(i) What does it mean to say that the process $\left(W_{t}\right)_{t \geqslant 0}$ is a Brownian motion? What does it mean to say that the process $\left(M_{t}\right)_{t \geqslant 0}$ is a martingale?

Suppose that $\left(W_{t}\right)_{t \geqslant 0}$ is a Brownian motion and the process $\left(X_{t}\right)_{t \geqslant 0}$ is given in terms of $W$ as

$$X_{t}=x_{0}+\sigma W_{t}+\mu t$$

for constants $\sigma, \mu$. For what values of $\theta$ is the process

$$M_{t}=\exp \left(\theta X_{t}-\lambda t\right)$$

a martingale? (Here, $\lambda$ is a positive constant.)

(ii) In a standard Black-Scholes model, the price at time $t$ of a share is represented as $S_{t}=\exp \left(X_{t}\right)$. You hold a perpetual American put option on this share, with strike $K$; you may exercise at any stopping time $\tau$, and upon exercise you receive $\max \left\{0, K-S_{\tau}\right\}$. Let $0<a<\log K$. Suppose you plan to use the exercise policy: 'Exercise as soon as the price falls to $e^{a}$ or lower.' Calculate what the option would be worth if you were to follow this policy. (Assume that the riskless rate of interest is constant and equal to $r>0$.) For what choice of $a$ is this value maximised?