### Browsing by Author "Tambue, Antoine"

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- ItemOpen AccessNovel fitted multi-point flux approximation methods for options pricing(2020) Koffi, Rock Stephane; Tambue, Antoine; Ebobisse, FrancoisIt 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.
- ItemOpen AccessNovel fitted schemes based on mimetic finite difference method for options pricing(2021) Attipoe, David Sena; Tambue, Antoine; Ebobisse, FrancoisNumerical methods have been increasingly important for finding approximate solutions of partial differential equations (PDEs) describing financial models since only a few of them have analytical solutions. Indeed, in the pricing of derivative securities such as European options, the underlying PDE, the so called Black-Scholes equation, is known to have a closed-form solution when the coefficients are constant. In the case of an American put option, however, there is no analytical solution, even for constant coefficients. In this thesis, we propose alternative schemes based on mimetic finite difference to overcome the known limitations of the finite difference method while pricing options. The standard mimetic finite difference method is known in fluid dynamics to preserve important properties of the continuous problem in the discrete case thereby resulting in more accurate approximations. The underlying Black-Scholes differential operator is known to be degenerate at the boundary when the stock price equals zero. At this singularity, important properties of the PDE are lost. A negative consequence here is that the classical finite difference scheme applied to such problems is no longer monotone and hence fails to give an accurate approximation when the stock price is small. Therefore, more sophisticated techniques that are adapted to handle the degeneracy must be sought. Our proposed scheme, a fitted local approximation method, is able to handle the degeneracy of the Black-Scholes differential operator near the boundary at zero. The novel combined schemes are called fitted mimetic finite difference methods and are used for spatial discretization of the Black-Scholes PDE in one and two dimensional domains. Furthermore, rigorous mathematical convergence proofs of the methods for the one dimensional case are provided where the standard Euler method is used for temporal discretization. Numerical simulations show that the proposed numerical methods (in one and two dimensional domains) applied to both European and American options are more accurate compared to the standard finite difference method and the standard fitted finite volume methods.
- ItemOpen AccessNumerical solution for subsurface reservoir simulation(2017) Etekpo, Kossi; Tambue, Antoine; Reddy, DayaTransport problems in porous media constitute an important field of scientific research in modern world, due to their broad applications in area such as petroleum engineering, water resources, pollutants transport and green- house gases sequestration to just mention few. The mathematical models that describe such problems have been developed and form one of the main classes of partial differential equations (PDEs) that scientists encounter in the real-world modeling. Nevertheless, in most of the cases, the exact solutions in the classical sense of those models are not available. The study of numerical approximation of PDEs is therefore an active research area and there is an extensive literature on numerical methods for PDEs. In this work, we review some numerical techniques, more precisely we present finite volume method with two-point flux approximation and mixed finite volume method for spatial discretization of elliptic and parabolic PDEs modeling transport flow in porous media. We then present some standard explicit and implicit methods, Rosenbrock schemes and exponential time stepping schemes for temporal discretization. We finally run some numerical simulations of advection-diffusion-reaction problems in a heterogeneous and an anisotropic porous media.