The perpetual American put option in jump-to-default models

Abstract

We study the perpetual American put option in a general jump-to-default model, deriving an explicit expression for the price of the option. We find that in some cases the optimal stopping boundary vanishes and thus it is not optimal to exercise the option before default occurs. Precise conditions for when this situation arises are given. Furthermore we present a necessary and sufficient condition for convexity of the option price, and also show that a nonincreasing intensity is sufficient, but not necessary, to have convexity. From this we also get conditions for when option prices are monotone in the model parameters.