(a) In the context of a multi-period model in discrete time, what does it mean to say that a probability measure is an equivalent martingale measure?

(b) State the fundamental theorem of asset pricing.

(c) Consider a single-period model with one risky asset $S^{1}$ having initial price $S_{0}^{1}=1$. At time 1 its value $S_{1}^{1}$ is a random variable on $(\Omega, \mathcal{F}, \mathbb{P})$ of the form

$$S_{1}^{1}=\exp (\sigma Z+m), \quad m \in \mathbb{R}, \sigma>0,$$

where $Z \sim \mathcal{N}(0,1)$. Assume that there is a riskless numéraire $S^{0}$ with $S_{0}^{0}=S_{1}^{0}=1$. Show that there is no arbitrage in this model.

[Hint: You may find it useful to consider a density of the form $\exp (\tilde{\sigma} Z+\tilde{m})$ and find suitable $\tilde{m}$ and $\tilde{\sigma}$. You may use without proof that if $X$ is a normal random variable then $\mathbb{E}\left(e^{X}\right)=\exp \left(\mathbb{E}(X)+\frac{1}{2} \operatorname{Var}(X)\right)$.]

(d) Now consider a multi-period model with one risky asset $S^{1}$ having a non-random initial price $S_{0}^{1}=1$ and a price process $\left(S_{t}^{1}\right)_{t \in\{0, \ldots, T\}}$ of the form

$$S_{t}^{1}=\prod_{i=1}^{t} \exp \left(\sigma_{i} Z_{i}+m_{i}\right), \quad m_{i} \in \mathbb{R}, \sigma_{i}>0$$

where $Z_{i}$ are i.i.d. $\mathcal{N}(0,1)$-distributed random variables on $(\Omega, \mathcal{F}, \mathbb{P})$. Assume that there is a constant riskless numéraire $S^{0}$ with $S_{t}^{0}=1$ for all $t \in\{0, \ldots, T\}$. Show that there exists no arbitrage in this model.