Alexander Kushpel

Alexander Kushpel, Jeremy Levesley, Xingping Sun

In this paper, we study the problem of interpolating a continuous function at $(n+1)$ equally-spaced points in the interval $[0,1]$, using shifts of a kernel on the $(1/n)$-spaced infinite grid. The archetypal example here is approximation using shifts of a Gaussian kernel. We present new results concerning interpolation of functions of exponential type, in particular, polynomials on the integer grid as a step en route to solve the general interpolation problem. For the Gaussian kernel we introduce a new class of polynomials, closely related to the probabilistic Hermite polynomials and show that evaluations of the polynomials at the integer points provide the coefficients of the interpolants. Taking cue from the classical Newton polynomial interpolation, we derive a closed formula for the Gaussian interpolant of a continuous function on a uniform grid in the unit interval.

Alexander Kushpel

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.