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


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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 general 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.

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    • Figure 1. The implied volatility obtained by the Ordinary Least Squares and our second order approximation by the Taylor series for the degenerate Heston model e^y = 0,249^2, T = 0.125, t = 0
    • Figure 2. The yield curve of first and second order approximation, the Taylor series for the degenerate Heston model e^y = 0, 249^2, T = 0.125, t = 0