Derivative pricing in lévy driven models

We consider an important class of derivative contracts written on multiple assets which are traded on a wide range of financial markets. More specifically, we are interested in developing novel methods for pricing financial derivatives using approximation theoretic methods which are not well-known to the financial engineering community. The problem of pricing of such contracts splits into two parts. First, we need to approximate the respective density function which depends on the adapted jump-diffusion model. Second, we need to construct a sequence of approximation formulas for the price. These two parts are connected with the problem of optimal approximation of infinitely differentiable, analytic or entire functions on noncompact domains. We develop new methods of recovery of density functions using sk-splines (in particular, radial basis functions), Wiener spaces and complex exponents with frequencies from special domains. The respective lower bounds obtained show that the methods developed have almost optimal rate of convergence in the sense of n-widths. On the basis of results obtained we develop a new theory of pricing of basket options under Lévy processess. In particular, we introduce and study a class of stochastic systems to model multidimensional return process, construct a sequence of approximation formulas for the price and establish the respective rates of convergence.