### CEV MODEL WITH STOCHASTIC VOLATILITY

#### Abstract

This paper develops a systematic method for calculating approximate prices for a wide range of securities implying the tools of spectral analysis, singular and regular perturbation theory. Price options depend on stochastic volatility, which may be multiscale, in the sense that it may be driven by one fast-varying and one slow-varying factor. The found the approximate price of two-barrier options with multifactor volatility as a schedule for own functions. The theorem of estimation of accuracy of approximation of option prices is established. Explicit formulas have been found for finding the value of derivatives based on the development of eigenfunctions and eigenvalues of self-adjoint operators using boundary-value problems for singular and regular perturbations. This article develops a general method of obtaining a guide price for a broad class of securities. A general theory of derivative valuation of options generated by diffusion processes is developed. The algorithm of calculating the approximate price is given. The accuracy of the estimates is established. The theory developed is applied to a diffusion operator, which is decomposed by eigenfunctions and eigenvalues. The purpose of the article is to develop an algorithm for finding the approximate price of two-barrier options and to find explicit formulas for finding the value of derivatives based on the development of self-functions and eigenvalues of self-adjoint operators using boundary-value problems for singular and regular perturbations. Price finding is reduced to the problem solving of eigenvalues and eigenfunctions of a certain equation. The main advantage of our pricing methodology is that, by combining methods in spectral theory, regular perturbation theory, and singular perturbation theory, we reduce everything to equations to find eigenfunctions and eigenvalues.

#### Keywords

derivative pricing, stochastic volatility, local volatility, spectral theory, singular perturbation theory, regular perturbation theory.

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