Systematic Approach to the Evaluation of Options Based on the CEV-Model

Abstract

The article is concerned with studying the derivative assets, using tools for spectral analysis, as well as of the singular and regular perturbation theory. Using the risk-neutral valuation, we obtain the Cauchy task, allowing to calculate an approximate price of derivative assets and their volatility based on a diffusion equation. In the overall diffusion we add two quickly and slowly changing factors of the nonlocal volatility to obtain a model with the multivariate stochastic volatility. Combining the methods of spectral theory of singular and regular perturbations, one can calculate the price of derivative assets as degradation by native functions and the own values of linear operators and solution of the Poisson equation. Prospects for further research in this direction will be improvement of spectral theory and dissemination of the results of the publication on the cases when the equation, from which the eigenvalues are found, has no discrete spectrum, as well as when the stochastic volatility depends on four or more disparate factors that are present in the stock markets.