Departement of mathematics and computer science
  Optimization and Financial mathematics

Current projects


Optimization and control theory:

Methods of Optimization and Optimal Control in Plane Geometry

Solution Branching in Partitioning Problems


Calculus of variations and partial differential equations:

A Generalisation of the Navier-Stokes Equation to Two-Phase Flow

Mathematical Modelling of Diffusion Induced Segregation

A Mathematical Model for Phase Transitions in Crystals


Stochastic analysis and financial mathematics:

(Credit) Risk Management and Dependence Modelling

(Credit Risk under incomplete information and nonlinear filtering

Risk Management for Derivatives under Market Frictions

 
 

Methods of Optimization and Optimal Control in Plane Geometry
Anita Kripfganz
A special kind of geometrical problems is concerned with extremal charakteristic parameters of plane convex figures. The considered figures are characterized by some other fixed charakteristic parameters. Such a type of problems can be formulated as constrained extremal problem in the space of convex figures. Some of them are successfully investigated by use of optimization methods and methods of optimal control like duality theory or Pontrjagin's Maximum Principle.

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Solution Branching in Partitioning Problems
Anita Kripfganz
We consider partitioning (allocation) problems with nonlinear total cost function and a linking constraint for the resource. Single projects are rated with the same convex-concave cost function, the total costs are additively composed. The structure of the optimal solution is to be find in dependence on the linking parameter. Under some additional conditions for the convex-concave partial cost function one can observe a branching behavior between symmetric and certain non-symmetric partitions.

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A Generalisation of the Navier-Stokes Equation to Two-Phase Flow

Thomas Blesgen
A modified Allen-Cahn equation is combined with the compressible Navier-Stokes system. It can be shown that, after a modification of the stress tensor, the second law of thermodynamics is valid for the resulting equations. A physical motivation of this alteration of the stress tensor can be given. The model can be used for describing cavitation in a flowing liquid. Finite volume calculations were carried out to illustrate the behaviour of the solutions. Global existence of a solution holds for instance if the system is assumed to be incompressible.

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Mathematical Modelling of Diffusion Induced Segregation
Thomas Blesgen, Stephan Luckhaus
Based on crystallographic experiments a mathematical model to describe the so-called chalcopyrite disease within sphalerite was developed. By choosing suitable free energies the significant properties of the physical process can be captured as they are observed in the laboratory. Numerical simulations stress the relevance of the model.

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A Mathematical Model for Phase Transitions in Crystals

Steffen Arnrich, Thomas Blesgen, Stephan Luckhaus
The goal of this project is to analyse under very weak assumptions the elastic theory of crystals with microstructure for a free energy density depending on the local particle density of one ore more varieties of molecules occuring in the crystal. This local partical density can change by diffusion. The model describes an isothermal situation, furthermore is is supposed that the mechanical deformation adapts instantenously to the diffusion and that there are no interstitials. In our model we consider the surface energy arising at the transition layers of the different phases.

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(Credit) Risk Management and Dependence Modelling
Rüdiger Frey
A major cause of concern in managing the credit risk of most financial institutions is the occurrence of disproportionately many joint defaults of different counterparties over a fixed time horizon. Joint default events also have an an important impact on the performance of derivative securities, whose payoff is linked to the loss of a whole portfolio of underlying bonds or loans such as collaterized debt obligations (CLOs) or basket credit derivatives. In fact, the occurrence of disproportionally many joint defaults is what could be termed "extreme credit risk" in these contexts. Clearly, precise mathematical models for the loss in a portfolio of dependent credit risks are needed to adequately measure and price this risk.
A key research area of the financial mathematics working group in Leipzig is therefore dependence modelling in credit risk management. More specifically we consider the following issues
  • We are working on models with interacting defaults, i.e. models where the default of one company has a direct impact on the default probability of other companies. Here we use some ideas from the probabilistic theory of interacting particle systems. To date we have studied pricing and hedging of credit derivatives in these models; relevant papers include Frey-Backhaus (2006) and Frey-Backhaus (2007) . In the future it is intended to analyse the impact of interactions between default events on the aggregate loss of large credit portfolios and the ensuing important implications for credit risk management. In the longer term, we aim at exploring the role of meta-stable states - created by interaction between economic agents - for the presence of disruptive financial phenomena such as "stock market crashes" or "credit crunches".
  • Rüdiger Frey has written a book on Quantitative Risk Management (together with Alexander McNeil, Herriott-Watt University Edinburgh and Paul Embrechts, ETH Zurich). Dependence modelling in finance and (credit) risk management plays a major role in this text.
  • Currently we are focusing on credit risk models under incomplete information; details are given below.

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    Credit Risk under Incomplete Information and Nonlinear Filtering
    Rüdiger Frey, Thorsten Schmidt
    The project is concerned with the development of realistic and tractable dynamic credit risk models for the valuation and the hedging of credit derivatives such as corporate bonds or related portfolio products. A new information-based approach mimicking the informational advantages of institutional investors is proposed. Under this approach, this informational advantage is reflected in prices of traded credit derivatives; secondary-market investors will therefore try to back out this information from observed prices. Our project will study the optimal solution of this problem, using advanced methods from stochastic calculus and in particular from nonlinear filtering theory. Moreover, we plan to analyze the ensuing \emph{dynamics} of prices of traded securities and credit spreads. Further research goals include calibration and testing of the model using market data; the development of suitable hedging strategies under incomplete information; the extension of our approach to equity-derivatives markets; and finally the analysis of portfolio-optimization problems in our context, with particular focus on incomplete information.
    This project is supported by the German Science Foundation (DFG); a more detailed description of the intended research can be found under proposal (pdf). Related papers coauthored by members of our group include Frey-Runggaldier (2006) , Frey-Schmidt-Gabih (2007) and Frey-Schmidt (2007) .
     

    Risk Management for Derivatives under Market Frictions
    Rüdiger Frey
    In recent years market liquidity has become an issue of high concern in risk management. In particular, risk managers realized that financial models which are based on the assumption that an investor can trade large amounts of an asset without affecting its price (perfectly liquid markets) may fail miserably in circumstances where market liquidity vanishes. This calls for additional research, extending traditional financial models to markets which are not perfectly liquid. In this project we focus on the risk management for derivative securities via dynamic hedging and study models of an illiquid market where the implementation of a dynamic hedging strategy has an impact on the price process of the underlying asset. From a mathematical viewpoint this lead to interesting nonlinear versions of the parabolic Black-Scholes pricing PDE. Currently we are focusing on qualitative properties of option prices in illiquid markets; an overview of this research can be found in the following slides.

    		 
    home Claudia Götz,
    Claudia.Go...@math.uni-leipzig.de