Wavelet pricing of contracts on assets driven by jump-diffusion processes

Christoph Schwab

Abstract: The analysis and implementation of fast, deterministic pricing schemes for general contracts on assets driven by general Lévy processes is addressed. Our approach is based on finite element solution of the associated deterministic differential equation. This equation involves the Dynkin operator of the semigroup generated by the price process which for Lévy processes is an integro-differential operator. Upon discretization, the linear systems to be solved in each implicit time step have dense and ill-conditioned stiffness matrices. We propose here a spline wavelet basis for discretization in the (logarithmic) price variable which allows to compress these matrices to sparse, well-conditioned ones that can be inverted by iterative schemes in almost linear complexity while not affecting the accuracy of the computed prices. For American style contracts, the wavelet basis allows to precondition the iterative solver for the associated Linear Complementarity Problems (LCPs) in each time step. The algorithm allows to treat any Lévy price process, even pure jump processes with infinite jump intensity. Moreover, general pay-off functions are admissible allowing in particular to handle compound options, with contracts of European or American style. For infinite activity jump processes, the process' Lévy measure has a density with respect to the Lebesgue measure which is nonintegrable and must be interpreted in the sense of distributions. We present a variational framework for the associated parabolic integro-differential inequality which accomodates nonintegrable Lévy densities.

The integro-differential inequality is discretized using the backward Euler scheme in time and a wavelet-based finite element discretization in (logarithmic) price. We show that using the wavelet type basis functions, the condition number of the resulting large matrix inequality problems remains bounded independent of the discretization level. This allows to conclude the convergence of a fixed point type iteration with a rate which is independent of the meshwidth.