Question
Consider the log normal model: S(T) = S(0)e^(mu)T + (sigma)*sqrt(T)*Z , where Z~N(0,1), with drift parameter mu=0.02 and volatility parameter sigma=0.2. If So = 100, find E[S(5)] and P(S(5)>100)
Explanation / Answer
Pr[ X+Y Y ~ Ga(n/2, 1/2) ==> get constant c_n for R_n ~ c_n in (*) If X_n ~ No(mu, sig^2) then SX_n = X_1 + X_2 + ... + X_n ~ No(n mu, n sig^2) X-bar = (SX_n)/n ~ No(mu, sig^2/n ) Mean = mu Variance = sig^2/n -> 0 (X_1-mu)^2 + (X_2-mu)^2 + ... + (X_n-mu)^2 ~ Ga(n/2, 1/2 sig^2) (1/n) * (") ~ Ga(n/2, n/2 sig^2), Mean = sig^2 Variance = 2sig^4 / n -> 0 Alas we don't know mu... but we can ESTIMATE it (by x-bar), and get (X_1 - Xbar)^2 + ... + (X_n - Xbar)^2 ~ Ga( (n-1)/2, 1/2 sig^2) 1/(n-1) * " has mean sig^2, variance 2*sig^4/(n-1) -> 0. Whee!