3 To keep matters simple, assume for now that the stock will pay no cash dividends during the life of the call. We also ignore transaction costs, margin requirements and taxes.

     3

    Table 1

    Arbitrage Table Illustrating the Formation of a Riskless Hedge

     expiration date

     present date S* = $25 S* = $100

    write 3 calls 3C 150

    buy 2 shares 100 50 200

    borrow 40 50 50

    total

If the call were not priced at $20, a sure profit would be possible. 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 accomplished by

    buying 3 calls, selling short 2 shares, and lending $40.

Table 1 can be interpreted as demonstrating that an appropriately levered position in stock will

    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 calls. In view of this, it should seem less surprising that all we needed to determine the exact value of the call was its strike

    price, underlying stock price, range of movement in the underlying stock price, and the rate of interest. What may seem more incredible is what we do not need to know: among other things, we do not need to know the probability that the stock price will rise or fall. Bulls and bears must

    agree on the value of the call, relative to its underlying stock price!

    This example is very simple, but it shows several essential features of option pricing. And we will soon see that it is not as unrealistic as it seems.

3. The Binomial Option Pricing Formula

    In this section, we will develop the framework illustrated in the example into a complete valuation method. We begin by assuming that the stock price follows a multiplicative binomial process over discrete periods. The rate of return on the stock over each period can have two possible values: u 1 with probability q, or d 1 with probability 1 q. Thus, if the current

    stock price is S, the stock price at the end of the period will be either uS or dS. We can

    represent this movement with the following diagram:

     uS with probability q

     S