# SC²S Colloquium - August 13, 2013

### From Sccswiki

Date: |
August 13, 2013 |

Room: |
02.07.023 |

Time: |
3:00 pm, s.t. |

## Pablo Gómez: POD-Galerkin für die Black-Scholes Gleichung

Solving the Black-Scholes equation to evaluate the price of a basket option is a numerically difficult task.

The nature of this parabolic partial differential equation (PDE) requires sophisticated methods or vast computing power.

There have been different approaches to this problem like e.g. the evaluation using Monte Carlo methods or the application of sparse grids. Both methods, however, are somewhat inconvenient to solve these high-dimensional problems, as the convergence rate of Monte Carlo methods is rather slow in these cases and the performance of sparse grids suffers from the so called "curse of dimensionality".

But since the evaluation of different options requires similiar calculations on sparse grids, it is possible to reduce the necessary computation time through the usage of model reduction techniques. One of these techniques is the Proper Orthogonal Decomposition (POD), which will be applied in this thesis to transform the system attained by the sparse grid methods to decrease the necessary degrees of freedom. The system of ordinary differential equations, that sparse grids use to solve the Black-Scholes equation, is thereby solvable with less computational power.