Originally Posted by

**shilz222** Consider a derivative that pays off $\displaystyle S_{T}^{n} $ at time $\displaystyle T $, where $\displaystyle S_T $ is the stock price at that time. When the stock price follows geometric Brownian motion, it can be shown that is price at time $\displaystyle t $ ($\displaystyle t \leq T $) has the form $\displaystyle h(t,T)S^{n} $ where $\displaystyle S $ is the stock price at time $\displaystyle t $ and $\displaystyle h $ is a function only of $\displaystyle t $ and $\displaystyle T $.

(a) Derive and ordinary differential equation satisfied by $\displaystyle h(t,T) $. I am guessing that I should use Black-Scholes here?

(b) What is the boundary condition for the differential equation for $\displaystyle h(t,T) $?

(c) Show that $\displaystyle h(t,T) = e^{[0.5 \sigma^{2} n(n-1) +r(n-1)](T-t)} $ where $\displaystyle r $ is the risk free interest rate and $\displaystyle \sigma $ is the stock price volatility. This need part (a) right?