Find a probability measure $\mathbb{Q}$, such that $\tilde{S} = e^{-rt}S_t$ is an $\mathbb{F}$-martingale

Question

I guess it's in the question but I want to create a probability measure such that $\tilde{S} = e^{-rt}S_t$ is a $\mathbb{F}$-martingale, where $dS_t = \mu S_t dt + \sigma S_t dW_t$ so is a Black-Scholes Market. I understand what each bit is but my problem is in actually creating this probability measure. I know that $\tilde{S}$ is a discounted stock price but I don't really know where to go from there. Any help would be much appreciated, thanks.

Answer 1

The key to these sorts of problems is Girsanov's theorem, which states that if $Z_t = \exp\left(\int_0^t \theta_s dW_s - \frac 12 \int_0^t \theta_s^2ds\right)$ and $d\mathbb{Q} = Z_T d\mathbb{P}$, then $\tilde W_t = W_t - \int_0^t \theta_s ds$ is a Brownian motion under $\mathbb{Q}$.

Here, we compute $$d(e^{-rt}S_t) = e^{-rt}(dS_t -rS_t dt) = e^{-rt}S_t((\mu - r) dt + \sigma dW_t) = \sigma e^{-rt}S_t\left(\frac{\mu - r}{\sigma} dt + dW_t\right).$$ We want this to be a martingale under $\mathbb{Q}$, so we want $d\tilde W_t = dW_t + \frac{\mu - r}{\sigma}dt$. This suggests setting $\theta_t = -\frac{\mu - r}{\sigma}$ for all $t$ in Girsanov's theorem, i.e. set $Z_t := \exp\left(\frac{r-\mu}{\sigma} W_t - \frac 12 \left(\frac{r-\mu}{\sigma}\right)^2t\right)$ and $d\mathbb{Q}=Z_T d\mathbb{P}$. Then $d(e^{-rt}S_t) = \sigma e^{-rt}S_t d\tilde W_t$ is a martingale under $\mathbb{Q}$, so $\mathbb{Q}$ is the desired probability measure.