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
Two-State Option Pricing Although the above example considers only four periods of time, one can lways choose an interval of time to recogmize price changes that more realistically tures expected stock price behavior. In Section Iv we demonstrate the sensitivity of option prices to the choice of the time differencing interval under the assumption that H and h" are chosen to hold the mean and variance of the distribution of stock price changes constant over the life of the option. In the Appendix, a generalized formula for the multiperiod case is derived for the situation where R, H and H- are constant. This formula is extended under the assumption that the two-state process evolves over an infinitesimally small interval of time u, Operationalizing the TSOPM In the TSoPM, the only parameters describing the probability distribution of returns of the underlying stock are the magnitudes of the holding period return, H and H. Although our examples assume that H+ and H remain constant l8.71 24.6 37,89 9.24 37.89 18.71 0 Figure 2. Price Path of European Call Option
The Journal of finance through time, this is not a necessary assumption for the implementation of the model. Thus, if one can simply specify the pattern of H* and H"through time it is possible to value the option The T'SoPM can be used as a method for obtaining exact values of opti hen the magnitudes of h* and h- are known in advance. As a practical matter, the values of H* and H will not be known, but must be estimated. For example if the probabilities associated with the occurrence of the + and -states remain stable over time along with the magnitudes of H* and H, then the two-state model implies a binomial distribution for the returns of the stock. It is well known that both the Normal and Poisson distributions can be viewed as limiting cases of the Binomial. Thus, the Binomial distribution can be employed as an approx imation procedure for deriving option prices when the actual distribution of returns is assumed to be either Normal or Poisson We will illustrate how the values of H and H can be determined when the binomial distribution is used as an approximation to the lognormal distribution If the magnitudes of the relative price changes in our model and their associated probabilities remain stable from one period to the next, then the distribution of returns which is generated after T time periods will follow a log-binomial distribution with a mean μ=T[h6+h(1-的]=T(h-h)日+h], and variance T(h-h)26(1-的), here 6= the probability that the price of the stock will rise in any period, h=ln(H+) h=In(h) and u for the entire four periods would be 324 and -003, respectively, if a f a In the last four-period example where H'= 1.175 and H-"=85, the value of of equal to 5 is assumed It is also possible to determine the values of H*and H that are implied by the values of 4, 0, 0, and T. By solving (8)and (9) in terms of these parameters and recognizing that H exp(h), we obtain the following implied values of H+ and H-exuT+T√ (10) H=exy{/T-(a/m)V(1-6) (11) As T becomes large, the log-binomial distribution will approximate a lognormal distribution with the same mean and variance
Two-State Option Pricing 1099 IV. Applications of the Model European Puts and Calls on Non Dividend Paying Stocks In this section we price European put and call options on non-dividend paying stocks using the two-state model as an approximation procedure for the case in which stock prices are assumed to follow a lognormal distribution. Given the assumptions of no dividends and lognormal returns, the Black-Scholes[3] model provides the exact values for both types of options, thereby serving as a bench mark to assess the accuracy of the two-state model as a numerical procedure In Table l, we present prices of one-year European put and call options with exercise prices of $75, $100, and $125 assuming a current stock price of $100. The riskless interest rate is assumed to be 5% per year. To conform with the black- Scholes model, continuous compounding of interest is assumed. Thus, R=e5/N 1, where N is the number of time intervals per year employed in the analysis The values of h+ and H- are chosen so that the annual standard deviation of the logarithmic return is, 324 as in the previous four-period example. The expected value of the logarithmic return is assumed to take on values of. 5,. 1, 0, -1 and 5 per year, and a value of 8 equal to 5 is assumed. Finally, option pric ces are calculated by partitioning the year into 12, 52, and 100 time periods Consider the panel of Table 1 in which the stock' s growth rate (u)is assumed to be 0%. When the year is divided into 100 time intervals, the two-state prices of the put and call options are quite close to their corresponding Black-Scholes prices. With these two parameters (u =0, T=100), the greatest absolute percentage difference between the black-Scholes and two-state prices is. 6% Even if only 12 time differencing intervals are assumed, the two-state and black Scholes prices are remarkably close For growth rates of 10% and-10%, the two-state prices do not appear to be significantly different from those obtained when a zero growth rate is assumed Thus, within this range of growth rates, the option price does not appear to be significantly affected by the growth rate If extreme growth rates are assumed (u= 5 and 4=-.5), the two-state model does not appear to provide an accurate approximation to the Black-Scholes price for low T values. However for 100 time intervals, the two-state and Black-scholes prices are reasonably close The entries in Table 1 reveal that the option price is slightly dependent upon the stock's growth rate. In addition, if 0 were varied we would also discover a slight dependence on investor probability beliefs. These findings seem to contra dict the earlier observation that two state prices are independent of both investor preferences(which would be revealed through p) and probability beliefs This dependence results from the fact that h* and H- are chosen in the two state model to conform with a given continuous distribution Since the two-stat model is only an approximation, the values of u and e implicit in the continuous distribution may be reflected in the two-state solution. In the limit as T-o, the two distributions will be identical, and therefore, preferences and probabilities 2 See Brennan and Schwartz [4] and Parkinson [7] for descriptions of alternative numerical procedures for solving option pncing equations
Comparison Between TSOPM and Black-Scholes Option Prices Black-Schole TSOPM 57122994 26-38.7 5230.57 0030651 1001214.239.3615.1410,26 621491007 0015.0310.l6 60524966582548-79 638 26.28 65025,41 1751230.58 I939074 5230.742.09 10030.72 10012153710.50151410.26+1.5+23 10015.1710.29 125126.78256965825.48+3. 的7545 572547 0751230852.1930.742.08 0 10030.752.10 0 100121488100015.141026-1.7-2.5 215081020 10015,1110.23 1251265525466.582548-3 255 1230601943074208 15 104315.1410.26+1.1 15 4l1 15.16 1028 1251265525.436.582548 526622548 10066025.43 1230251.5930.74208 16 3065200 3069 1385 89715.1410.26 8.6-126 521487 10o0 1499 10.12 25124.6523.556.582548 S,= 100,R=eo/N-1, a=324, 8=.5. In this table, N=T in all cases puted according to: (TSOPM-BS)/BS, rounded to the nearest one-tenth of on l10
Two-State Option Pricing should not be reflected in option prices. In the appendix, we derive the Black holes equation using the two-state model. As expected, neither the growth rate nor probabilities enter the final solution. For practical applications, the two-state model appears to provide an accurate approximation to the black-Scholes model if 100 or more time differencing intervals are assumed along with any reasonable growth rate. As we show below, however, it is possible to select a growth rate that will closely approximate the value of u that minimizes the error in the two-state Finding the Best approximation According to equation(A. 11)in the Appendix, the price of a call option in the wo-state model can be stated in terms of two binomial pseudo probability distributions, In each distribution, y and are the pseudo probabilities that the price of the underlying stock will rise. These pseudo probabilities are not neces- sarily equal to the true probability, 0, but nevertheless, the mathematics of probability theory are still applicable According to the Laplace- DeMoivre Limit Theorem, it can be shown that the best fit between the binomial and normal distributions occurs when the binomial probability (or pseudo probability in this caseis -. As a general rule and will not be identical. Therefore it will usually be impossible to simultaneously set both pseudo probabilities to However, since y=p(H/(1+R)), and the term in parenthesis will generally be close to unity, the parameters of the underlying distribution that sets o to will set y to approximately By expanding o in Taylor's series, we find that o is approximately when / If the true probability, 8, is this expression simplifies to (13) For the parameters underlying Table 1, we find that(approximately) the best two-state approximation occurs when A =-.002488. The reasonableness of this result is confirmed by the u=0 panel of Table 1. We wish to acknowledge the referee for suggesting that the best approximation would occur We repeated the analysis of Table 1 by setting g to-002488. Although the prices were almost identical to those abtained by setting u to zero they were slightly more accurate