PRisMa 2009

Prof. Dr. Uwe Schmock
(FAM @ TU Wien)
Welcome

Prof. Dr. Alexander Schied
(Universität Mannheim)
Order Book Resilience, Price Manipulations, and the Positive Portfolio Problem

Abstract: We consider a class of limit order book models, extending the block-shaped, exponential-resilience model introduced by Obizhaeva and Wang (2005). An important question is the viability of these models. Examples show that the requirement of the absence of price manipulation strategies as introduced by Huberman & Stanzl (2004) is not sufficient to guarantee that the model is indeed well-behaved, as these examples admit price manipulation strategies in a weak sense. For block-shaped limit order book models we can characterize those resilience functions for which there are no price manipulation strategies in both the usual and the weak sense. We also discuss the case of nonlinear price impact and exponential resilience. The talk is based on joint work with A. Alfonsi and A. Slynko.

Download the slides of the talk.

Prof. Dr. Walter Schachermayer
(Universität Wien)
Representation Results for Law Invariant Time Consistent Functions

Abstract: We show that the only dynamic risk measure which is law invariant, time Consistent and relevant is the entropic one. Moreover, a real valued function c on L^∞ (a,b) is normalized, strictly monotone, continuous, law invariant, time consistent and has the Fatou property if and only if it is of the form c(X)=u^-1(E[u(X)]), where u:(a,b) → R is a strictly increasing, continuous function. The proofs rely on a discrete version of the Skorohod embedding theorem.

Abstract: We study risk assessment for cash flows using dynamic convex risk measures for processes introduced in Cheridito, Delbaen, and Kupper (2006). These risk measures account not only for the amount but also for the timing of a cash stream. We show that ambiguity on the discounting factor becomes a part of model uncertainty in this case, and we make the role of discounting visible in the robust representation of the risk measure. In particular uncertainty about discounting leads to cash sub-additivity of the risk measure. Risk assessment for cash flows can be embedded into the framework of risk measures for random variables by considering processes as random variables on an appropriate product space. Using this relation we extend various characterizations of time consistency properties of dynamic convex risk measures for random variables to the framework of risk measures for processes. The results will be illustrated by some examples. (This talk is based on a joint work in progress with Hans Foellmer and Irina Penner.)

Dipl.-Math. Barbara Dengler
(PRisMa Lab, FAM @ TU Wien)
On the Asymptotic Variance of the Estimator of Kendall's Tau for the t-Distribution

Abstract: We are interested in the performance of different estimators of dependence measures. Our work is motivated by the fact, that for elliptical distributions there is a unique connection between the linear correlation coefficient and the rank-based measure Kendall's tau. This offers two ways for estimation: using the standard estimator of the linear correlation directly or estimating Kendall's tau and transforming it into linear correlation. The second way is promising as one has the advantages of a rank-based measure like e.g. robustness concerning outliers. As the estimator of Kendall's tau is a U-statistic, it is asymptotically normal. Also the standard estimator of the linear correlation has this property under appropriate conditions, so we use the asymptotic variance to compare the performance of the two estimators. The asymptotic variance of the estimator of Kendall's tau originally has a non-trivial form. For spherical distributions that have a density we found a representation that consists of several nested 2-fold definite integrals, which are generally very complicated to solve and can especially not be computed in closed form e.g. in Mathematica. Nonetheless we found further simplifications for the uncorrelated t-distribution. Using the fact that its density can be seen as a mixture of normal densities with the gamma density as mixing function, we show that the asymptotic variance equals an integral involving the square of the arctangent and a hypergeometric function. After involved and tricky integration strategies we end up with a surprisingly simple closed form for the asymptotic variance for all integer-valued degrees of freedom. It turned out that especially for small degrees of freedom the alternative estimation via Kendall's tau performs much better than the standard estimator. (This is joint work with U. Schmock.)

Dipl.-Math. Verena Goldammer
(FAM @ TU Wien)
Generalization of the Dybvig-Ingersoll-Ross Theorem and Asymptotic Minimality

Abstract: The long-term limit of zero-coupon rates with respect to the maturity does not always exist. In this case we use the limit superior and prove corresponding versions of the Dybvig–Ingersoll–Ross, which says that long-term spot and forward rates can never fall in an arbitrage-free model. Examples of models needing this generalization are presented. In addition, we discuss several definitions of arbitrage, prove asymptotic minimality of the limit superior of the spot rates, and illustrate our results by several continuous-time short-rate models. (Joint work with U. Schmock.)

Dr. Gregory Temnov
(University College Cork)
Natural Exponential Family with Stability Property with Application to Financial Modelling

Abstract: We investigate a class of distributions originating from an exponential family and having a property related to the strict stability. The characteristic functions representation for this family is obtained and its properties are investigated. The parallels with so-called Truncated Levy flights often used to model the increments of financial prices are analyzed, regarding the possible advantages of the exponential family with stability property. Finally, illustrating the results with financial data, we find that the introduced distribution can attain a reliable fit to empirical data and has more flexibility in fitting than Stable distributions or the Truncated Levy flights. (This work is a joint work with professor Lev B. Klebanov from Charles University Prague.)

Dipl.-Ing. Christa Cuchiero
(ETH Zürich)
Affine Processes on Positive Semidefinite Matrices

Abstract: Motivated by applications in mathematical finance and in particular by multivariate stochastic volatility models, we study stochastically continuous affine processes on S⁺_d, the cone of positive semidefinite symmetric matrices. A complete characterization of this class of Markov processes is provided through a detailed parameter specification of the infinitesimal generator. The derivation of these necessary and sufficient conditions on the parameters relies on stochastic invariance results for closed convex sets. It is further shown that affine processes on S⁺_d are automatically regular and admit the Feller property. The subclass of processes without diffusion component is characterized by infinite divisibility.