June 18,  19,  20   2003
 
 
 

02:30 PM - Wednesday, June 18, 2003 - Amphitheater Darboux

Christoph Schwab

Seminar für Angewandte Mathematik
ETH Zentrum
HG G 58.1 Ch-8092 Zürich, Switzerland

schwab@sam.math.ethz.ch

Wavelet - Galerkin Asset Pricing in Jump Diffusion Models

Prices of contracts on assets with jump diffusion and Levy process price models are solutions of parabolic integrodifferential equations (PIDEs). If the jump intensity of the price process is infinite, the infinitesimal generators of the Levy process are hypersingular integral operators of order in [0,2]. In logarithmic price, the equations are posed on all of R with exponentially growing data, the payoff functions. We show well-posedness of the PIDE and time analyticity of the corresponding semigroup in Sobolev spaces on R with exponential weights. The efficient numerical solution of these PIDEs uses a wavelet Galerkin discretization of the hypersingular integral operators and hp-discontinuous Galerkin time-stepping. We prove that this algorithm allows to price European Vanillas for general Levy price processes in O(N(\log N)^c) work where N denotes the number of spatial degrees of freedom. American put contracts lead to parabolic variational inequalities. We introduce a wavelet preconditioned iterative solver for the LCPs at each implicit time step. American contracts for general Levy price processes and general pay-off functions can be solved. Numerical examples are given.