Pricing of high-dimensional options

Abstract

Pricing of high-dimensional options is one of the most important problems in Mathematical Finance. The objective of this manuscript is to present an original self-contained treatment of the multidimensional pricing. During the past decades the Black-Scholes this model, which essentially is based on the log-normal assumption, has been increasingly criticised. In particular, it was noticed by Mandelbrot that empirical log-returns distributions are more concentrated around the origin and have considerably heavier tails. This suggests to adjust the Black-Scholes model by the introduction of the Levy processes instead of Brownian ones. This approach has been extensively studied in a univariate setup since the nineties. In the multivariate settings the theory is not so advanced. We present a general method of high-dimensional option pricing based on a wide range of jump-diffusion models. Namely, we construct approximation formulas for the price of spread options. It is important to get an efficient approximation for the respective density function, since the reward function has usually a simple structure. Instead of a commonly used tabulation approach, we use the respective m-widths to compare a wide range of numerical methods. We give an algorithm of almost optimal, in the sense of the respective m-widths, reconstruction of density functions. To demonstrate the power of our approach we consider in details a concrete class of Levy driven processes and present the respective rates of convergence of approximation formulas. The interrelationship between the theory and tools reflects the richness and deep connections in Financial Mathematics, Stochastic Processes, Theory of Martingales, Functional Analysis, Topology and Harmonic Analysis.