Black Scholes/American Put/Martingale Condition

Question

Consider a Black Scholes model with $r \geq 0$. Show that the price of an American Put Option with maturity $T > 0$ is bounded by $\frac{K}{1 + \alpha} {(\frac{\alpha K}{1 + \alpha})}^{\alpha}{S_{0}^{-\alpha}}$. Hint: find $\alpha > 0$ such that $e^{-rt}S_{t}^{-\alpha}$ is a martingale.

I am trying to work through the hint and when rewriting $e^{-rt}S_{t}^{-\alpha}$ I get $e^{-rt}S_{t}^{-\alpha} = exp(-\alpha \sigma W_t - t(r + \alpha (r-\frac{{\sigma}^2}{2})))$

I know that in case of the GBM the drift has to equal zero for it to be a martingale, which property has to be fulfilled here?

Edit: the condition that my professor wrote:

$r + \alpha (r-\frac{{\sigma}^2}{2}) = \frac{\alpha^2 \sigma^2}{2}$

which leads to a quadratic equation for $\alpha$, but O do not quite understand where is equation comes from.

Answer 1

Too long for a comment.

\begin{align} S_t&=S_0\,e^{rt+\sigma W_t-\sigma^2t/2}\,,&S_t^{-\alpha}=S_0^{-\alpha}\,e^{-\alpha\, r\,t\,-\,\alpha\,\sigma\, W_t\,+\,\alpha\,\sigma^2\,t/2}\,, \end{align} \begin{align} e^{-rt}S_t^{-\alpha}&=S_0^{-\alpha}e^{-(1+\alpha)\,r\,t\,-\,\alpha\,\sigma\, W_t\,+\,\alpha\,\sigma^2\,t/2}\\ &=S_0^{-\alpha}e^{-(1+\alpha)\,r\,t\,+\,(\alpha^2+\alpha)\,\sigma^2\,t/2}\,\underbrace{e^{-\,\alpha\,\sigma\, W_t\,-\,\alpha^2\sigma^2\,t/2}}_{\text{martingale}}\,. \end{align} For the whole expression to be a martingale we must have (your professor's quadratic equation): \begin{align} -(1+\alpha)r+(\alpha^2+\alpha)\sigma^2/2=0\,. \end{align} One solution of that is $\alpha=-1$ but your professor wants a positive solution. This is \begin{align} \alpha&=\frac{-\frac{\sigma^2}{2}+r+\sqrt{(\frac{\sigma^2}{2}-r)^2+2\sigma^2r}}{\sigma^2}\\ &=\frac{-\frac{\sigma^2}{2}+r+\frac{\sigma^2}{2}+r}{\sigma^2}=\frac{2r}{\sigma^2}\,. \end{align}