Two-State Option Pricing TOR Richard J. Rendleman Jr, Brit J. bartter Journal of Finance, Volume 34, Issue 5(Dec, 1979), 1093-1110 Your use of the IStoR database indicates your acceptance of ISTOR's Terms and Conditions of Use. A copy of IsTor'sTermsandConditionsofUseisavailableathttp://www.jstororg/about/terms.htmlbycontactingJsTor at jstor-info @umich. edu, or by calling JSTOR at(888)388-3574, (734)998-9101 or(FAX)(734)9989113 No part of a IStOR transmission may be copied, downloaded, stored, further transmitted, transferred, distributed, altered, or otherwise used, in any form or by any means, except: (1)one stored electronic and one paper copy of any article solely for your personal, non-commercial use, or(2)with prior written permission of jSTOR and the publisher of the article or other text Each copy of any part of a IStoR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission Journal of Finance is published by American Finance Association. Please contact the publisher for further permissions regarding the use of this work. Publisher contact information may be obtained at http://www.jstor.org/journals/afina.html aal of finance 9 American Finance Association ISTOR and the IStoR logo are trademarks of JSTOR, and are Registered in the U.S. Patent and Trademark Office For more information on STOR contact jstor- info@umich. edu @2000 JSTOR Sat dec160951:04200
THE JOURNAL OF FINANCE. VOL XXXIV. NO 5. DECEMBER 1979 The fournal of finance VOL. XXXIV DECEMBER 1979 Two-State Option Pricing RiChard J, RENDLEMAN, JR, and brIt. barttER' I. Introduction IN THIS PAPER WE present an elemental two- state option pricing model (TSoPM) which is mathematically simple, yet can be used to solve many complex option pricing problems. In contrast to widely accepted option pricing models which require solutions to stochastic differential equations, our model is derived alg braically. First we present the mathematics of the model and illustrate its application to the simplest type of option pricing problem. Next, we discuss the statistical properties of the model and show how the parameters of the model can be estimated to solve practical option pricing problems. Finally, we apply the model to the pricing of European and American put and call options on both non-dividend and dividend paying stocks. Elsewhere, we have applied the model to the valuation of the debt and equity of a firm with coupon paying debt in its capital structure [9], the valuation of options on debt securities [7], and the pricing of fixed rate bank loan commitments[1, 2]. In the Appendix we derive the Black-scholes [3] model using the two-state approach I. The Two-State Option Pricing Model Consider a stock whose price can either advance or decline during the next period Let Hi and Hi represent the returns per dollar invested in the stock if the price rises(the state)or falls(the - state), respectively, from time t-l to time t and Vi and Vi the corresponding end-of-period values of the option. With the assumption that the prices of the stock and its option follow a two-state process it is possible to form a riskless portfolio with the two securities. [See Black and Scholes [3] for the continuous time analog of riskless hedging. Since the end-of- period value of the portfolio is certain, the option should be priced so that the portfolio will yield the riskless interest rate The riskless portfolio is formed by investing one dollar in the stock and Both Assistant Professors of Finance, Graduate School of Management, Northwestern University Since the original writing of this paper, the authors have learned that a similar procedure has been suggested by Rubinstein [10], Sharpe [11], and Cox, Ross, and Rubinstein[5]
The Journal of finance purchasing a units of the option at a price of Pr-1. The value of a is chosen so that the portfolio payoffs are the same in both states, or Vi=Hi +av lving for a we obtain the number of units of the option to be held in the rtfolio per $1 invested in the stock. A negative value of a implies that the option is sold short (written)with the proceeds being used to partially fund the purchase of the stock The time t- 1 value of portfolio is 1 +aP-I. The end-of-period value is given by either side of (1). Discounting the left-hand side by the riskless interest rate, R, and setting the discounted value equal to the present value of the portfolio, a pricing equation for the option is obtained Hi +av 1 +aP- Substituting the value of a from(2)into(3), the price of the option can be solved in terms of its end-of-period values. +R-Hd)+v( P (H-H;)(1+R) Equation 4 is a recursive relationship that can be applied at any time t-l to determine the price of the option as a function of its value at time t Note that in equation (4)we make a notational distinction between an option,s value (V) and its price(P). Assuming that an investor will exercise an option when it is in his best interest to do so V,= MAXIPI, VEXER where VEXER, is the value of exercising the option at time t The distinguishing feature among American and European puts and calls is in he definition of their exercisable values. American options can be exercised at any time whereas European options can only be exercised at maturity. Calls are options to buy stock at a set price whereas puts are options to sell. Letting S, represent the time t price of the stock, X the option,s exercise price, and t the maturity date of the option, we obtain American Call VEXEl t, Put VEXER,=X-S, for all t Call VEXER, =S,-X for t=T vEⅹER,=0 for t< T Put veXer=X-s, for t VEXER,=0 Recognizing that for both American and Eupopean puts and calls Pr=0
Two-State Option Pricing since there is no value associated with maintaining an option position beyond maturity, (4-7)represent the formal specification of the two-state model. Through repeated application of (4), subject to (5-7), one can begin at an option s maturity date and recursively solve for its current price. To illustrate the model, consider a call option on a stock with an exercise price of $100. The current price of the stock is $100 and the possible prices of the stock on the option s maturity date are $110 and $90 implying Hf 1.10 andHi =.90 Assuming that the option is exercised if the stock price rises to $110 and is allowed to expire worthless if the stock price falls to $90, the present prices and the end-of- period payoffs of the stock and option can be represented by the following two-branched tree diagram. Stock Option Stock Option () S10 s9Q$0 Today Option s Maturity Dat If an investor purchases the stock and writes two call options, the end-of-period portfolio value will be $90 in both states. Equivalently, for every $l invested in the stock, a riskless hedge requires that a=(90-1. 10)/(10-0)=-02, or that 02 options are written. Assuming a risk free interest rate of 5%, the present value of the riskless portfolio should be $90/1.05 or $85. 71 to ensure no riskless arbitrage opportunities between the stock-option portfolio and a riskless security. Since he riskless portfolio involves a $100 investment in the stock which is partially offset by the two short options, an option price of $7. 14 is required to obtain an $85.71 portfolio value. The option price can also be obtained directly from(4) Pn=5005-90)+0.10-106 10(15) $7.14. Although this example is unrealistic, it nevertheless illustrates two of the most important features of the TSOPM. We can observe that the option price does not depend upon the probabilities of the up(+)or down (-) states occurring or the risk preferences of the investor. Two investors who agreed that the stock price is n equilibrium, but had different probability beliefs and preferences, would both view $7. 14 as the equilibrium option price. As long as they agreed on the magnitudes of the underlying stock's holding period returns(H*and H),they would agree on the price of the option
1096 The Journal of finance The example can be extended to a multiperiod framework in which the price of the underlying stock can take on only one of two values at any time t given the price of the stock at t-1. Consider the case in which a non-dividend paying stock's holding period return is 1. 175 in all up states and. 85 in all down states Given an initial stock price of $100, these return parameters imply the four-period price pattern shown in Figure 1 Assume that we wish to value a call option which matures at the end of peric and has an exercise price of $100. Given a riskless interest rate of 1.25%per period(5% per year, assuming a one-year maturity), the sequence of option valt corresponding to the stock prices in Figure 1 is given in Figure 2. In Figure 2 the prices $90.61 and $37. 89 are the values of the call obtainable by exercising at maturity. For those states at maturity where the price of the stock falls below the exercise price of $100, the option expires worthless. Each of the time 3 option prices is obtained from (4). Similarly, the prices at time 2, 1, and 0 are obtained by recursive application of (4)resulting in a current call option price f1441 99.88 99.75 100.00 85.00 99.75 72.16 61.41 Figure 1. Price Path of Underlying Stock