14 Now, the latter bracketed expression is the complementary binomial tribution function B[a; n, P]. The first bracketed expression can also be interpreted as a complementary binomial distribution function bla; n, p'],where P’三(u/x)pand1-p's(d/r)(1-p) P is a probability measure, since 0< p<1. To see this a-py32:[=[生a-]-pap n-了 In summary: BINOMIAL OPTION PRICING FORMULA C= SB[a;n, p']-kr B[ain, pI where P= (r-d)/(u-d) and p's(u/r)p ln(K/Sd") a<1+ ln(u/d) where a is an integer. to be interpreted necessarily as a subjective probability, asure and not Caution: Despite this, p is only a probability me
Before analyzing this formula in detail, we should mention an al- ternative shortcut derivation. Suppose the call value c depends on the concurrent stock price s. and the number of periods n remaining to expiration. We express the call value as c(s n). As before,we can set up a riskless hedge and derive c(s, n)=[pc(us, n-1)+(1-p)(ds, n-1)]=I, where, at n=0, C(S,, 0)=max[0, Sn-K]. Since this relationship doe not involve investor attitudes toward risk, it must be true regardless of what attitudes we assume. For computational purposes, the most con- venient choice is risk neutrality. In this case, present values are just expected future values discounted back to the present at the riskless in- terest rate. since the stock is assumed not to pay dividends, it would never be optimal to exercise the call before its expiration date There- fore, with risk neutrality, its present value must be its expected value on its expiration date discounted back to the present. Recall from our previous discussion, 5oud-is and, with risk neutrality, q Since Etmax[0,So-kl) we can then derive equation (3). The proof then proceeds as before For some readers, an alternative complete markets interpreta- tion of our binomial approach may be instructive, Suppose that d epresent the state-contingent discount rates to states u and d