Mathematical Researches

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In financial markets, volatility decreases with rising stock prices. The constant elasticity of variance (CEV) model is a good model to show this inverse relationship between stock price and its volatility in the market. In this paper, we assume that stock price dynamics follows the CEV model. But this model cannot show the trend memory effect in financial markets. Given that fractional derivatives are suitable tools for describing the trend memory effect, they can interpret and express the hereditary characteristics of the options well. Hence, under the assumption that the price change of the underlying asset follows a fractal transmission system, we investigate the pricing of the European option. The main objective of this paper is to numerically solve the time-fractional Black-Scholes equation based on the meshless local Petrov-Galerkin (MLPG) and implicit finite difference methods for discretizing the option price and time variable, respectively. In this study, MLPG type 2 (MLPG2) is developed based on the moving Kriging interpolation method to construct shape functions that have the Kronecker delta property, and the Kronecker delta is the test function. Also, we analyze the stability of the proposed method using the matrix method. Numerical examples show the accuracy and efficiency of the method.

Keywords: European option, Time-fractional Black-Scholes equation, Meshless Local Petrov-Galerkin method, Moving Kriging interpolation

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