Numerical methods for finance

The first part of the course aims to provide a good knowledge of stochastic calculus and no arbitrage principles that constitute the foundations in the pricing of financial derivatives. We first discuss the Wiener process, then we move to the construction of stochastic integrals. We also introduce the concept of a martingale measure and its connection with the Fundamental Theorem of Asset Pricing.
The second part of the course aims to introduce students to the main numerical methods for the estimation of stochastic processes and to the numerical evaluation of contingent claims. The main topics presented are: Monte Carlo simulation, parameter estimation of stochastic processes, model selection and calibration.

At the end of the course, students should have acquired the fundamentals of stochastic calculus and the main numerical methods for the evaluation of contingent claims. Students should be able to produce scripts in the R programming language for the estimation of a stochastic process that describes the asset price dynamics and evaluate numerically contingent claims based on no arbitrage principles.