1. ## Derivatives and Securities

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?

2. 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?
Which Black-Scholes equation would you be using here? The stochastic difeq that is used for geometric Brownian motion is: $\displaystyle \frac{dS(t)}{S(t)}=\alpha dt+\sigma dZ(t)$ where $\displaystyle \alpha$ is the drift factor and $\displaystyle \sigma$ is the volatility or variance factor. That exponential function is derived from the differential equation.