KT_37_Sun.qxd 30.05.2008 09:35 Seite 2

Forschung Lehre Wirtschaft fuks intern

A New Solution for - Stable Family Models

Von Framework. In contrast to normal or Gaus- yields superior risk-adjusted returns relative Prof. Dr. Svetlozar Rachev, Dr. Wei Sun, sian distributions, stable distributions captu- to what is known as the mean-variance or Prof. Dr. Frank J. Fabozzi re the fat tails and the asymmetries of real- Markowitz efficient portfolios. Institut für Statistik und Mathematische Wirt- world risk factor distributions. In addition, we Finally, in our research, we introduce an schaftstheorie make use of copulas, a generalization of alternative performance measure tools for overly restrictive linear correlation models, evaluating professional asset managers: to account for the dependencies between the stable tail adjusted return ratio (STARR), entral to is the risk factors during extreme events, and mul- which is a generalization of the well-known assumption about the return dis- tivariate ARCH-type processes with stable Sharpe ratio with the stable distribution fra- C tribution of assets. Conventional innovations to account for another stylized mework. approaches to modeling asset returns are fact observed for asset returns: joint volatili- We also present a new approach for based either on a historical or a Gaussian ty clustering. We demonstrate that the ap- using fractional stable models to compute distribution for returns. There is overwhel- plication of these techniques results in more VaR by using high-frequency data based ming empirical evidence suggesting that accurate modeling of extreme risk event on the requirement of finding day-trading neither approach adequately captures the probabilities, and consequently behavior of asset prices and returns. The delivers more accurate risk me- historical model is bounded by the extent of asures for both trading and risk the available observations and the proper- management. ties of the Gaussian distribution model are Using these superior models, such that it cannot capture extreme returns commonly used measures of that have been observed in real-world fi- risk used by institutional inves- nancial markets. In order to overcome the tors such as value-at-risk (VaR) inadequacy of these models, we have sug- become a much more accurate gested alternative return distribution models measure of downside risk. More in our research papers. More specifically, importantly stable expected tail we have proposed modeling asset returns loss (SETL) can be accurately using stable family models which includes calculated and used as a more the stable Paretian model, tempered stable informative measure of market model, and fractional stable model. We app- risk and credit risk for portfolios lied these models in finance to the areas of and trading positions. Along with , portfolio optimization, being a superior risk measure, funds management, financial products pri- SETL enables an elegant appro- cing, and strategic trading optimization. ach to portfolio optimization via We introduce a practical alternative to convex optimization that can be Gaussian risk factor distributions based on solved using standard scalable Svetlozar Rachev's book coauthored with linear programming software. In Stefan Mittnik, Stable Paretian Models in Fi- our research, we demonstrate nance, and called the Stable Distribution that SETL portfolio optimization

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Forschung Lehre Wirtschaft fuks intern

red to those of a standard nonparametric estimation method that captures the empiri- cal distribution function, and with models where tail events are modeled using Gaus- sian distribution and fractional Gaussian no- ise. The results suggest that the proposed parametric approach yields superior predic- tive performance. We introduce an alternative class of tem- pered stable distributions which we call mo- dified tempered stable (MTS) distribution model. The model is sufficiently flexible in describing the skewness and kurtosis of as- set returns and has all moments that are fi- nite. The Lévy process derived from the MTS distribution is included in the class of regular Lévy processes of exponential (RLPE). Fur- thermore, we obtain the MTS distribution applying the exponential tilting to the sym- metric MTS distribution. Based on the MTS model, we introduced an enhanced GARCH model, namely the MTS-GARCH model, by applying MTS inno- strategies. Our ap- vations to the classical proach is a parame- GARCH model. As a result, tric model using the MTS-GARCH time series an ARMA(1,1)- model for stock returns ex- GARCH(1,1) model plains the volatility clustering where the tail phenomenon, the leverage ef- events are modelled fect, and both conditional using the fractional skewness and leptokurtosis. stable model. Using The risk-neutral measure is high-frequency data obtained by applying a change for the German DAX of measure to the MTS distri- Index, the VaR esti- bution. The MTS-GARCH mo- mates from this ap- del is a more realistic model proach are compa- than the normal-GARCH mo- del. Currently, we are resear- ching optimal intraday trading strategies (which calls for the application of sophisticated data mining methods), analy- zing performance of fund of funds, studying dynamic risk measures and asset/liability management, and building an early warning system for bankruptcy. In addition, we are doing research on behavioral finance, for example, investi- gating the equity premium puzzle.

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