Friday, October 4, 2019

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[((1)τ(STK))+erT] 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  er(Tt)max(E((STK)+Ft),E((KST)+Ft))= er(Tt)E((KST)+Ft)+er(Tt)max(E((STK)Ft),0)= er(Tt)E((KST)+Ft)+max(StKer(Tt),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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