A Proof of the Smoothness of the Finite Time Horizon American Put Option for Jump Diffusions

We give a new proof of the fact that the value function of the finite time horizon American put option for a jump diffusion, when the jumps are from a compound Poisson process, is the classical solution of a free boundary equation. We also show that the value function is $C^1$ across the optimal stopping boundary. Our proof, which only uses the classical theory of parabolic partial differential equations of [7,8], is an alternative to the proof that uses the the theory of vicosity solutions [14]. This new proof relies on constructing a monotonous sequence of functions, each of which is a value function of an optimal stopping problem for a geometric Brownian motion, converging to the value function uniformly and exponentially fast. This sequence is constructed by iterating a functional operator that maps a certain class of convex functions to classical solutions of corresponding free boundary equations. On the other handsince the approximating sequence converges to the value function exponentially fast, it naturally leads to a good numerical scheme. We also show that the assumption that [14] makes on the parameters of the problem, in order to guarantee that the value function is the \emph{unique} classical solution of the corresponding free boundary equation, can be dropped.