1. Derivatives and Securities

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

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

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

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

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

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

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

(c) Show that $h(t,T) = e^{[0.5 \sigma^{2} n(n-1) +r(n-1)](T-t)}$ where $r$ is the risk free interest rate and $\sigma$ is the stock price volatility. This need part (a) right?
Which Black-Scholes equation would you be using here? The stochastic difeq that is used for geometric Brownian motion is: $\frac{dS(t)}{S(t)}=\alpha dt+\sigma dZ(t)$ where $\alpha$ is the drift factor and $\sigma$ is the volatility or variance factor. That exponential function is derived from the differential equation.