(I tried posting the below last night but it's not showing) Hi, I am having trouble understanding how we get from E[exp(sigma B_t)] to exp(0.5 sigma^2 t) using the MGF on pg 11 of the tables. If we compare exp(sigma B_t) to M(t) do we get: exp(sigma B_t) to equal exp(0.5 sigma^2 t^2?)? mu=0 but how we get to the end result puzzles me! Could someone please help? Thanks!
M(t) = E[exp(tX)] where X is a Normally distributed variable and t is a const. Here compare t to sigma a const. Bt is you N(0,1) variable Now M(t) = exp(mean.t + 0.5var.t^2) Here mean = 0 Var = 1 t = sigma => E[exp(sigma.Bt)] = exp(0.sigma + 0.5 x 1 x sigma^2) = exp(0.5sigma^2)
