Pricing Of The American Put Under Lévy Processes

We consider the American put with finite time horizonT, assuming that, under an EMM chosen by the market, the stock returns follow a regular Lévy process of exponential type. We formulate the free boundary value problem for the price of the American put, and develop the non-Gaussian analog of the method of lines and Carr's randomization method used in the Gaussian option pricing theory. The result is the (discretized) early exercise boundary and prices of the American put for all strikes and maturities from 0 toT. In the case of exponential jump-diffusion processes, a simple efficient pricing scheme is constructed. We show that for many classes of Lévy processes, the early exercise boundary is separated from the strike price by a non-vanishing margin on the interval[0, T), and that as the riskless rate vanishes, the optimal exercise price goes to zero uniformly over the interval[0, T), which is in the stark contrast with the Gaussian case.