916 Irrelevance of a stockS e(r) Proof:( continued Lets call pu the probability of an increase in the stock price and p=1-pu the probability of a stock decrease S0△-f=[心(Su△-f)+p(S0d-f6)]eKT where k is the appropriate rate for the risk involved However, A is chosen such that SouA-f= SodA-fg and we know that p=1-pu Substituting S0△-f=[(Sou△-fu+(1-p)(S0U-f)]ekT (SouA-fye-rt as since this is risk-free, k=r No pus or pas left, thus probability of stock increase is irrelevant Options, Futures, and Other Derivatives, 4th edition@ 2000 by John C. Hull Tang Yincai, Shanghai Normal UniversityOptions, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, Shanghai Normal University 9.16 Irrelevance of a Stock’s E(R) Proof: (continued) • Let’s call pu the probability of an increase in the stock price and pd=1- pu the probability of a stock decrease S0D - f = [pu (S0uD-fu )+ pd (S0dD-fd )] e-kT where k is the appropriate rate for the risk involved • However, D is chosen such that S0uD-fu= S0dD-fd and we know that pd=1- pu • Substituting, S0D - f = [pu (S0uD-fu )+ (1- pu )(S0uD-fu )] e-kT = (S0uD-fu )e-rT as since this is risk-free, k = r • No pu ’s or pd ’s left, thus probability of stock increase is irrelevant •