Numerical Aspects of Optimization in Finance
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Thu, 25/02/2010 14:00 |
Prof. Ekkehard Sachs (University of Trier) |
Computational Mathematics and Applications  |
3WS SR |
| There is a widespread use of mathematical tools in finance and its importance has grown over the last two decades. In this talk we concentrate on optimization problems in finance, in particular on numerical aspects. In this talk, we put emphasis on the mathematical problems and aspects, whereas all the applications are connected to the pricing of derivatives and are the outcome of a cooperation with an international finance institution. As one example, we take an in-depth look at the problem of hedging barrier options. We review approaches from the literature and illustrate advantages and shortcomings. Then we rephrase the problem as an optimization problem and point out that it leads to a semi-infinite programming problem. We give numerical results and put them in relation to known results from other approaches. As an extension, we consider the robustness of this approach, since it is known that the optimality is lost, if the market data change too much. To avoid this effect, one can formulate a robust version of the hedging problem, again by the use of semi-infinite programming. The numerical results presented illustrate the robustness of this approach and its advantages. As a further aspect, we address the calibration of models being used in finance through optimization. This may lead to PDE-constrained optimization problems and their solution through SQP-type or interior-point methods. An important issue in this context are preconditioning techniques, like preconditioning of KKT systems, a very active research area. Another aspect is the preconditioning aspect through the use of implicit volatilities. We also take a look at the numerical effects of non-smooth data for certain models in derivative pricing. Finally, we discuss how to speed up the optimization for calibration problems by using reduced order models. | |||