Lectures and courses

Stefan Ankirchner

The Skorokhod embedding problem - applications to economics and finance

Abstract: The course deals with the Skorokhod embedding problem (SEP): Let W be a Brownian motion and µ a centered distribution. The SEP consists in finding an integrable stopping time τ such that the Brownian motion W stopped at τ has the distribution µ.
Research on the SEP has recently got some stimulus from new economic applications. For example, the SEP appears in winner-take-all contests, where players aim at stopping a BM, possibly with drift, at a highest possible value. Such contests possess Nash equilibrium distributions that need to be embedded with suitable stopping times.
In some financial applications Skorokhod embeddings are used for the derivation of model independent bounds for exotic option prices.
The aim of the course is to present some well-known solutions of the SEP and to illustrate the applications.

Søren Asmussen

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](with H. Albrecher, S D. Kortschak) (2006) Tail asymptotics for the sum of two heavy-tailed dependent risks. Extremes 9, 107–130.

[6] P. Glasserman (2004) Monte Carlo Methods in Financial Engineering. Springer-Verlag

[7] S. Foss, A. Richards (2010) On sums of conditionally independent subexponential random variables. Math. Oper. Res. 35, 102-119      

[8] B. Ko, Q. Tang (2008) Sums of dependent nonnegative random variables with subexponential tails. J. Appl. Prob. 45, 85-94      

[9] D. Kortschak, H. Albrecher (2009) Asymptotic results for the sum of dependent nonidentically distributed random variables. Methodol. Comput. Appl. Probab. 11, 279-306.


Vicky Fasen


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.

Jean Jacod

Statistical methods of high-frequency data II

Abstract: This course is a following of Mark Podolskij's course. Building upon the tools developed in the first part, we will consider in more details some specific problems, in two directions: one
is the analysis of jumps, with emphasis on the so-called degree of activity of jumps. The other direction is about spot volatility: we will show how it can estimated, how its jumps can
be located, and how it allows for efficient estimations of various functionals, such as the quarticity.

Georg Pflug

Risk measures

Abstract: We review well known and some not so well known properties of single period risk functionals and then go over to conditional risk functionals. A particular emphasis is put on convex dual representations and  subgradients. These representations allow to find sensitivities and worst case directions.  Model ambiguity may be incorporated by finding the maximal risk within a family of possible models.
We then pass to multiperiod risk functionals. Time consistency, information monotonicity and compound convexity are crucial properties for these functionals. Also decomposition properties are studied, which link multiperiod risk management to dynamic optimization. Again, model ambiguity may be incorporated.
Finally, we demonstrate how risk and acceptability functionals may replace the superreplication requirement in pricing of derivative contracts, which leads to new pricing principles.      

[1] P. Artzner, F. Delbaen, D. Heath, and H. Ku. Coherent multiperiod risk adjusted values and
Bellman's principle. Annals of Operations Research, 152(1):5_22, 2007.
[2] P. Cheridito and M. Kupper. Composition of time-consistent dynamic monetary risk measures in discrete time. International Journal of Theoretical and Applied Finance, 14(1):137_162, 2011.
[3] K. Detlefsen and G. Scandolo. Conditional and dynamic convex risk measures. Finance and
Stochastics, 9:539_561, 2005.
[4] A. Eichhorn and W. Römisch. Polyhedral risk measures in stochastic programming. SIAM J.
Optim., 16:69 _ 95, 2005.
[5] H. Föllmer and A. Schied. Convex measures of risk and trading constraints. Finance and Stochastics, 6:429_447, 2002.
[6] M. Fritelli and G. E. Rosazza. Putting order in risk measures. Journal of Banking and Finance,
26, 2002.
[7] R. M. Kovacevic and G. C. Pflug. Time consistency and information monotonicity of multiperiod acceptaility functionals. In H. Albrecher, W. Runggaldier, and W. Schachermayer, editors, Advanced Financial Modelling, volume 8 of Radon Series on Computational and Applied Mathematics, pages 347_370. de Gruyter, 2009.
[8] M. Kupper and W. Schachermayer. Representation results for law invariant time consistent
functions. Mathematics and Financial Economics, 2:189 _ 210, 2009.
[9] G. Pflug and W. Römisch. Modeling, Measuring and Managing Risk. World Scienti_c, August
[10] G: Pflug and A. Pichler. Decomposition of Risk Measures. Manuscript.

Mark Podolskij

Statistical methods of high-frequency data I

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. Barndorff-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 and high frequency data. In "Statistical Methods for Stochastic Differential Equations", M. Kessler, A. Lindner, M. Sorensen eds, Taylor and Francis: London. 191-310 (2012)
[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.

Alexander Schied

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. Further topics may include the mathematical modeling of electronic limit order books,  stability issues with dark pools, and multi-player equi

[1] Gatheral, J. and Schied, A. (2012). Dynamical models for market impact and algorithms for optimal order execution. To appear in: Handbook on Systemic Risk (eds.: J.-P. Fouque and J. Langsam), Cambridge University Press.
[2] Gökay, S., Roch, A., and Soner, H. M. (2010). Liquidity models in continuous and discrete time. Preprint.

Volker Schmidt

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.

Robert Stelzer

Multivariate stochastic volatility modelling

Abstract: The use of stochastic volatility models is very popular in finance, as they allow to capture most of the so-called stylised features of the price series observed at financial markets. In particular, they model the occurrence of upheated (volatile) phases (at times of “crises”) followed by calm phases where the typical price changes are much smaller by specifying a latent stochastic process which describes the time-dynamic variance of the price process.
In a multivariate setting one needs to model a whole covariance matrix changing randomly over time, i.e. one needs a stochastic process in the positive semi-definite matrices. In this course we explain the challenges of defining stochastic processes in the positive semi-definite matrices and analyse several possibilities in detail.
Thereafter we study the arising stochastic volatility models focusing on the Wishart and the Ornstein-Uhlenbeck stochastic volatility models. We study l the properties of these models, the pricing of derivatives as well as statistical estimation based on historical data and calibration to option prices.
If time permits, we will also give an introduction to other multivariate stochastic volatility models, e.g. the multivariate COGARCH and supOU model, and into the extremal behaviour of multivariate stochastic volatility models.

[1] O. E. Barndorff-Nielsen and R. Stelzer. Positive-definite matrix processes of finite variation. Probab. Math. Statist., 27:3–43, 2007.
[2] O. E. Barndorff-Nielsen and R. Stelzer. The multivariate supOU stochastic volatility model. Math. Finance, 2012. to appear.
[3] R. Cont and P. Tankov. Financial Modelling with Jump Processes. Chapman & Hall, London, 2004.
[4] C. Cuchiero, D. Filipović, E. Mayerhofer, and J. Teichmann. Affine Processes on Positive Semidefinite Matrices. Ann. Appl. Probab., 21:397–463, 2011.
[5] J. Da Fonseca, M. Grasseli, and C. Tebaldi. Option Pricing when Correlations are Stochastic: An Analytical Framework. Review of Derivatives Research, 10:151–180, 2007.
[6] C. Gourieroux. Continuous time Wishart process for stochastic risk. Econometric Rev., 25:177–217, 2006.
[7] C. Gourieroux and R. Sufana. Derivative Pricing with Wishart Multivariate Stochastic Volatility. J. Bus. Econom. Statist., 28:438–451, 2010.
[8] J. Muhle-Karbe, O. Pfaffel, and R. Stelzer. Option pricing in multivariate stochastic volatility models of OU type. SIAM J. Financial Math., 3:66–94, 2011.
[9] C. Pigorsch and R. Stelzer. A multivariate Ornstein-Uhlenbeck type stochastic volatility model. submitted for publication, 2009. available here.
[10] R. Stelzer. Multivariate COGARCH(1,1) Processes. Bernoulli, 16:80–115, 2010.

Jochen Ruß /
Hans-Joachim Zwiesler

Stochastic methods in modern life insurance

Abstract: Life Insurance contracts are complex financial instruments involving both financial as well as insurance related risk factors. Their valuation and risk analysis requires the use of stochastic models. Due to various features which are typically embedded in such products (e.g. profit participation, surrender options, long-term interest rate guarantees), relatively complex models are necessary. Therefore the process of modeling as well as the mathematical analysis and the numerical treatment of these models pose numerous difficult problems, some of which we will discuss in our presentation.

[1] Bauer, D., Börger, M., Ruß, J., Zwiesler,H.: The Volatility of Mortality, Asia-Pacific Journal of Risk and Insurance, 2008, Vol. 3 (1)
[2] Bauer, D., Kling, A. und Ruß, J.: A Universal Pricing Framework for Guaranteed Minimum Benefits in Variable Annuities. ASTIN Bulletin, Volume: 38, Issue: 2 (2008), 621-651
[3] Bergmann, D.: Nested Simulations in Life Insurance, ifa-Verlag, 2011
[4] Graf, S., Kling, A. und Ruß, J.: Financial Planning and Risk-return profiles. will appear in European Actuarial Journal
[5] Kling, A., Ruez, F. und Ruß, J.: The Impact of Stochastic Volatility on Pricing, Hedging and Hedge Efficiency of Variable Annuity Guarantees. ASTIN-Bulletin 41(2), 511-545

Additionally there will be short talks given by the participants (application necessary).