– Random number generation and variance reduction techniques.
– Calibration and estimation of financial models from the data using method of moments, maximum likelihood estimation and least squares method.
– The main stochastic processes used to model stock prices in finance, such as the Geometric Brownian motion and Lévy processes.
– Girsanov theorem and the risk neutral transformation (Esscher transform and mean correcting martingale).
– Introduction to stochastic calculus (Ito's lemma and quotient rule, multiplicative rule, Radon-Nikodym derivative, Girsanov kernel etc.) and derivation of the Black and Scholes formula and of Margrabe's formula.
– Monte Carlo methods with applications to (European, American and Multi-Asset) option pricing.
– Fast Fourier transform method with applications to European option pricing under different stochastic processes.
– Finite difference methods with applications to American option pricing.
– Multi-Asset options: how to capture the dependence between the multiple underlying assets using a linear correlation coefficient, copula methods and factor models. Both under the assumption of stock prices following a Geometric Brownian motion and Lévy processes.
– The course is highly focused on the use of the R statistical environment.