A recruiter asked me this question:
Suppose you have the following contract:
What is the price of such contract ?
I begin with $E[\left((-1)^{\tau}(S_{T}-K)\right)^{+}e^{-rT}]$ with $\tau$ a random variable that equals $+1$ if we change the call into a put and $-1$ if we don't change it, but i'm stuck with this...
Let $t=1$ and $T=2$. The value at time $t$ is given by \begin{align*} &\ e^{-r(T-t)}\max\left(E\left((S_T-K)^+\mid \mathcal{F}_{t}\right), \, E\left((K-S_T)^+\mid \mathcal{F}_{t}\right)\right) \\ =&\ e^{-r(T-t)}E\left((K-S_T)^+\mid \mathcal{F}_{t}\right) +e^{-r(T-t)}\max\left(E\left((S_T-K)\mid \mathcal{F}_{t}\right), \, 0\right)\\ =&\ e^{-r(T-t)}E\left((K-S_T)^+\mid \mathcal{F}_{t}\right) +\max\left(S_{t}-Ke^{-r(T-t)}, \, 0\right). \end{align*} That is, the value is for a portfolio with a put at $T$ and a call at $t$, and can be computed using formulas of Black-Scholes type.
Let's define $t=0$, $T_1 = 1$ and $T_2 = 2$.
I believe the interviewer is looking for the price of the "global" option $V_t$ for $t \leq T_1 \leq T_2 $.
Let's define the payoff at time $T_1$: it is the maximum between the value of a call or a put on the same underlying with maturity at $T_2$.
$$\text{Payoff}_{T_1} = \max( c_{T_1}, p_{T_1} )$$
where $c_{T_1}$ and $p_{T_1}$ are respectively the price at time $T_1$ of a call and a put under the same underlying with expiry at time $T_2$.
As noted by Gordon, we know that $\max( a, b ) = b + ( a - b )^+$, hence:
$$\text{Payoff}_{T_1} = p_{T_1} + ( c_{T_1} - p_{T_1} )^+$$
By the put call parity, we know that $c_{T_1} - p_{T_1} = S_{T_1} - e^{-r(T_2-T_1)}K$ and therefore we get:
$$\text{Payoff}_{T_1} = p_{T_1} + ( S_{T_1} - e^{-r(T_2-T_1)}K )^+$$.
Let's write the general statement: $$V_t = \mathbb{E}_Q \left[ e^{-r(T_1 - t)} \text{Payoff}_{T_1} | \mathcal{F}_t \right] = \mathbb{E}_Q \left[ e^{-r(T_1 - t)} \left( p_{T_1} + ( S_{T_1} - e^{-r(T_2-T_1)}K )^+ \right) | \mathcal{F}_t \right]$$
We can split the value into two main terms:
$$V_t = \mathbb{E}_Q \left[ e^{-r(T_1 - t)} p_{T_1} | \mathcal{F}_t \right] + \mathbb{E}_Q \left[ e^{-r(T_1 - t)} ( S_{T_1} - e^{-r(T_2-T_1)}K )^+ | \mathcal{F}_t \right]$$
We see that the second term is simply the price at time $t$ of a call option on $S$ expiring at $T_1$ with strike $K' = e^{-r(T_2-T_1)}K$.
Furthermore, we know that:
$$ p_{T_1} = \mathbb{E}_Q \left[ e^{-r(T_2 - T_1)} (K - S_{T_2}) | \mathcal{F}_{T_1} \right]$$
So the first term can be seen as:
$$\mathbb{E}_Q \left[ e^{-r(T_1 - t)} \mathbb{E}_Q \left[ e^{-r(T_2 - T_1)} (K - S_{T_2})^+ | \mathcal{F}_{T_1} \right] | \mathcal{F}_t \right] = \mathbb{E}_Q \left[ e^{-r(T_1 - t)} e^{-r(T_2 - T_1)} (K - S_{T_2})^+ | \mathcal{F}_t \right] = \mathbb{E}_Q \left[ e^{-r(T_2 - t)} (K - S_{T_2})^+ | \mathcal{F}_t \right] $$
because of the iterated conditoning rule and since $\mathcal{F}_t \subseteq \mathcal{F}_{T_1}$.
This second term is the value at time $t$ of a put expiring at time $T_2$ with strike $K$.
So the answer is that at time $t$ the value of the option is the value of the put plus the value of a call expiring at $T_1$ with strike $K' = e^{-r(T_2-T_1)}K$.
