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[((−1)τ(ST−K))+e−rT] with τ 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 e−r(T−t)max(E((ST−K)+∣Ft),E((K−ST)+∣Ft))= e−r(T−t)E((K−ST)+∣Ft)+e−r(T−t)max(E((ST−K)∣Ft),0)= e−r(T−t)E((K−ST)+∣Ft)+max(St−Ke−r(T−t),0).
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.
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