Heavy-tailed sums: simulation and dependence

Abstract: Sums of the type X_1+...+X_n with heavy-tailed summands show up
in a number of contexts, in particular insurance risk, credit risk and  operational risk. With starting point in the classical approximations from the subexponential area, we proceed to present a variety of methods for efficient simulation of the tail probabilities, including importance sampling and conditional Monte Carlo. Further the extension from  independence to dependence is studied: when does the classical subexponential asymptotics stay in force, and what else could happen?

[1] Applied Probability and Queues, Second Edition. Springer–Verlag, New York (2003).
[2] (with P.W. Glynn) Stochastic Simulation: Algorithms and Analysis. Springer-Verlag (2007).
[3] (with H. Albrecher) Ruin Probabilities (2nd ed.). 606+x pp. Advanced Series in Statistical
Sciences & Applied Probability 14, World Scientific Publishing Co.
[4] (with H. Albrecher D. Kortschak) Tail asymptotics for dependent subexponential differences. Submitted.
[5] 1 13 16 17 from http://imf.au.dk/publication/publid/919/
[6] P. Glasserman (2004) Monte Carlo Methods in Financial Engineering. 
Springer-Verlag.

Extremal behaviour of stochastic processes

Abstract: Extreme value theory is a statistical discipline that develops techniques and models for describing rare events of extremes. The design of protection systems against the effects of extreme events in such diverse fields as financial risk management, insurance mathematics, engineering sciences and environmental engineering use extreme value models and techniques. The classical theory investigates extremal events, e.g., quantiles and return periods, for independently and identically distributed random variables. However, mostly the independent assumption is violated such that more sophisticated models and statistics are necessary. In particular, the sample autocovariance function is not a sufficient measure to describe clustering in extremes, a typical phenomenon in financial time series and to distinguish between different models. For heavy tailed models the autocorrelation function does already not exists. The aim of this lecture is to analyze the extremal behavior of stationary time series models, e.g., ARMA and GARCH models and their continuous-time counterparts, and to study different kind of extremal dependence measures as the extremogramm, a correlation function for extreme events. 

[1] J. Beirlant, Y. Goegebeur, J. Segers and J.L. Teugels (2004) Statistics of Extremes: Theory and Applications, Wiley.
[2] R. Davis and T. Mikosch (2009) The extremogramm: a correlogram for extreme events. Bernoulli 15, pp. 977-1009.
[3] S.G. Coles (2001) An Introduction to Statistical Modeling of Extreme Values. Springer.
[4] P. Embrechts, C. Klüppelberg and T.Mikosch (1997) Modelling Extremal Events for Insurance and Finance. Springer.
[5] V. Fasen, C. Klüppelberg and M. Schlather (2010) High-level dependence in time series models, extremes 13, pp. 1-33.
[6] Fasen (2009) Extremes of continuous-time processes. In: T.G. Andersen, R. A. Davis, J.-P. Kreiss and T. Mikosch (Eds.), Handbook of Financial Time Series, Springer, pp. 653-667.
[7] V. Fasen, C. Klüppelberg and Lindner, A. (2006) Extremal behavior of stochastic volatility models  In: A. Shiryaev, M.d.R. Grossinho, P. Oliviera, M. Esquivel (Eds.), Stochastic Finance, Springer, New York, pp. 107-155.
[8] A.J. McNeil, R. Frey and P. Embrechts (2005) Quantitative Risk Management: Concepts, Techniques, and Tools. Princton University Press.
[9] T. Mikosch and C. Starica (2000) Limit theory for the sample autocorrelations and extremes of a GARCH(1,1) process. Ann. Statist. 28, pp. 1427-1451.

Statistical methods of high-frequency data I

Abstract: In this course we will explain the basic concepts of statistical methods for high frequency observations of semimartingales. Semimartingales is a class of processes which are used in finance to model asset prices or derivatives. Using the availability of vast amount of intraday data we would like to extract as much information about the characteristics of the (true) price process as possible. For this purpose we will develop some theoretical tools including some non-standard laws of large numbers and central limit theorems. Then we will apply those methods to derive estimation and testing procedures for the characteristics of a semimartingale.

[1] O.E. Barndor-Nielsen and N. Shephard: Variation, jumps, market frictions and high  frequency data in financial econometrics: In R. Blundell, T. Persson and W.K. Newey (Eds.), Advances in Economics and Econometrics. Theory and Applications, Ninth World Congress, Econometric Society Monographs, Cambridge University Press 2007, pp. 328-372.
[2] J. Jacod: Statistics for high frequency data. Lecture notes, 2008.
[3] M. Podolskij and M. Vetter: Understanding limit thoerems for semimartingales: a short survey. Statistica Nederlandica 64(3), 329-351, 2010.
[4] P. Protter: Stochastic integration and differential equations. Springer, 2nd edition, 2003.

Mathematical aspects of electronic trading and market impact

Abstract: Recent years have seen dramatic changes in the way financial markets are functioning. For instance, most trades are no longer executed by brokers but by automated trading algorithms that operate in electronic limit order books. In addition, "dark" trading venues are being created in which large trades can  be executed without information leakage to the public.
In this course, we will present overview on some mathematical problems arising in this context. In particular, we will look at the situation in which trading strategies can move the underlying asset price. In dealing with this situation, we first need a suitable modeling framework. We will thus discuss several model classes proposed in the literature. In characterizing the viability of such models, it turns out that requiring the absence of arbitrage opportunities in the usual sense may not be enough. We illustrate this fact by several examples. In particular, we will come accross the possibility of profitable price manipulation strategies. We will take the point of view that a viable viable market impact model should not encourage such price manipulation strategies and, for some model classes, we will characterize their absence mathematically. Other topics will include the mathematical modeling of electronic limit order books and of dark pools.

[1] Gatheral, J., Schied, A. & Slynko, A. (2011), ‘Exponential resilience and decay of market impact’, In: Econophysics of Order-driven Markets. F. Abergel, B.K. Chakrabarti, A. Chakraborti, M. Mitra (Eds.), pp. 225-236, Springer (2011). 
[2] Gökay, S., Roch, A., and Soner, H. M. (2010). Liquidity models in continuous and discrete time. Preprint.
[3] Schied, A. and Slynko, A. (2011). Some mathematical aspects of market impact modeling. In J. Blath, P. Imkeller, and S. Roelly, editors, Surveys in Stochastic Processes. Proceedings of the 33rd SPA. EMS Series of Congress Reports.

Methods of spatial stochastic modelling, with applications to analysis and simulation of insurance risks

Abstract: For assessing the probabilities and intensities of geo-risks, like (re-) insured losses caused by global warming, continental storms, tropical cyclones, flooding, and earthquakes, it is necessary to analyze spatially resolved data regarding the occurrence of these risks. Unfortunately, the historical data available for risk assessment is relatively limited, in particular with respect to the occurrence of extreme events. For instance, historical data on tropical cyclones making landfall in the North Atlantic cover a time span of about 150 years, whereas reinsurers need to include wind speeds with much lower exceedance frequencies (< 0.001 p.a.) in their calculations. Mathematical models from stochastic geometry, like random marked point processes and random geometric graphs [1, 2], can help to solve this problem. Fitting these models to historical data, synthetic but realistic tracks of tropical cyclones can be simulated for much larger time horizons than provided by the historical data [3].

[1] Illian J., Penttinen A., Stoyan H., and Stoyan D. (2008) Statistical Analysis and
Modelling of Spatial Point Patterns. J. Wiley & Sons, Chichester.
[2] Kendall W.S. and Molchanov I. (eds.) (2010) New Perspectives in Stochastic Geometry.
Springer, Berlin.
[3] Rumpf J., Weindl H., Hoeppe P., Rauch E. and Schmidt V. (2009) Tropical cyclone hazard assessment using model-based track simulation. Natural Hazards 48, 383-398.

Jochen Ruß /
Hans-Joachim Zwiesler