Meshfree Approximation in Nonlinear Black-Scholes Option Pricing Equation with Transaction Cost

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Differential equations play a very important role in the world of finance since most problems in finance are modelled by means of differential equation. These problems are sometimes nonlinear which can be solved by using numerical techniques. We review option pricing models in general, the formulation of the Black-Scholes model, and the Black-Scholes model with transaction costs. This thesis takes a look at Black-Scholes model with transaction costs with special reference to the model by Guy Barles and Halil Mete Soner. The resulting model is a nonlinear Black-Scholes equation in which the variable volatility is a function of the second derivative of the option price. We solve this nonlinear equation with a special class of numerical technique, called, the meshfree approximation using radial basis function. The method is analysed for stability and thorough comparative numerical results are provided. The numerical results showed that, the value of the option in the case of transaction cost is found to be higher than the analytical value of the standard Black-Scholes model.
A Thesis submitted to the Department of Mathematics, Kwame Nkrumah University of Science and Technology in partial fulfilment of the requirements for the degree of Master Of Philosophy in Applied Mathematics College of Science