GBM and Derivatives

sreeharimenon · November 28th, 2011, 5:28 pm

Hi AllHow does one proceed on thisA stock price follows the geometric brownian motion: dSt =mu Stdt+sigmaStdWt. Let r be the continously compounded risk-free rate and D(t; St) = St[ln St +(r+sigma^2/2)(T-t)]Prove that D(t; St) is the price at time t of a derivative security with a maturity payoff at time T of f(ST) =S(T) ln S(T) .Prove that this derivative is riskier than the underlying stock.RegardsSreehari Menon

Alan · November 29th, 2011, 1:07 am

Well, since this is homework, here is just a hint on the first part.You want to evaluate e^(-r T) E[f(ST)] = e^(-r T) int x e^x e^{-(x-m)^2/(2 sig^2 T}} dx, using x = log ST,where m = r T - (1/2) sig^2 T + log S0There is a formula for doing Gaussian integrals that all quants should memorize, namely X ~ N(m,sig^2) => E[e^(a X)] = e^(a m + (1/2) a^2 sig^2). A clever use of that will get you your answer without much real work.

secret2 · November 29th, 2011, 1:50 am

Interesting, I was thinking along the line of pricing it with other numeraire.

pgarciajaramillo · May 19th, 2013, 3:48 pm