Given the time decay and the fact that there is no price scale other than the stock price, I will guess thesolution is the trivial one: exercise immediately. p.s. Proof.At time-0, with time T to expiration, and stock price S0, the critical exercise price must be Sc = f S0, withf dimensionless. So f = f(mu T, sig^2 T). Suppose S0=100 and f <> 1. For example, say f = 1.12Then Sc = 1.12 x 100 = 112. But if, instead, S0=112, then Sc = f 112 = 1.12 x 112 > 112. So we havenonsense unless (i) f=1, or (ii) f = 0 or infinity. But case (ii) means never exercise, which is clearlywrong since you would end up with 0. So the solution is f=1, which means "exercise immediately".Then, the value of the derivative is the same as the value of the ATM option you receive from exercising. To make the problem non-trivial, you need another price scale K.
Last edited by Alan
on February 23rd, 2011, 11:00 pm, edited 1 time in total.