The double call option can be exercised at time t with strike price k, or at time T with strike price K and T>t. The risk free interest rate is r. Show that it is not optimal to exercise at t if ke^(-rt) > Ke^(-rT).
There is a hint: compare exercising the call at t with selling short.
My answer:
Exercising at T gives a return of min{ (S(T)-K) , 0 }
Selling short: borrow a share at time t and sell it for S(t) and put this money in a bank. At time T, our return is e^r(T-t) * S(t) + min{ (S(T)-K , 0 } -S(T).
So it is not optimal to exercise at t if
min{ (S(T)-K) , 0 } < e^r(T-t) * S(t) + min{ (S(T)-K , 0 } -S(T)
=> ... => S(T)e^-rT < S(t)e^-rt
This is the right answer with S(T) instead of K and S(t) instead of k. I don't understand how to get to K and k from here.