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


     [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)]


    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.


    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.


    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


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


     dS with probability 1 q

    We also assume that the interest rate is constant. Individuals may borrow or lend as much as they wish at this rate. To focus on the basic issues, we will continue to assume that there are no


taxes, transaction costs, or margin requirements. Hence, individuals are allowed to sell short any 4security and receive full use of the proceeds.

Letting r denote one plus the riskless interest rate over one period, we require u > r > d. If

    these inequalities did not hold, there would be profitable riskless arbitrage opportunities 5involving only the stock and riskless borrowing and lending.

To see how to value a call on this stock, we start with the simplest situation: the expiration date

    is just one period away. Let C be the current value of the call, C be its value at the end of the u

    period if the stock price goes to uS and C be its value at the end of the period if the stock d

    price goes to dS. Since there is now only one period remaining in the life of the call, we know that the terms of its contract and a rational exercise policy imply that C= max[0, uS K] and u

    C= max[0, dS K]. Therefore, d

     C= max[0, uS K] with probability q u


     C= max[0, dS K] with probability 1 q d

    Suppose we form a portfolio containing shares of stock and the dollar amount B in riskless 6bonds. This will cost S + B. At the end of the period, the value of this portfolio will be

     uS + rB with probability q

     S + B

     dS + rB with probability 1 q

    Since we can select and B in any way we wish, suppose we choose them to equate the end-of-period values of the portfolio and the call for each possible outcome. This requires that

    uS + rB = C u

    dS + rB = C d

Solving these equations, we find

    CCuCdCuddu,B (1) (ud)S(ud)r

     4 Of course, restitution is required for payouts made to securities held short.

     5 We will ignore the uninteresting special case where q is zero or one and u = d = r.

     6 Buying bonds is the same as lending; selling them is the same as borrowing.


With and B chosen in this way, we will call this the hedging portfolio.

If there are to be no riskless arbitrage opportunities, the current value of the call, C, cannot be

    less than the current value of the hedging portfolio, S + B. If it were, we could make a riskless

    profit with no net investment by buying the call and selling the portfolio. It is tempting to say that it also cannot be worth more, since then we would have a riskless arbitrage opportunity by reversing our procedure and selling the call and buying the portfolio. But this overlooks the fact that the person who bought the call we sold has the right to exercise it immediately.

    Suppose that S + B < S K. If we try to make an arbitrage profit by selling calls for more than S + B, but less than S K, then we will soon find that we are the source of arbitrage profits rather than the recipient. Anyone could make an arbitrage profit by buying our calls and exercising them immediately.

    We might hope that we will be spared this embarrassment because everyone will somehow find it advantageous to hold the calls for one more period as an investment rather than take a quick profit by exercising them immediately. But each person will reason in the following way. If I do not exercise now, I will receive the same payoff as a portfolio with S in stock and B in bonds.

    If I do exercise now, I can take the proceeds, S K, buy this same portfolio and some extra

    bonds as well, and have a higher payoff in every possible circumstance. Consequently, no one would be willing to hold the calls for one more period.

    Summing up all of this, we conclude that if there are to be no riskless arbitrage opportunities, it must be true that

    CCuCdCrdur??????uddu; (2) CS;BC;C/r????ud??ud(ud)rudud??????

     7if this value is greater than S K, and if not, C = S K.

Equation (2) can be simplified by defining

    urrd1pp and udud

    so that we can write

    C = [pC+ (1 p)C]/r (3) u d

It is easy to see that in the present case, with no dividends, this will always be greater than S K

    as long as the interest rate is positive. To avoid spending time on the unimportant situations where the interest rate is less than or equal to zero, we will now assume that r is always greater

     7 In some applications of the theory to other areas, it is useful to consider options that can be exercised only on the expiration date. These are usually termed European options. Those that can be exercised at any earlier time as well, such as we have been examining here, are then referred to as American options. Our discussion could be easily modified to include European calls. Since immediate exercise is then precluded, their values would always be given by (2), even if this is less than S K.


    than one. Hence, (3) is the exact formula for the value of a call one period prior to the expiration in terms of S, K, u, d, and r.

To confirm this, note that if uS ( K, then S < K and C = 0, so C > S K. Also, if dS K, then

    C = S (K/r) > S K. The remaining possibility is uS > K > dS. In this case, C = p(uS K)/r.

    This is greater than S K if (1 p)dS > (p r)K, which is certainly true as long as r > 1.

This formula has a number of notable features. First, the probability q does not appear in the

    formula. This means, surprisingly, that even if different investors have different subjective probabilities about an upward or downward movement in the stock, they could still agree on the relationship of C to S, u, d, and r.

Second, the value of the call does not depend on investors’ attitudes toward risk. In constructing

    the formula, the only assumption we made about an individual’s behavior was that he prefers more wealth to less wealth and therefore has an incentive to take advantage of profitable riskless arbitrage opportunities. We would obtain the same formula whether investors are risk-averse or risk-preferring.

    Third, the only random variable on which the call value depends is the stock price itself. In particular, it does not depend on the random prices of other securities or portfolios, such as the market portfolio containing all securities in the economy. If another pricing formula involving other variables was submitted as giving equilibrium market prices, we could immediately show that it was incorrect by using our formula to make riskless arbitrage profits while trading at those prices.

    It is easier to understand these features if it is remembered that the formula is only a relative pricing relationship giving C in terms of S, u, d, and r. Investors’ attitudes toward risk and the

    characteristics of other assets may indeed influence call values indirectly, through their effect on these variables, but they will not be separate determinants of call value.

Finally, observe that p (r d)/(u d) is always greater than zero and less than one, so it has

    the properties of a probability. In fact, p is the value q would have in equilibrium if investors

    were risk-neutral. To see this, note that the expected rate of return on the stock would then be the riskless interest rate, so

    q(uS) + (1 q)(dS) = rS


    q = (r d)/(u d) = p

    Hence, the value of the call can be interpreted as the expectation of its discounted future value in a risk-neutral world. In light of our earlier observations, this is not surprising. Since the formula does not involve q or any measure of attitudes toward risk, then it must be the same for any set of preferences, including risk neutrality.

    It is important to note that this does not imply that the equilibrium expected rate of return on the call is the riskless interest rate. Indeed, our argument has shown that, in equilibrium, holding the call over the period is exactly equivalent to holding the hedging portfolio. Consequently, the risk


    and expected rate of return of the call must be the same as that of the hedging portfolio. It can be shown that 0 and B ( 0, so the hedging portfolio is equivalent to a particular levered long position in the stock. In equilibrium, the same is true for the call. Of course, if the call is currently mispriced, its risk and expected return over the period will differ from that of the hedging portfolio.

    Now we can consider the next simplest situation: a call with two periods remaining before its expiration date. In keeping with the binomial process, the stock can take on three possible values after two periods,

     2S u


    S duS


     2 dS

Similarly, for the call,

     2 C= max[0, uS K] uu

     C u

    C C= max[0, duS K] du

     C d

     2 C= max[0, dS K] dd

C stands for the value of a call two periods from the current time if the stock price moves uu

    upward each period; C and C have analogous definitions. dudd

    At the end of the current period there will be one period left in the life of the call, and we will be faced with a problem identical to the one we just solved. Thus, from our previous analysis, we know that when there are two periods left,

    C= [pC+ (1 p)C]/r u uu ud

    and (4)

    C= [pC+ (1 p)C]/r d du dd

Again, we can select a portfolio with S in stock and B in bonds whose end-of-period value

    will be C if the stock price goes to uS and C if the stock price goes to dS. Indeed, the ud


functional form of and B remains unchanged. To get the new values of and B, we

     and C. simply use equation (1) with the new values of Cud

    Can we now say, as before, that an opportunity for profitable riskless arbitrage will be available if the current price of the call is not equal to the new value of this portfolio or S K, whichever

    is greater? Yes, but there is an important difference. With one period to go, we could plan to lock in a riskless profit by selling an overpriced call and using part of the proceeds to buy the hedging portfolio. At the end of the period, we knew that the market price of the call must be equal to the value of the portfolio, so the entire position could be safely liquidated at that point. But this was true only because the end of the period was the expiration date. Now we have no such guarantee. At the end of the current period, when there is still one period left, the market price of the call could still be in disequilibrium and be greater than the value of the hedging portfolio. If we closed out the position then, selling the portfolio and repurchasing the call, we could suffer a loss that would more than offset our original profit. However, we could always avoid this loss by maintaining the portfolio for one more period. The value of the portfolio at the end of the current period will always be exactly sufficient to purchase the portfolio we would want to hold over the last period. In effect, we would have to readjust the proportions in the hedging portfolio, but we would not have to put up any more money.

    Consequently, we conclude that even with two periods to go, there is a strategy we could follow which would guarantee riskless profits with no net investment if the current market price of a call differs from the maximum of S + B and S K. Hence, the larger of these is the current value

    of the call.

    Since and B have the same functional form in each period, the current value of the call in terms of C and C will again be C = [pC+ (1 p)C]/r if this is greater than S K, and C = udu d

    S K otherwise. By substituting from equation (4) into the former expression, and noting that C= C, we obtain du ud

     222 C = [pC+ 2p(1 p)C+ (1 p)C]/ruu ud dd

     (5) 22222 = [pmax[0, uS K] + 2p(1 p)max[0, duS K] + (1 p)max[0, dS K]]/r

A little algebra shows that this is always greater than S K if, as assumed, r is always greater 8than one, so this expression gives the exact value of the call.

    All of the observations made about formula (3) also apply to formula (5), except that the number of periods remaining until expiration, n, now emerges clearly as an additional determinant of the

    call value. For formula (5), n = 2. That is, the full list of variables determining C is S, K, n, u,

    d, and r.

     8 In the current situation, with no dividends, we can show by a simple direct argument that if there are no arbitrage opportunities, then the call value must always be greater than S K before the expiration date. Suppose that the

    call is selling for S K. Then there would be an easy arbitrage strategy that would require no initial investment and would always have a positive return. All we would have to do is buy the call, short the stock, and invest K dollars

    in bonds. See Merton (1973). In the general case, with dividends, such an argument is no longer valid, and we must use the procedure of checking every period.


    We now have a recursive procedure for finding the value of a call with any number of periods to go. By starting at the expiration date and working backwards, we can write down the general valuation formula for any n:

    n????n!jnjjnjnCp(1p)max[0,udSK]/r (6) ???????j!(nj)!????0j??

    This gives us the complete formula, but with a little additional effort we can express it in a more convenient way.

    Let a stand for the minimum number of upward moves that the stock must make over the next n periods for the call to finish in-the-money. Thus a will be the smallest non-negative integer an-asuch that udS > K. By taking the natural logarithm of both sides of this inequality, we could nwrite a as the smallest non-negative integer greater than log(K/Sd)/log(u/d).

For all j < a, jn-jmax[0, udS K] = 0

    and for all j a, jn-jjn-jmax[0, udS K] = udS K


    n????n!jnjjnjn ??Cp(1p)[udSK]/r?????j!(nj)!ja????

    Of course, if a > n, the call will finish out-of-the-money even if the stock moves upward every period, so its current value must be zero.

By breaking up C into two terms, we can write

    jnjnn????????n!ud??n!jnjnjnj??CSp(1p)?? ?? Krp(1p)????????n????j!(nj)!j!(nj)!r????jaja????????

Now, the latter bracketed expression is the complementary binomial distribution function [a;

    n, p]. The first bracketed expression can also be interpreted as a complementary binomial distribution function [a; n, p′], where

    p′ (u/r)p and 1 p′ (d/r)(1 p)

p′ is a probability, since 0 < p′ < 1. To see this, note that p < (r/u) and

    jnjjnj??udud????jnjjnj????p(1p)p(1p)p(1p) n??????rrr??????


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