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Filtering techniques in the modeling, pricing and hedging of interest rate and credit risk.
Rüdiger Frey
In this
project stochastic filtering methodology will be employed for
solving pricing, hedging and calibration problems in interest rate and
credit risk models. Stochastic filtering is concerned with the
detection of signals from noisy observations. In interest rate and
credit risk modeling, filtering problems arise naturally since
important state variables such as firm values cannot be observed
directly by investors.
Existing filtering results are not yet sufficient for the application
to complicated problems in model calibration and derivative pricing.
The mathematical contribution of this project will therefore be the
generalization of filtering results from the literature and the
development of new numerical methods. On the financial side the project
will contribute to a better understanding of dynamic credit risk
models, including counterparty credit risk and credit contagion.
Moreover, risk management techniques for derivatives such as dynamic
hedging will be analyzed with the help of filtering. The practical
relevance of these issues has been highlighted during the current
financial crisis.
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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
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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".
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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.
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Currently we are focusing on
credit risk models under incomplete information; details are given below.
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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.
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