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 uses only the classical theory of parabolic partial differential equations of [A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, NJ, 1964] and [A. Friedman, Stochastic Differential Equations and Applications, Dover, Mineola, NY, 2006], is an alternative to the proof that uses the theory of viscosity solutions (see [H. Pham, Appl. Math. Optim., 35 (1997), pp. 145-164]). 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 of the American put option for the jump diffusion 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 hand, since the approximating sequence converges to the value function exponentially fast, it naturally leads to a good numerical scheme.