as long as the interest rate is not negative (i.e, r>1), premature ex- ercise is not optimal. To see this, suppose the worst, that us>& and dS < K. Then, if he does not exercise imnediately, C= P max[0, uS-Kl +(1-p)max[0, ds-KJ)+r (uS-k) Our proble the money he would receive if he exercIsed immediatel a little algebra easily confirms this inequality. Since it is then not optimal to exercise the call, formula (1)is unambiguously its correct value one to expiration. This formula has a number of notable features First, the only assumption mposed on investor behavior is the otivation to climinate all opportunities for profitable riskless arbitray For example, il vestors can be risk-averse or risk-preferring, and we would derive the same formula. The assumption that no profitable riskless arbitrage oppor- tunities exist is particularly acceptable from a practical point of view, since we gain whether it is true or false. If it is true, we have ex- plained call prices f it is false, we will be the first to take advan tage of its falsity by transacting in the market Second, probability q does not appear in the Formula. This means, surprisingly, that even if different investors have different sub ective probabilities about an upward or downward movement in the stock
price, they would still agree on the relationship of c to s andI This can be understood if it is remembered that the formula is only a relative pricing relationship between c on one side and s and r on the other. The probability. q will affect the values of s and r and only transmitted through them, will indirectly affect the value of c Third, P E (r-d)/(u-d)has all the properties of a probability measure: that is 0<p< 1 lowever, p is generally not a subiective probability (i.e, a probability in an investor 's mind). Indeed, to re- quire p to be a subjective probability would unnecessarily restrict the context of the formula The current call value would then be equal to its expected future value discounted at the riskless rate with no adjust ment for risk. Only if investors were risk-neutral or the risk of the call costlessly eliminated by diversification would such an interpretation be reasonable. Nonetheless, and this will be important subsequently, the same formula for c in terms of S,K, u, d, and r holds even if we in- terpret p as a subjective probability. That 1s, had we made this inter- pretation initially, we would not have been misled, since we would have derived the same formula Even in a risk-averse or risk-preference envi- ronment, the relationship of c to S, K,u, d, and r would be deter- mined as if investors were risk-neutral. Incidentally, if investors were risk-neutral, then q =p. To see this, since then q(us)+(1-g)(dS)=rS then (r-d)/(u-d)
Finally, the only random variable the call value depends upon is the stock price itself. In particular, it does not depend, in addi- tion, on the random prices of other securities or portfolios, such as the "market" portfolio containing all securities in the economy. Indeed, if c depended separately on some other random variable, since its price would then be different than formula (1), we know from the derivation this formula that profitable riskless arbitrage would be possible In reality, we can sell or exercise a call at many dates prior to its expiration. Stepping backward one morc period, we will now examine what happens to the call value two periods before expiration. In keeping with our binomial stochastic process, for the stock: 2>dus In other words, s follows a stationary random binomial process with step size u or d with probability q or 1-q, respectively. Addition- ally, we assume one pIts the interest rate r is stationary period to period. Then, for the call:
C.三max[o,us-K x[O, duS-KJ [0,d2s From our previous analysis with one period left, applying formula (1 +(1-p)c,/rand d= pc, +(1-p)C,1/r Again we construct a riskless hedge at the beginning of the first period by choosing a so that, investing as-C, we are certain to receive ads-c d aus-Cu Following the same reasoning as before, to prevent profitable riskless arbitrage, again C= [pC +(1-P)C,J/r, Substitut ing in the above equations for and C C=[pC+2p(1-p) Note that C, =c,. Again, we can use this and compare it to s-x to d show that the call wilI not be exercised prematurely. Moreover, all other observations made about formula (1) also apply to formula (2),'except that the f periods n remaining to expiration now emerges clearly:as an additional determinant of the call value. For formula (2),n= 2 That is, the full list of varlables determining c sS, h, K, u, d, d
13 Working backward in time deductively, we can write down the general pricing formula for any n 1!(n-j) (1-p) x[O K 0 S⊥ ds s<u ds< ing the opt⊥onw11not expire in-the-money for sure, or out-of-the-money for sure, there must exist an integer 0< a <n, such that <K<u s For all j <a, max[O,uds-x]=0 and for all j>a, ax[o,u jan)s-K] K a e can solve for a by taking natural logarithms cf the above inequality a is an integer such that In(u/d)<a<1+In(K/sd") In(K/sd Breaki up c into two terms C=s j!(a-j)!