Métodos Numéricos em Finanças
Nuno Filipe Marcelino Martins
Weekly - 4
Total - 68
1. Simulation of random variables
Inverse transformation method. Acceptance-rejection method. Simulation of variables with normal distribution. Box-Muller and Marsaglia''''s variant. Multivariate normal distributed variables.
2. Numerical integration
Rectangle rules for numerical integration. Monte Carlo methods. Error analysis. Variance reduction techniques for Monte Carlo. Antithetic variates, importance sampling, control variates and stratified sampling techniques.
Quasi Monte Carlo methods. Discrepancy. Koksma-Hlawka inequality. Low discrepancy sequences. Van der Corput and Halton sequences.
3) Numerical integration of ordinary and stochastic differential equations.
Review of methods for first order differential equations. Euler (explicit and implicit). Taylor methods. Runge-Kutta methods. Convergence and stability.
Simulation of unidimensional Brownian motions: Random walk and Brownian bridges. Fourier methods. Fast Fourier transform. Multidimensional Brownian motions. Itô integral. Euler-Maruyama''''s method for SDE. Strong and weak convergence. Weak Euler-Maruyama method. Milstein''''s method. Stability. Stochastic Runge-Kutta methods.
4) Finite differences method for the heat equation.
Finite differences for the heat equation with Dirichlet boundary equations and for Cauchy problems. Progressive, regressive and Crank-Nicolson schemes. Theta schemes. Convergence and stability. Finite differences for one dimensional obstacle problems SOR methods with projection . Aplications to pricing for European and American options.
Programs where the course is taught: