Consider a multi-period model with asset prices $\bar{S}_{t}=\left(S_{t}^{0}, \ldots, S_{t}^{d}\right), t \in\{0, \ldots, T\}$, modelled on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ and adapted to a filtration $\left(\mathcal{F}_{t}\right)_{t \in\{0, \ldots, T\}}$. Assume that $\mathcal{F}_{0}$ is $\mathbb{P}$-trivial, i.e. $\mathbb{P}[A] \in\{0,1\}$ for all $A \in \mathcal{F}_{0}$, and assume that $S^{0}$ is a $\mathbb{P}$-a.s. strictly positive numéraire, i.e. $S_{t}^{0}>0 \mathbb{P}$-a.s. for all $t \in\{0, \ldots, T\}$. Further, let $X_{t}=\left(X_{t}^{1}, \ldots, X_{t}^{d}\right)$ denote the discounted price process defined by $X_{t}^{i}:=S_{t}^{i} / S_{t}^{0}, t \in\{0, \ldots, T\}, i \in\{1, \ldots, d\}$.

(a) What does it mean to say that a self-financing strategy $\bar{\theta}$ is an arbitrage?

(b) State the fundamental theorem of asset pricing.

(c) Let $\mathbb{Q}$ be a probability measure on $(\Omega, \mathcal{F})$ which is equivalent to $\mathbb{P}$ and for which $\mathbb{E}_{\mathbb{Q}}\left[\left|X_{t}\right|\right]<\infty$ for all $t$. Show that the following are equivalent:

(i) $\mathbb{Q}$ is a martingale measure.

(ii) If $\bar{\theta}=\left(\theta^{0}, \theta\right)$ is self-financing and $\theta$ is bounded, i.e. $\max _{t=1, \ldots, T}\left|\theta_{t}\right| \leqslant c<\infty$ for a suitable $c>0$, then the value process $V$ of $\bar{\theta}$ is a $\mathbb{Q}$-martingale.

(iii) If $\bar{\theta}=\left(\theta^{0}, \theta\right)$ is self-financing and $\theta$ is bounded, then the value process $V$ of $\bar{\theta}$ satisfies

$$\mathbb{E}_{\mathbb{Q}}\left[V_{T}\right]=V_{0}$$

[Hint: To show that (iii) implies (i) you might find it useful to consider self-financing strategies $\bar{\theta}=\left(\theta^{0}, \theta\right)$ with $\theta$ of the form

$$\theta_{s}:= \begin{cases}\mathbf{1}_{A} & \text { if } s=t \\ 0 & \text { otherwise }\end{cases}$$

for any $A \in \mathcal{F}_{t-1}$ and any $t \in\{1, \ldots, T\}$.]

(d) Prove that if there exists a martingale measure $\mathbb{Q}$ satisfying the conditions in (c) then there is no arbitrage.