Being a student at the master program was a great experience. I hold a B.Sc. in Engineering Physics and my interest in computing and science lead me to the CSiS master. I was looking for something more focused on the numerical analysis and the development of high performance algorithms rather than a program focused on the development of general purpose systems, like is the case in some IT master courses.

I had the opportunity to get taught by great professors from a wide range of areas of science: from physics to computer science to mathematics. This surely gives the student a better approach to the topics taught, the possibilities of research, and the connection between areas of science that, at first sight, seem disparate, like economics and physics.

After the completion of the master I was offered a job at EON, an energy company which also engages in trading for hedging purposes, in the area of risk management as developer of mathematical algorithms to support decision making and risk compliance.

The computational representation of partial differential equations (PDE) can be challenging in some cases and special numerical algorithms must be created to achieve reliable results. My thesis deals with the convection-dominated behavior present in the computational simulation of convection-diffusion PDEs. This behavior is caused by the parameters of the PDE and lead to incorrect approximations of the function of interest. Moreover, the derivatives of this approximation are even worst. In financial mathematics, the derivatives of the price of the option are required to measure the sensibility of the price to different parameters like the volatility.

In computational finance, there exist various methods for the pricing of Options, and one of them is the numerical simulation of the celebrated model called the Black-Scholes equation. This model is a convection-diffusion PDE and in many cases the parameters are such that convection-dominated behavior is present.
Three methods are presented and emphasis is put on the Kurganov-Tadmor (KT) scheme. The KT scheme delivers excellent results and handles discontinuities or non-smoothness on the initial condition satisfactorily. In comparison to other methods like Crank-Nicolson or F inite Volume Methods, the KT scheme performs competitively and, additionally, it is easy to implement.

In the image, the approximation to the first derivative of price of the option is shown. Unrealistic oscillations artificially introduced by the method appear.

In many cases, Monte Carlo methods are used because of the ease of implementation and flexibility. Nevertheless, when the dimensionality of the problem is not high, using an scheme like the KT represents advantages and a considerable optimization in terms of computing time. This method could help the financial analyst to obtain approximations to the price of an option and its derivatives in an accurately manner.