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.