Call option on a put option ($[p(T_1,S_{T_1};K_2,T_2)-K_1{]}^{+}$)

This compound European contingent claim gives the right to buy a later-expiring put option at time $T_1$ for $K_1$. Its payoff is the positive part of the put value minus this exercise cost. Under the Black-Scholes model the put value decreases strictly with the stock spot, so a nondegenerate positive exercise cost gives a unique upper spot threshold for exercise. With zero exercise cost, the compound claim has exactly the underlying put's price by the tower property.