Encyclopedia of Quantitative Finance, IV Volume Set

Wavelet Galerkin Method

Wavelet methods in finance are a particular realization of the finite element method (see Finite Element Methods) that provides a very general PDE-based numerical pricing technique. The methods owe their name to the choice of a wavelet basis in the finite element method. This particular choice of basis allows the method to solve partial integro-differential equations (PIDEs) arising from a very large class of market models. Therefore, wavelet-based finite element methods are well suited for the analysis of model risk and pricing in multidimensional and exotic market models. Since wavelet methods are mesh-based methods they allow for the efficient calculation of Greeks and other model sensitivities.

As for any finite element method, the general setup for wavelet methods can be described as follows. Consider a basket of d ≥ 1 assets whose log returns X_t are modeled by a Lévy or, more generally, a Feller process with state space and X₀ = X. By the fundamental theorem of asset pricing, the arbitrage-free price u of a European contingent claim with payoff g(·) on these assets is given by the conditional expectation

under an a priori chosen equivalent martingale ...