Exercise 5.10 (Chooser option). Consider a model with a unique riskneutral measure $\widetilde{\mathbb{P}}$ and constant interest rate $r$. According to the risk-neutral pricing formula, for $0 \leq t \leq T$, the price at time $t$ of a European call expiring at time $T$ is
$$
C(t)=\tilde{\mathbb{E}}\left[e^{-r(T-t)}(S(T)-K)^{+} \mid \mathcal{F}(t)\right],
$$
where $S(T)$ is the underlying asset price at time $T$ and $K$ is the strike price of the call. Similarly, the price at time $t$ of a European put expiring at time $T$ is
$$
P(t)=\tilde{\mathbb{E}}\left[e^{-r(T-t)}(K-S(T))^{+} \mid \mathcal{F}(t)\right] .
$$
Finally, because $e^{-r t} S(t)$ is a martingale under $\tilde{\mathbb{P}}$, the price at time $t$ of a forward contract for delivery of one share of stock at time $T$ in exchange for a payment of $K$ at time $T$ is
$$
\begin{aligned}
F(t) & =\tilde{\mathbb{E}}\left[e^{-r(T-t)}(S(T)-K) \mid \mathcal{F}(t)\right] \\
& =e^{r t} \widetilde{\mathbb{E}}\left[e^{-r T} S(T) \mid \mathcal{F}(t)\right]-e^{-r(T-t)} K \\
& =S(t)-e^{-r(T-t)} K .
\end{aligned}
$$
Because
$$
(S(T)-K)^{+}-(K-S(T))^{+}=S(T)-K,
$$
we have the put-call parity relationship
$$
\begin{aligned}
C(t)-P(t) & =\tilde{\mathbb{E}}\left[e^{-r(T-t)}(S(T)-K)^{+}-e^{-r(T-t)}(K-S(T))^{+} \mid \mathcal{F}(t)\right] \\
& =\widetilde{\mathbb{E}}\left[e^{-r(T-t)}(S(T)-K) \mid \mathcal{F}(t)\right]=F(t) .
\end{aligned}
$$
Now consider a date $t_0$ between 0 and $T$, and consider a chooser option, which gives the right at time $t_0$ to choose to own either the call or the put.
(i) Show that at time $t_0$ the value of the chooser option is
$$
C\left(t_0\right)+\max \left\{0,-F\left(t_0\right)\right\}=C\left(t_0\right)+\left(e^{-r\left(T-t_0\right)} K-S\left(t_0\right)\right)^{+} .
$$
(ii) Show that the value of the chooser option at time 0 is the sum of the value of a call expiring at time $T$ with strike price $K$ and the value of a put expiring at time $t_0$ with strike price $e^{-r\left(T-t_0\right)} K$.