(a) State the fundamental theorem of asset pricing for a multi-period model.

Consider a market model in which there is no arbitrage, the prices for all European put and call options are already known and there is a riskless asset $S^{0}=\left(S_{t}^{0}\right)_{t \in\{0, \ldots, T\}}$ with $S_{t}^{0}=(1+r)^{t}$ for some $r \geqslant 0$. The holder of a so-called 'chooser option' $C\left(K, t_{0}, T\right)$ has the right to choose at a preassigned time $t_{0} \in\{0,1, \ldots, T\}$ between a European call and a European put option on the same asset $S^{1}$, both with the same strike price $K$ and the same maturity $T$. [We assume that at time $t_{0}$ the holder will take the option having the higher price at that time.]

(b) Show that the payoff function of the chooser option is given by

$$C\left(K, t_{0}, T\right)= \begin{cases}\left(S_{T}^{1}-K\right)^{+} & \text {if } S_{t_{0}}^{1}>K(1+r)^{t_{0}-T} \\ \left(K-S_{T}^{1}\right)^{+} & \text {otherwise }\end{cases}$$

(c) Show that the price $\pi\left(C\left(K, t_{0}, T\right)\right)$ of the chooser option $C\left(K, t_{0}, T\right)$ is given by

$$\pi\left(C\left(K, t_{0}, T\right)\right)=\pi(E C(K, T))+\pi\left(E P\left(K(1+r)^{t_{0}-T}, t_{0}\right)\right),$$

where $\pi(E C(K, T))$ and $\pi(E P(K, T))$ denote the price of a European call and put option, respectively, with strike $K$ and maturity $T$.