Journal of Applied Mathematics & Bioinformatics

Nonlinear Black-Scholes Equation Through Radial Basis Functions

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• Abstract

  The Black-Scholes equation is a partial differential equation (PDE) characterizing the price evolution of a European call option and put option on a stock. In this work, we use Multi-Quadratic Radial Basis Functions (Multi-Quadratic RBF) for approximating the solution of the Black- Scholes equation and show how it can be applied to a case in which the volatility is not constant but is dependent on the transaction costs (a nonlinear case). That is, volatility satisfies an Ornstein-Uhlenbeck stochastic process. These nonlinear models are also presented in the financial markets characterized by lack of liquidity. After discretizing the problem with the Crank-Nicholson algorithm, we expose a numerical example to validate the developed method.