SFB Workshop
SFB Workshop |
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- Integration of Operational Risk in the Risk Management Framework
- Quantification of Operational Risk
- Input Data
- Control Risk Assessment
- Combining input data
- Backtesting
Default distributions of credit portfolios with high granularity often can be approximated by means of analytic and semi-analytic techniques. If a portfolio is not too inhomogeneous, (semi-)analytic approximations yield remarkably good results. Typically in such approximations the portfolio default quote can be written as a function of just one single random variable. In our talk we discuss a comparable semi-analytic approach which works also for portfolios with low granularity. Hereby we rely on the upper Frechet or comonotonic copula for the default quote path of the portfolio. Applications include the evaluation of basket credit derivatives and other structured finance products.
In order to capture features of financial time series such as tail heaviness and persistence of the associated volatility, Lévy driven non-linear time series models provide a useful framework. We discuss several such models, including continuous-time threshold, stochastic volatility and GARCH models, which have been found useful in the modelling of asset returns.
References:
- Barndorff-Nielsen, O.E. and Shepherd, N. (2001): Non Gaussian Ornstein-Uhlenbeck-based models and some of their uses in financial economics, JRSS B, 63, 167 - 241.
- Brockwell, P.J. (2004): Representations of continuous-time ARMA processes, J. Appl. Prob., 41A, 375-382.
- Brockwell, P.J. and Williams, R.J. (1997): On the existence and application of continuous-time threshold autoregressions of order two, Adv. Appl. Prob., 29, 205-227.
- Klüppelberg, C., Lindner, A. and Maller, R. (2004): Stationarity and second-order behaviour of discrete and continuous time GARCH(1,1) processes, J. Appl. Prob., to appear.
We are interested in models for financial tick-by-tick data. Since such data are observed at irregularly spaced time points, we suggest a continuous time model as a convenient way to cope with this feature. We consider a nonparametric regression model, where the response is a function of the log-return and the error process is a discretely sampled Continuous time ARMA(p,q) process. In this talk we concentrate on estimating the parameters of the error process. For this we introduce second order Levy driven CARMA(p,q) processes (Brockwell (2001)). Maximum likelihood and Whittle type estimation - in case of Poisson sampling - is presented and compared in a simulation study.
Discussions concerning Basel II recently focus on the problem of procyclicality and especially the impact of bank capital regulation. The question is whether fluctuations in risk capital amplify the problem of procyclicality. In this talk a hypothetic loan portfolio is used to analyse variations in bank risk capital during a business cycle. Based on empirical default rates an easy modelling setup is developed to investigate fluctuations and potential macroeconomic reactions. In addition the relevance of economic and regulatory capital is examined.
This talk deals with the modelling of financial data. A "good" model should fit the data, but also be able to be used for option pricing. Particular attention will be paid to complete models with stochastic volatility. We will present both continuous time and discrete time versions, and detail their main probalistic and financial properties. We will compare these models with more classical ones, such as stochastic volatility models and ARCH models. Statistical issues will be also discussed, with applications to real data sets coming from european markets.
We consider a portfolio credit risk model in the spirit of CreditMetrics. The multivariate normally distributed underlying risk factors in the model are replaced by more general multivariate elliptical factors with heavy-tailed marginals. We consider a full-scale version of the model, i.e. we incorporate not only the default risk, but also rating migrations, credit spread volatility and recovery risk. We derive an upper bound of the portfolio loss distribution, which works particularly well at high loss levels. As an application, we provide an approximation of VaR and a method to determine the contributions of the individual credits to the risk of the portfolio.
Logistic regression and discriminant analysis are widely used methods in credit scoring to find linear combinations of covariates which discriminates between defaults and non-defaults. A popular technique of assessing the performance of the resulting score is the receiver operating characteristic (ROC) as well as its summary statistic, the area under the curve (AUC).
In this talk, we compare a standard mechanism of variable selection within the framework of logistic regression to an approach which directly connects the choice of variables to the AUC criterion. Given a certain subset of variables, we extend this idea by seeking the linear combination which maximises the accuracy index associated with the resultant score. We propose a distribution-free approach rather than counting on the assumption of a multivariate normal distribution.
Abstract in pdf-format.
In the light of Basel II, redesigning rating systems has been becoming an important issue for banks and other financial institutions. The available data base for this task typically contains only the accepted credit applicants and is thus censored. To evaluate existing and alternaive rating systems, we would actually need the full data base of all past credit applicants. In this talk we discuss how to assess the performance of credit ratings under the assumption that for credit data only a part of the defaults and non-defaults is observed. The talk investigates criteria that are based on the difference of the score distributions under default and non-default such as the accuracy ratio. We show how to estimate bounds for these criteria in the usual situation that the bank storages only data of the accepted credit applicants.
Dependence modellingCombining qualitative and quantitative data
- Severity and frequence distributions
- Size effects
- Factor model versus Copula
- Mixing probability distributions
- Bayesian approaches
Two new estimators for the first-order autoregressive coefficient in a linear regression model with arbitrary exogenous regressors are proposed and extended to the ARMAX(1,q) model. The estimators are similar to the method of median unbiased estimation of Andrews (1993), but exhibit smaller mean squared error over the entire parameter space, most notably in the high-persistence and unit root cases. The relative properties of the estimators appear to be virtually invariant to choice of design matrix, sample size and even the distribution of the innovation process, so that general conclusions and recommendations can be made. Several applications in finance are considered.
Using Lévy processes in modelling financial time series data can capture the fat-tail or non-gaussianity behavior of returns, but is unable to reproduce the observed phenomenon of volatility clustering. Therefore, Carr et al. (2003) and Carr and Wu (2004) proposed to subordinate a pure jump Lévy process of infinite activity by a nondecreasing process. This can be interpreted as a time-change of calendar time to economic activity. So far literature has focused on pricing options within this framework. In contrast we examine time-changed Lévy processes under the physical measure, being inter alia important for risk assessment. Based on the efficient method of moments proposed by Gallant and Tauchen (1996), we develop a method to estimate time-changed Lévy processes. Our empirical results are contrasted to the more common alternative models for stock price dynamics.
References:
- Gallant, A. Ronald and George Tauchen (1996): Which Moments to Match? Econometric Theory 12, 657-681.<(li>
- Carr,Peter and Hélyette Geman and Dilip B. Madan and Marc Yor (2003): Stochastic Volatility for Lévy Processes. Mathematical Finance 13, No. 3, 345-382.
- Carr, Peter and Liuren Wu (2004): Time-changed Lévy processes and option pricing. Journal of Financial Economics 71, No. 1, 113-141.
Under the symmetric alpha-stable distributional assumption for the disturbances, Blattberg et al (1971) consider unbiased linear estimators for a regression model with non-stochastic regressors. We consider the situation where both the regressors are stochastic, and the regressors and/or the disturbances can be heavy-tailed with either finite or infinite variances, or even means. We get rid of the assumption of stability. Furthermore, the tail-thickness parameters of the regressors and disturbances may be different.
We consider both the rate of convergence to the true value and the asymptotic distribution of the normalized error . It turns out that a comparison of different estimators depends heavily of the tail exponents of the regressors and the disturbances. We address the nontrivial issue of selecting good estimators when these tail exponents are not known precisely.
The amount of loss a bank suffers with a defaulted credit is, in general, not known in advance and must be modelled as a random variable. Recent research presented evidence for stochastic dependence of default events and realized loss given default (LGD) rates. We consider two approaches to dependence modelling for LGD rates: a first one based on an assumption of conditional independence of defaults and LGD rates, and a second one which may be characterized as a copula approach. The copula approach turns out to result in higher risk measurements but -- compared to the conditional independence approach -- reduces estimation complexity as the set of required parameters is smaller. This observation is illustrated with a Basel-II-style one factor credit portfolio model.