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From September 2011, the MRM seminars will
be integrated in the new STO(chastics) Seminar Series.
Multivariate Risk Modelling Seminar 2010 Dorota Kurowicka (Delft University of Technology), December 8, 2010 Regular Vines - New developments Copulae (distributions on unit hypercube with uniform margins) have become
very popular in dependence modeling in financial as well as other engineering
contexts. Bivariate copulae are well studied, understood and applied.
Multivariate copulae, however, are often limited in range of correlation
structures and other properties as e.g. tail dependence that they can handle. A
new graphical model introduced in 1997, called regular vines, allows
specification of a joint distribution on n variables with given margins by
specifying n- choose-2 bivariate copulae and conditional copulae. A landmark
advance in associating bivariate copulae to a vine and estimating copula
parameters from data via maximum likelihood principle (pair-copula construction)
demonstrated the superiority of vines and opened large areas of application in
mathematical finance, risk analysis and uncertainty modeling in engineering. Florin Avram (Université de Pau)
There is considerable interest lately in statistics and in mathematical finance
in the class of diffusions models with a second order polynomial variance and
a first order polynomial drift, which was already considered in Kolmogorov’s
founding paper in 1931. The generator of this process is a second order
differential operator well known for its connections to Gauss's hypergeometric
function, orthogonal polynomials, etc. Albert Ferreiro Castilla (Universitat Autònoma de Barcelona), November 16, 2010
We prove that certain quotients of entire functions are characteristic functions. Under some conditions, the probability measure corresponding to a characteristic function of that type has a density which can be expressed as a generalized Dirichlet series, which in turn is an infinite linear combination of exponential or Laplace densities. These results are applied to several examples. We also explore the Wiener-Hopf factorization under Levy processes with such jumps. Sami Umut Can (Cornell University, Ithaca, USA), June 2, 2010 Random Rewards, Brownian Additive Functionals and Stable Self-Similar Processes We describe a class of symmetric alpha-stable processes with stationary increments that arise as a large time scale limit in a situation where many users are performing random walks and collecting random rewards. The resulting models are asymptotically or exactly self-similar. This is a generalization of an earlier result published by S. Cohen and G. Samorodnitsky in 2006. Alexander Herbertsson (University of Gothenburg, Sweden), May 25, 2010 Pricing basket default swaps in a tractable shot-noise model We value CDS spreads and k-th-to-default swap spreads in a tractable shot noise model. The default dependence is modelled by letting the individual jumps of the default intensity be driven by a common latent factor. The arrival of the jumps is driven by a Poisson process. By using conditional independence and properties of the shot noise processes we derive tractable closed-form expressions for the default distribution and the ordered survival distributions in a homogeneous portfolio. These quantities are then used to price and study CDS spreads and k-th-to-default swap spreads as function of the model parameters. We study the k-th-to-default spreads as function of the CDS spread, as well as other parameters in the model. All calibrations lead to perfect fits. Abhimanyu Mitra (Department of OR&IE, Cornell University, Ithaca, NY), April 13, 2010 Two problems in tail probability estimation 2009 Andrea Krajina, Universiteit van Tilburg An M-Estimator of Tail Dependence in Arbitrary Dimensions Let X1, X2,…, Xn be a d-dimensional random sample from a distribution function F. Assume that F is in the max-domain of attraction of an extreme-value distribution and assume that the dependence structure of the extreme-value comes from some parametric family. We propose an M-estimator of the unknown vector of parameters. The estimator is defined as the value of the parameter vector that minimizes the distance between the vector of weighted integrals of the tail dependence function on the one hand and empirical counterparts of these integrals on the other hand. We show that the estimator, with probability tending to one, exists and is a unique, global solution of the minimization problem. Under natural conditions we prove that the estimator is consistent and asymptotically normal. Since the differentiability of the tail dependence function is not required, the method applies to discrete models as well. The small sample behaviour of the estimator and its applicability are demonstrated on examples. A special case of this estimator, when d = 2, and the number of equations cannot exceed the number of parameters, is the one proposed in J.H.J. Einmahl, A. Krajina and J. Segers [Bernoulli, 14(4), 2008, 1003-1026]. Alexander Herbertsson (department of Economics/Centre for finance School of Business, Economics and Law at the University of Gothenburg, Sweden), May 29, 2009 Default contagion in large homogeneous portfolios We study default contagion in large homogeneous credit portfolios. Using data from the iTraxx Europe series, three synthetic CDO portfolios are calibrated against their tranche spreads, index CDS spreads and average CDS spreads, all with five year maturity. After the calibrations, which render perfect fits, we investigate the implied expected ordered defaults times, implied default correlations, and implied multivariate default and survival distributions, both for ordered and unordered default times. Many of the numerical results differ substantially from the corresponding quantities in a smaller inhomogeneous CDS portfolio. Furthermore, the studies indicate that market CDO spreads imply extreme default clustering in upper tranches. The default contagion is introduced by letting individual intensities jump when other defaults occur, but be constant between defaults. The model is translated into a Markov jump process. Expressions for the investigated quantities are derived by using matrix-analytic methods. Mitja Stadje (Department of Operations Research & Financial Engineering, Princeton University), February 3, 2009 Stochastic Calculus and Dynamic Risk Measures The main aim of this talk is to present an approach for the transition from risk measures in discrete time to their counterparts in continuous time. After a general introduction to risk assessment in mathematical finance it is shown that a large class of risk measures in continuous time can be obtained very naturally as limits of time-consistent risk measures in a discrete setting. The discrete-time risk measures are constructed from properly rescaled ('tilted') one-period risk measures, using a d-dimensional random walk converging to a Brownian Motion. Under suitable conditions (covering the classical one-period risk measures) we obtain convergence of the discrete risk measures to the solution of a backward stochastic differential equation, defining a risk measure in continuous time, whose driver can then be viewed as the continuous-time analogue of the discrete 'driver' characterizing the one-period risk. We derive the limiting drivers for the semi-deviation risk measure, Value at Risk, Average Value at Risk, and the Gini risk measure in closed form.
Last modified:
14-10-11
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