Numerical Analysis of Partial Derivative Equations in Finance

Objectives

At the end of this course the student will have acquired knowledge and skills in numerical methods for numerical analysis of Partial Differential Equations, parabolic equations, and apply these methods to models in Financial Mathematics (European and American options).

General characterization

Code

12969

Credits

6.0

Responsible teacher

Magda Stela de Jesus Rebelo

Hours

Weekly - 4

Total - 56

Teaching language

Português

Prerequisites

Available soon

Bibliography

  1. P.A. Raviart and J.M. Thomas, Introduction a l'' Analyse Numérique des Equations aux Derivées Partielles, Masson, Paris, 1983.
  2. J.M. Steele, Stochastic Calculus and Financial Applications, Applications of Mathematics: Stochastic Modelling and Applied Probability, 45, Springer, Berlin, 2004.
  3. E. Zauderer, Partial Equations of Applied Mathematics (second ed.), John Wiley and Sons, New York, 1989.
  4. M.Braun, Differential Equations and their applications (4th edition). Springer-Verlag, 1993.
  5. Y. Achdou, O. Pironneau, Computational Methods for Option Pricing, SIAM, Frontiers in Applied Mathematics, 2005.
  6. P. Wilmott, J. Dewynne, S. Howison, Option Pricing – Mathematical models and computation, Oxford Financial Press, 1995.

Teaching method

Available soon

Evaluation method

Available soon

Subject matter

  1. Partial differential equations Equações com Derivadas Parciais (EDPs)

1.1.    Introduction to partial differential equations

1.2.    . The heat equation; separation of variables

1.3.     Fourier series

1.3.1.A Application to the heat equation

1.4.     Fourier transform

1.4.1.Definition. Properties of the Fourier transform

1.4.2.Fourier transform invers. Aplication to the heat equation

1.5.    The Black-Sholes equation and applications to pricing for European  options.

2. Numerical methods for partial differential equations

2.1.    Finite differences for the heat equation with Dirichlet boundary equations and for Cauchy problems.

2.2.    Progressive, regressive and Crank-Nicolson schemes. Theta schemes. Convergence and stability.  

2.3.    Applications to pricing for European.

2.4.    Finite differences for one dimensional obstacle problems

2.5.     SOR methods with projection

2.6.    Applications to pricing for American options

Programs

Programs where the course is taught: