How to price an option allowing to change a call into a put?

A recruiter asked me this question:

Suppose you have the following contract:

• a call option with maturity $T$ = 2 years


• the possibility to change this call into a put at $t$ = 1 year

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...

Answer

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.