CEV MODEL WITH STOCHASTIC VOLATILITY

. 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.


INTRODUCTION
Spectral theory was widely used in the second half of the 20 th century by many economists. In recent years spectral analysis has become an increasingly popular tool for use in financial mathematics to analyze diffusion models which are based on the expansion of eigenfunctions and eigenvalues of linear operators. For example, it is used to find the price of a European option using Black-Scholes model [8]. Among the scientific problems that can be solved by applying spectral methods: predicting option prices, [5] securities interest rates [11], modeling the volatility of financial assets [4].
Assets estimation problems are solved analytically by methods of spectral theory [5]. Spectral theory as well as stochastic volatility models has become an indispensable tool in financial mathematics, for the matter of that, two barrier option prices are subjected to Brownian motion and are CEV Model with Stochastic Volatility 23 correlated with volatility [6]. The study of stochastic volatility, volatility assets in particular, underlies the derivative and is controlled by nonlocal diffusion.
In this article we continue the area of our research [1; 2], expending it on the theory model CEV (constant elasticity of variance model), which was designed by John Cox in 1975, employing his methods [3; 9; 10].
Combining the methods of spectral theory and regular perturbance, we are able to calculate approximately the opportunity cost as expansion of eigenfunctions. We will work with infinitesimal generators of three-dimensional diffusion.

PROBLEM STATEMENT
First, consider the one-dimensional diffusion = ( ) + ( ) which has the possibility to show default jump at a speed ℎ( ) ≥ 0, -geometric Brownian motion, X is always strictly positive. We add two nonlocal volatility factors to the total diffusion: ( ) → ( ) ( , ). The first factor Y is dynamic. The second factor Z changes slowly. So, our model is a multidimensional volatile stochastic model.
Let (Ω, F, P) denote probability space that supports correlated Brownian motion (W , W , ) and an exponential random variable ~(1), which is not independent of (W , W , ). We assume that the economy with three factors is described by homogeneous time, continuous Markov process χ = (X, Y, Z), which takes values in some state space E = × R × R, = ( 1 , 2 ), − ∞ ≤ 1 < 2 ≤ ∞. Suppose that χ begins in E and instantly disappears once ∉ , that is: The dynamics of χ according to the physical value ℙ , is as follows: where (X,Y,Z) are assigned 2 ( ) 2 + 21 ( ) ), Therefore, and have an internal timeline > 0 and 1/ > 0. We assume that << 1 and << 1, the internal timeline is small, whereas the internal timeline is large. So, is volatility of fast variable factor, whereas is vitality of a slow variable factor. Note that Μ and Μ t are 24 Ivan Burtnyak, Anna Malytska Suppose you have to pay share dividends = { > } , > 0. then the state space X will be 1 , 2 = (0, ∞). Consider a multidimensional diffusion process at Killing (default jumps) of constant variable model. In particular, P dynamics of X default is set as To simplify calculations assume that the risk-free interest rates = 0, > 0, > 0, and are fast and slow variables of volatility, which are defined In our study, there may be two possible ways of default when X is beyond the time-frame , or at random time ℎ , (ℎ( ) ≥ 0 stochastic value (the so-called level of danger). Mathematically default time can be expressed as follows [2]. Volatility X includes the local component and nonlocal component of multidimensionality ( , ). We assume, η <0, that is local volatility component increases when , decreases. It means that prices and volatility have negative correlation. Stochastic danger level ℎ( ) increases when X decreases. Now let's calculate the approximate price of European option for assets S. The European option price can be defined by the formula (2).

RESULTS
The results of this research result from the article [2] The European option profit with strike price K > 0 can be decomposed as follows [9]: The first item on the right hand side (3) profit option is submitted to default at time . The second item is profit option which is submitted after the default, which occurs at time . So, the value of the option with strike price -is denoted as , ( , ; ) and can be expressed as the sum of: Expressions for 0, and 1, can be found in [10].

Ivan Burtnyak, Anna Malytska
The estimated value of the European option can now be calculated using the theorems 1, 2, 3 [2].  As expected, ε and δ which move to the zero, volatility moves to volatility price implied by full value.

CONCLUSIONS
This paper, extends the method of finding approximate price for a wide range of derivative assets. One of the main advantages of our pricing methodology is that by combining methods of the spectral theory of singular and regular perturbance, the calculation of asset prices leads to solving the equation by eigenvalues and eigenfuntion methods as well as by solving Poisson equation. Once this equation is solved, the approximate price of a derivative asset may be calculated formulaically.
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.