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  5. Métodos Numéricos em Finanças

Métodos Numéricos em Finanças

Objectives

Available soon

General characterization

Code

11582

Credits

6.0

Responsible teacher

Nuno Filipe Marcelino Martins

Hours

Weekly - 4

Total - Available soon

Teaching language

Português

Prerequisites

Available soon

Bibliography

1. Y. Achdou, O. Pironneau, Computational Methods for Option Pricing,  SIAM, Frontiers in Applied Mathematics, 2005.
 
2. P. Glasserman,  Monte Carlo Methods in Financial Engineering, Applications of Mathematics, Stochastic Modelling and Applied Probability, 53, Springer, 2003.
 
3. D. J. Higham, An Algorithmic Introduction to Numerical Simulation of Stochastic Differential Equations, SIAM REVIEW, 43 (3), 525-546, 2001.
 
4. H. Niederreiter,  Random Number Generation and Quasi-Monte Carlo Methods, SIAM, CBMS-NSF Regional Conference Series in Applied Mathematics, 63, 1992.
 
5. C.P. Robert, G. Casella,  Introducing Monte Carlo Methods with R, Springer,  2010.
 
6. P. Wilmott, J. Dewynne, S. Howison, Option Pricing – Mathematical models and computation,  Oxford Financial Press, 1995.

Teaching method

Available soon

Evaluation method

There are two handouts T1 , T2 and a final project (Tf).

 

Final grade is  computed considering 25 % of T1, 25 % of  T2   and 50 % for Tf .

Subject matter

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. Erro 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

Programs where the course is taught: