Price of European Call Option

Question

A ECC called $X$. The buyer of $X$ has, at any time $t$, the option to receive a European Call option, $C$ or to receive an European Put option, $P$. Both with the same maturity $T$ and strike price $K$. The market is a multi-period binomial model.

At what price should $X$ be sold?

I'm struggling to answer this question.

I think I should use that the buyer of $X$ will only choose the call optin iff $C_t \geq P_t$ in order to compute the payoff $X_T$. But I don't know how to proceed.

Thanks for any help.

Answer 1

The payoff at $T$ equals the $C_T+P_T$. Hence you can perfectly hedge $X$ by buying a call and a put option. Hence the price has to be the sum of the price of the put option and the price of the call option. You would never exercise $X$ early.