Reduced Order Models with Applications in Finance

Improved option pricing models, namely jump-diffusion models which are gaining importance in practice, are an interesting field of research from a numerical point of view. Calibrating these models to real market data results in optimization problems with partial integro differential equation (PIDE) contraints. Solving these PIDEs numerically leads to dense systems of equations, which are hard to solve.

Proper orthogonal decomposition (POD) can be used to derive a reduced order model for the PIDE. While an optimization algorithm is working, the need for an update of the model might arise, what is due to the fact that the reduced order model depends on the parameters that are to be calibrated. The problem of updates can efficiently be solved through an embedding into a trust region framework. However, updates could prove to be a costly part in the overall computing budget. In order to alleviate this effect, a model technique can be employed handling coarse and fine grid models of the PIDE in the optimization phase. This is based on a multi-level trust region technique.

Publications:
[1] Sachs, E.W. and Strauss, A.K.  Efficient Solution of a Partial Integro-Differential Equation in Finance.
Applied Numerical Mathematics. Vol. 58, pp. 1687-1703, 2008
[2] Sachs, E.W. and Schu, M.  Reduced order models (POD) for calibration problems in finance
K. Kunisch, G. Of, and O. Steinbach, editors, Numerical Mathematics and Advanced Applications, ENUMATH 2007, pages 735-742, Heidelberg, 2008. Springer-Verlag
[3] Sachs, E.W. and Schu, M.  Reduced order models in PIDE constrained optimization
submitted 2009

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