Adaptive wavelets computational method for pricing multi-asset derivatives

Abstract

In this thesis, we have constructed a B-spline wavelet numerical method (referred to as the BSWE method hereinafter) to compute the price of two-asset European options. This method is numerically effective and very robust, in the sense that it can be adapted to different particularities of the problem. The original one-asset B-spline wavelet numerical method was first proposed by Ortiz and Oosterlee. Our results can be served as a lead on how the method can be generalized to multi-asset cases, and how difficulties can be resolve when facing multi-asset problems. We focus on problems where the assets are modelled by stochastic models driven by affine processes, and find out that our BSWE method, for payoff functions with jump discontinuities, can outperform the Fourier-cosine method, which does not seem to converge in this case due to the "Gibbs phenomenon". The efficiency and effectiveness of our method is further confirmed via numerical illustrations.

A thorough and complete error analysis is also obtained for our numerical method. Since we adopt an integration-by-parts technique to analyze the errors, we can obtain our error estimates under weaker assumptions, i.e. the joint probability density function is Lipschitz continuous, than the Taylor-expansion-based argument used say by Oriz and Oosterlee, which requires both the probability density function and the payoff function to be at least twice continuously differentiable. We also explain in the thesis how a Fast Fourier Transform (FFT) technique can be incorporated into the computation of wavelet coefficients to speed up the numerical calculations.

Finally, it is common that many transformed-based pricing methods, such as the Fourier-cosine method and the wavelet expansion method used in this thesis, exhibit non-trivial sensitivity to an exogenous parameter required to truncate the computational domain. This parameter is part of the cumulant-based truncation rule first introduced by Fang and Oosterlee and subsequently used by many researchers. Some modifications have been proposed to alleviate such sensitivity, but without much success. In this thesis, we have proposed a new adaptive truncation rule based on the Cornish-Fisher expansion series, which significantly improves the existing methods.