“Taylor expansion for derivative securities pricing as a precondition for strategic market decisions”

The strategy of managing the pricing processes, in particular managing the dynamics of the price of the underlying asset and its volatility, the prices of indices, shares, options, the magnitude of financial flows, in the method of calculating the company’s rating based on the available quotations of securities, is developed. The article deals with the study of pricing and calculating the volatility of European options with gen-eral local-stochastic volatility, applying Taylor series methods for degenerate diffusion processes. The application of this idea requires new approaches caused by degradation difficulties. Price approximation is obtained by solving the Cauchy problem of partial differential equations diffusion with inertia, and the volatility approximation is completely explicit, that is, it does not require special functions. If the payoff of options is a function of only x, then the Taylor series expansion does not depend on t and an analytical expression of the fundamental solution is considerably simplified. Applied an approach to the pricing of derivative securities on the basis of classical Taylor series expansion, when the stochastic process is described by the diffusion equation with inertia (degenerate parabolic equation). Thus, the approximate value of options can be calculated as effectively as the Black-Scholes pricing of derivative securities. On the basis of the solved problem, it is possible to calculate their turns step-by-step. This enables to predict the dynamics of the pricing of derivatives and to create a strategy of behavior at options according to the passage of the process. For each approximation, price volatility is calculated , which makes it possible to take into account all changes in the market and to calculate possible situations. The step-by-step finding of the change in yield and volatility in the relevant analysis enables us to make informed strategic decisions by traders in the financial markets.


INTRODUCTION
Modeling and forecasting of pricing dynamics by using financial instrument are an important element of investment activity, as they have a high level of financial leverage and fulfillment of obligations is realized in the future. Derivative prices tend to change and predicting their behavior is becoming more and more complicated. Stochastic processes that are described by the equation of diffusion with inertia are widely used in the theory of mass service, in particular in the theory of queues. Often, such processes occur in financial markets with the pricing of European and Asian options. The theory of pricing derivatives and the study of the behavior of volatility for the analysis of profitability are necessary for flexibility in the adoption of managerial strategic decisions by managers. The validity of strategic decisions allows managers to make step-by-step additional investments in order to maintain strategic positions of the company in the stock market. Typically, a high level of volatility gives managers more opportunities to change their decisions in the future. Volatility is important for trad-ers when pricing several different series of options with different execution rates and maturities. Since it is stable in the Black-Scholes model (Black & Scholes, 1973), speculative price changes can occur in these cases, which cannot be verified without the presence of a certain kind of specialized knowledge. This is one example of how volatility pricing options are related to the principles of modeling and risk management in the financial market. Therefore, it is necessary to use models with variable volatility.
The purpose of this article is to introduce a unified approach to pricing and find the implied volatility of European options, which are described by degenerate diffusion processes of diffusion type with inertia, based on classical Taylor series approximation. Therefore, the approximate value of options can be calculated as effectively as the price of Black-Scholes (Black & Scholes, 1973) for European options.

LITERATURE REVIEW
Various operations with financial products suitable for selling and buying are carried out using derivative financial instruments (Aboulaich et al., 2013). Securities market participants should have a good understanding of derivative pricing to achieve successful financial results. Derivative securities transactions occupy an important place in the stock market financial activity, since each participant must hedge the risks to obtain extra profit based on stock market speculations (Anderson, 2011). Therefore, the derivatives are one of the major instruments in the securities market. An important task is to study the state and dynamics of the domestic stock market in close interrelation with other countries' stock markets and analyze the volatility of financial instruments to increase the efficiency of investment operations. Nowadays many approaches have been developed to calculate local and stochastic volatility that describe the overall dynamics of underlying price using CEV models (Lindsay & Brecher, 2012; JDCEV Carr & Linetsky, 2006), Heston model (Heston, 1993), SABR-model (Ren et al., 2007), but the application of these models requires the use of special functions and numerous integrations of complex functions. Hence, this may lead to miscalculations, but considering the time law (dependency), a considerable number of the models become unstable. Other methods for pricing derivative securities are required to get direct calculations (Forde et al., 2012). The use of Taylor series expansion depends on the model structure, specifically, on function properties, which are used in the model, and model ability to maintain the stability at time change.
The article considers models without default of diffusion process with inertia, with coefficients depending on the variables ( ) ,, ; txy for derivatives pricing we use Taylor series expansion for degenerate diffusion processes. In particular, complete correlation matrices are non-degenerate in the works by Mendoza-Arriagaetal (2010), Lorig (2013), and Lorigetal (2013), and in our work they are degenerate, so the application of this idea requires new approaches caused by degradation difficulties.

Consider
where τ is the stopping time Let U be an on-arbitrary price of the European option, which at time T is a gain ( ) T S  (Jeanblanc et al., 2009) that: To calculate the price of the European option, we need to find the mathematical expectation from where operator P has the form: ,, . !! .
Operator P of a parabolic type x and has a degenerate parabolicity on y (since there is no second order derivative on y ), which usually operates in financial spaces, that is, in the financial markets.
A fundamental solution was constructed on the homogeneous Cauchy problem (7) and it uses the properties of the fundamental solution (7)

RESULTS
The article considers the degenerate diffusion process that describes price dynamics and im- . Using approximate approximations considering the Black-Scholes price function as an initial approximation, we obtain explicit formulas for finding the initial approximations of implicit volatility. We note that they in the form of the record match the formulas , but in this case they are more complicated by the degeneration of the equation and the formula for the approximation of the price.
The obtained implications of implicit volatility and consistent price approximations make it possible to analyze the process of passage in the financial market. Let as make corrections and concrete steps to improve the situation for optimizing financial strategies.   Knowledge of the approximate price and implied volatility at each step at a fixed time gives an opportunity to develop a strategy for managing the dynamics of derivative prices in financial markets and to avoid speculative changes in pricing.
The density of distribution for degenerate diffusion processes is constructed when the coefficients depend only on time (7).
The method of constructing the density of distribution in the case of coefficients dependence on time and spatial variables is developed (3), when the correlation matrix is not strictly definite positive.
Knowing the density of distribution, one can always find the price by the formula (13) where the function of 0  is fulfilled by the density of distribution of probabilities of the investigated economic process.

CONCLUSION
This paper expands the methodology of approximate pricing for a wide range of derivative assets. Price approximation is obtained by solving the Cauchy problem for differential equations in partial derivatives of diffusions with inertia. If the payoff of options is a function of only x, then the Taylor series expansion does not depend on t and analytical expression of the fundamental solution of the distribution density of the stochastic process is considerably simplified. We have applied an approach to the pricing of European options on the basis of classical Taylor series expansion, when the stochastic process is described by the diffusion equation with inertia (degenerate parabolic equation). For a degenerate parabolic equation, the approximate price of options is fairly simple since it uses only estimates of derivatives of the distribution density of diffusion with inertia obtained by explicit formulas for European options price approximations.
The approach to the pricing of derivatives of securities and the finding of implicit volatility based on the classical approximation of the Taylor series when the stochastic process is described by the equation of diffusion with inertia (degenerate parabolic equation) is introduced. For a degenerate parabolic equation, finding the approximate price of options is fairly simple since it uses only estimates of derivatives of the distribution density of diffusion with inertia.
The step-by-step finding of the change in yield and volatility in the relevant analysis enables us to make informed strategic decisions by traders in financial markets.