“Spectral study of options based on CEV model with multidimensional volatility”

This article studies the derivatives pricing using a method of spectral analysis, a theory of singular and regular perturbations. Using a risk-neutral assessment, the authors obtain the Cauchy problem, which allows to calculate the approximate price of derivative assets and their volatility based on the diffusion equation with fast and slow variables of nonlocal volatility, and they obtain a model with multidimensional stochastic volatility. Applying a spectral theory of self-adjoint operators in Hilbert space and a theory of singular and regular perturbations, an analytic formula for approximate asset prices is established, which is described by the CEV model with stochastic volatility dependent on -fast l variables and -slowly r variables, and a local variable. Applying the Sturm-Liouville theory, Fredholm’s alternatives, as well as the analysis of singular and regular perturbations at different time scales, the authors obtained explicit formulas for derivatives price approximations. To obtain explicit formulae, it is necessary to solve 2l Poisson equations.


INTRODUCTION
The constant elasticity of variance (CEV) model is a generalization of geometric Brownian motion models. This model has been introduced by Cox and Ross (1976) for pricing European call options. The CEV model is based on the assumption that the risk-neutral process that describes the stock price S has the form: α > then, with the increase in the asset value, its volatility also increases. This corresponds to volatility smile and means that volatility is an increasing function depending on the asset price.
Comparing with the models of the geometric Brownian motion, the advantages of a CEV model lie in the fact that the instability ratio correlates with the price of risky assets and may explain empirical bias, such as the volatility smile (Schroder, 1989). The CEV model is usually applied to calculate the theoretical price, sensitivity and expected volatility of options (Emanuel & MacBeth, 1982;Fouqueet at al., 2000). In recent years, the problem of a pension fund investment is very urgent, it turns out that the CEV model has been successfully applied to study the effective investment strategy (Davydov & Linetsky, 2001; Davydov & Linetsky, 2003).
volatility (Schroder, 1989). It was found that the asset price behaves like volatility. According to a Black-Scholes model, volatility is constant (Black & Scholes, 1973). As a result, this led to a series of works to expand this model. Empirical studies have found that volatility is a random variable dependent on time. Hull and White (1989), Stein and Stein (1991), Heston (1993) introduced analytical models with stochastic volatility. Carr and Linetsky (2006), Aboulaich et al. (2013) investigated the stochastic volatility model with jumps. The CEV model is a generalization of dynamic volatility models. In particular, it provides an opportunity to examine the asset price that changes continuously over time. There is a series of works dedicated to this problem (Andersen & Piterbarg, 2007;Lindsay & Brecher, 2012). In this article, we consider the CEV model with multidimensional stochastic volatility. Using a spectral theory of selfadjoint operators in Hilbert space and a theory of singular and regular disturbances, an analytic formula for the approximate asset prices is established, which is described by the CEV model with stochastic volatility dependent on -fast l variables and -slowly r variables, 1, r 1, l ≥≥ , lN rN ∈∈ and a local variable. The theorem of closeness in estimation of financial instruments prices approximation is proved.

RESEARCH METHODOLOGY
The purpose of the article is to establish the approximate derivative prices, which are defined by CEV model with stochastic multidimensional volatility and depend on many factors of a spectral theory and a theory of perturbations.
Thus, when a CEV model with stochastic multidimensional volatility is adequate for describing the dynamics of an underlying, the spectral method outlined above serves as a powerful tool for analytically pricing derivatives on that underlying. Among the topics that have been addressed by applying spectral methods with multidimensional diffusions are option pricing (both vanilla and exotic), mortgages valuation, interest rate modeling, volatility modeling, and credit risk.
The necessity to have stock price diffusions that don't jump to zero in order to default and still have a non-zero probability of falling to zero leads us to naturally consider CEV processes. Moreover, CEV models have the advantage to provide closed-form formulae for European vanilla options and for the probability of default.
In particular, many problems related to the pricing of derivative assets have been solved analytically by using methods from spectral theory. An overview of the spectral method applied to derivative pricing is as follows.
is the interval in R with points 1 , a and 2 , a such that 12 . aa −∞ < < < ∞ Let that χ has a beginning at R and instantly disappears as soon as X goes beyond H. In particular, the dynamics of χ with physical measure P is as follows: ( ) 11 , , …, , ,…, , , In changing from the physical probability measure to the risk-neutral pricing measure, we consider a class of market prices of risk that is general enough to treat credit, equity, and interest rate derivatives in a single framework.
Process X can represent many economic phenomena and processes, which describe the optimal investment strategies. For example, the stockpiles, the index price, a risk-free shortterm interest rate, etc. Even more broadly, X is an external factor that characterizes the cost of any of the abovementioned processes. We are considering the process X with stochastic volatility ( ) ( ) 11 , …, , ,…, 0, where HR ∈ is the interval terminating at 1 a and 2 a and p is a diffusion density rate.
The boundary conditions for 1 a and 2 a are implemented on the output, input and regular bounds.
We evaluate the derivative security with payoff at time 0, t > which may depend on trajectory . X In particular, we will consider the forms of payoff: where θ is a random moment of time during which there is a failure to make a payment of premium. Since we are interested in derivatives estimates, we must determine the dynamics .
, , , where the operator , P ηγ′ has the form:

APPLICATION OF THE DESCRIBED METHODOLOGY
The models developed by these scholars have their advantages and disadvantages, but each of them is used to increase the liquidity of financial markets. The findings are credit spread of credit market instruments, calculating option prices for interest rates, determining the risk and derivatives' rate of return of the stock market financial instruments.
Using the method of Eigen function expansions, we derive analytical solutions for zero-coupon bonds and bond options under CEV processes for the shadow rate. This class of models can be used to model low interest rate regimes.
Let's assume that the asset is defined by Since S must be positive, the space of states X will be ( ) ( ) 12 , 0, . aa = ∞ A multiscale diffusion is formed on the default leap using a method of continuous variations (Carr & Linetsky, 2006 The first item on the right-hand side (11) is the option payoff before the default at time t. The second item represents option pay off after the default, which may occur at time t. Thus, the option value with execution price K is designated by The approximate European option price which are described by system (10)