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

Abstract

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 quasi-variational inequality and it is C1 across the optimal stopping boundary. Our proof only uses the classical theory of parabolic partial differential equations of Friedman 2006 and does not use the theory of vicosity solutions, since our proof relies on constructing a sequence of functions, each of which is a value function of an optimal stopping time for a diffusion. The sequence is constructed by iterating a functional operator that maps a certain class of convex functions to smooth functions satisfying variational inequalities (or to value functions of optimal stopping problems involving only a diffusion). The approximating sequence converges to the value function exponentially fast, therefore it constitutes a good approximation scheme, since the optimal stopping problems for diffusions can be readily solved. Our technique also lets one see why the jump-diffusion control problems may be smoother than the control problems with piece-wise deterministic Markov processes: In the former case the sequence of functions that converge to the value function is a sequence of value function of control problems for diffusions, and in the latter case the converging sequence is a sequence of the value functions of deterministic optimal control problems. The first of these sequences is known to be smoother than the second one.