1 9 M ar 2 00 7 Remarks on the American Put Option for Jump Diffusions ∗ †

We prove that the perpetual American put option price of an exponential Lévy process whose jumps come from a compound Poisson process is the classical solution of its associated quasi-variational inequality, that it is C except at the stopping boundary and that it is C everywhere (i.e. the smooth pasting condition always holds). We prove this fact by constructing a sequence of functions, each of which is a value function of an optimal stopping problem for a diffusion. The sequence is constructed sequentially using a functional operator that maps a certain class of convex functions to smooth functions satisfying some quasivariational inequalities. This sequence converges to the value function of the American put option uniformly and exponentially fast, therefore it provides a good approximation scheme. In fact, the value of the American put option is the fixed point of the functional operator we use.