A pr 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. This sequence, which converges to the value function of the American put option for jump diffusions, is constructed sequentially using a functional operator that maps a certain class of convex functions to smooth functions satisfying some quasi-variational 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.