Novel fitted multi-point flux approximation methods for options pricing

Doctoral Thesis


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It is well known that pricing options in finance generally leads to the resolution of the second order Black-Scholes Partial Differential Equation (PDE). Several studies have been conducted to solve this PDE for pricing different type of financial options. However the Black-Scholes PDE has an analytical solution only for pricing European options with constant coefficients. Therefore, the resolution of the Black-Scholes PDE strongly relies on numerical methods. The finite difference method and the finite volume method are amongst the most used numerical methods for its resolution. Besides, the BlackScholes PDE is degenerated when stock price approaches zero. This degeneracy affects negatively the accuracy of the numerical method used for its resolution, and therefore special techniques are needed to tackle this drawback. In this Thesis, our goal is to build accurate numerical methods to solve the multidimensional degenerated Black-Scholes PDE. More precisely, we develop in two dimensional domain novel numerical methods called fitted Multi-Point Flux Approximation (MPFA) methods to solve the multi-dimensional Black-Scholes PDE for pricing American and European options. We investigate two types of MPFA methods, the O-method which is the classical MPFA method and the most intuitive method, and the L-method which is less intuitive, but seems to be more robust. Furthermore, we provide rigorous convergence proofs of a fully discretized schemes for the one dimensional case of the corresponding schemes, which will be well known on the name of finite volume method with Two Point Flux Approximation (TPFA) and the fitted TPFA. Numerical experiments are performed and proved that the fitted MPFA methods are more accurate than the classical finite volume method and the standard MPFA methods.