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Einige Forschungsthemen

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.

Calibration of Hestons Stochastic Volatility Model

Since the ground-breaking work of Black, Scholes and Merton the development of financial market models has gone a long way. Nowadays quite sophisticated models are employed in the financial market industry to price and hedge options. But before the models can be applied in practice, one has to implicitely identify the unknown model parameters from given market data. Usually, as an industry standard, a set of standard option prices serves as an appropriate set of market data. A good model must hence be able to provide a suitable fit of this so-called volatility surface.

In a joint research project, the University of Trier developed an algorithm for the identification of the underlying parameters of Hestons stochastic volatility model.

Index Forecasting with Neural Networks

In a joint research project Schröder Münchmeyer Hengst Research GmbH, Frankfurt/Main, and the University of Trier, Germany, developed a software which uses latest numerical techniques to achieve an efficient design of neural networks for forecasting stock and indices in the financial market.

The financial markets are based on a complex combination of mutually interacting factors. Since the markets behave neither in a uniform nor in an ordered way, small changes of a few input factors can result in large reactions of the market. Neural networks are used to improve the modelling of these nonlinear dependencies.

The number of the parameters to be determined in a neural network can lead very quickly to exceeding the computing time resources even for the most powerful computers. The use of modern numerical techniques for the training of neural networks reduces the computing time substantially compared to known methods. This allows to train networks with more substantial data input and improves the accuracy of the forecasting.