In our research one focus lies on portfolio optimization, i.e. on finding optimal investment and consumption strategies in financial markets. Here, "optimal" means that the expected utility from terminal wealth and/or consumption is maximized or that a certain risk is minimized. The parameters in the market that the influence stock prices are usually not observable and have to be estimated based on stock returns.
In particular, our group works on portfolio optimization under model uncertainty, that means that no distribution of model parameters is known. This leads to a worst-case optimization problem and to the question which strategies are particularly robust with respect to uncertainty. Besides, we analyze optimization problems with crash szenarios and investigate how expert opinions affect the estimation of market parameters.
Pricing of financial and actuarial products
Our group works on various techniques for determining option prices in an efficient way, for example by approximation via new binomial models for multivariate price processes, by improved multi-level Monte Carlo methods or by developing price systems under transaction costs. In cooperation with the Department of Electrical and Computer Engineering we also work on hardware acceleration.
Another focus lies on designing new rating systems for pricing financial products by means of utility-based risk measures or risk-chance-classes for comparing pension products. Tariff arrangements under regulations in actuarial markets, e.g. gender neutral tariffs, are investigated by equilibrium models. These can also be applied for agricultural risk.
Financial time series analysis
In financial markets a careful analysis of historical returns is crucial for product pricing, portfolio optimization and risk analysis. To improve the understanding of the underlying return process mechanisms one can set up models which yield a best possible fit to the observed data. On the other hand one can use different methods of data analysis to determine special properties of the return series.
In particular we work on the development of appropriate models, e.g. regime switching models, together with the design and analysis of model dependent parameter estimation methods, e.g. based on stochastic filtering. We also use clustering methods for examining return processes to aim at risk minimal asset allocation.
Monte Carlo methods
Monte Carlo methods are very useful in situations where it is impossible to derive closed-form solutions for problems arising in financial markets. For pricing and risk assessment of complex financial products they are indispensable and hence widely used both in practical applications and in research.
The foundation for Monte Carlo simulation is the Strong Law of Large Numbers which states that the arithmetic mean of independent and identically distributed random variables converges almost surely to their expectation. This immediately suggests to determine option prices by simulating independent realizations of the option payoff and approximating the discounted expectation by the arithmetic mean.
Reliability of buildings made of reinforced concrete
Within the RTG 1932 and together with the "Institute of Concrete Structures and Structural Engineering" in the Department of Civil Engineering we analyse the reliability of buildings made of reinforced concrete. Our focus especially lies on system relevant and time dependent aspects which also appear in a similar form in financial markets (e.g. credit default risks in banks).
On the mathematical side we especially use Monte Carlo methods (importance sampling) and Bayesian inference methods.