C.W.: On American options under the Variance Gamma process

ABSTRACT American options are considered in a market where the underlying asset follows a Variance Gamma process. A sufficient condition is given for the failure of the smooth fit principle for finite horizon call options. A second-order accurate finite-difference method is proposed to find the American option price and the exercise boundary. The problem is formulated as a Linear Complementarity Problem and solved numerically by a convenient splitting. Computations have been accelerated with the help of the Fast Fourier Transform. A stability analysis shows that the scheme is conditionally stable, with a mild stability condition of the form k5O(|log(h)|21). The theoretical results are verified numerically throughout a series of numerical experiments. KEY WORDS: Integro-differential equations, variance gamma, finite differences, FF