  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.
