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Mathematics for Finance and Economics


Courses, professors and summaries

Lévy driven financial models - C1 Course 1 (8 hours)
prof. Ernst Eberlein

Summary:
Empirical analysis of financial data reveals that standard diffusion models do not generate sufficiently accurate return distributions. To reduce model risk more powerful classes of driving processes are appropriate.
In this course exponential Lévy models and models driven by semimartingales in general are considered. Plain vanilla as well as exotic options are priced in this new model class. As a further application in risk management we show that estimates of the value at risk of a portfolio of securities are improved.
In the second part we develop a Lévy term structure theory. Three basic approaches are introduced: the forward rate model, the forward process model, and the LIBOR or market model. Pricing formulae for interest rate derivatives as well as efficient numerical algorithms to evaluate these formulae are derived. The LIBOR model is extended to a multi-currency and a credit setting. As an application pricing of cross-currency and a variety of credit derivatives is discussed.
  • Exponential Lévy models
  • Pricing derivatives
  • Risk management
  • Lévy term structure theory
  • Lévy forward rate model
  • Lévy Libor model
  • Multi-currency model
  • Credit portfolio model
  • Credit derivates


References:
  • E. Eberlein: Application of generalized hyperbolic Lévy motions to finance. In: Lévy Processes: Theory and Applications, O.E. Barndorff-Nielsen, T. Mikosch, and S. Resnick (eds.), Birkhäuser Verlag (2001) 319-337
  • E. Eberlein, E. A. von Hammerstein: Generalized hyperbolic and inverse Gaussian distributions: limiting cases and approximation of processes. In: Seminar on Stochastic Analysis, Random Fields and Applications IV, Progress in Probability 58, R.C. Dalang, M. Dozzi, F. Russo (eds.), Birkhäuser Verlag (2004) 221-264
  • E. Eberlein, J. Jacod, S. Raible: Lévy term structure models: no-arbitrage and completeness. Finance and Stochastics 9 (2005) 67-88
  • E. Eberlein, U. Keller: Hyperbolic distributions in finance. Bernoulli 1 (1995) 281-299
  • E. Eberlein, U. Keller, K. Prause: New insights into smile, mispricing and value at risk: the hyperbolic model. Journal of Business 71 (1998) 371-405
  • E. Eberlein, W. Kluge: Exact pricing formulae for caps and swaptions in a Lévy term structure model. Journal of Computational Finance 9 (2) (2006) 99-125
  • E. Eberlein, W. Kluge, P. J. Schönbucher: The Lévy Libor model with default risk. Journal of Credit Risk 2 (2) (2006) 3-42
  • E. Eberlein, N. Koval: A cross-currency Lévy market model. Quantitative Finance 6 (2006) 465-480
  • E. Eberlein, F. Özkan: The Lévy Libor Model. Finance and Stochastics 9 (2005) 327-348
  • E. Eberlein, K. Prause: The generalized hyperbolic model: financial derivatives and risk measures. In: Mathematical Finance- Bachelier Congress 2000, H. Geman, D. Madan, S. Pliska, T. Vorst (eds.), Springer Verlag (2002), 245-267
  • E. Eberlein, S. Raible: Term structure models driven by general Lévy processes. Mathematical Finance 9 (1999) 31-54

Pricing and hedging jump risk - C2 Course 2 (8 hours)
prof. Peter Tankov

Summary:
The recent period of extreme volatility in financial markets has once more drawn the attention of academics and practitioners to the insufficiency of Gaussian modelling and the importance of taking into account the extreme market moves. The aim of this course is to show that Lévy processes now offer an easy to use toolkit for pricing and hedging the jump risk in financial markets. After a brief overview of the mathematical aspects of Lévy processes, we concentrate on their uses in risk management, exploring the financial applications where using jump processes really makes a difference.
  • Introduction. Compound Poisson processes and jump-diffusions. Poisson random measures and the path structure of a Lévy process. Characteristic functions and the Lévy-Khintchine formula. Basic examples of Lévy processes used in financial modelling.
  • Exponential Lévy models. Stochastic vs. ordinary exponentials. Measure changes, Esscher transform and the absence of arbitrage. Market incompleteness. Option pricing by Fourier methods and the behaviour of implied volatility in exponential Lévy models.
  • Risk management with exponential Lévy models. Constant proportion portfolio insurance and other leveraged strategies in the presence of jumps. Option hedging in the presence of jumps: mean-variance hedging, hedging with options, from continuous to discrete rebalancing. Pricing and hedging gap options.
Primary references:
  • R. Cont and P. Tankov, Financial modelling with jump processes, Chapman & Hall / CRC (2004)
  • R. Cont and P. Tankov, Constant Proportion Portfolio Insurance in presence of Jumps in Asset Prices, Mathematical Finance (to appear) - available from www.math.jussieu.fr/~tankov
  • R. Cont, P. Tankov and E. Voltchkova, Hedging with options in models with jumps (with R. Cont and E. Voltchkova), Stochastic Analysis and Applications - the Abel Symposium 2005, Springer (2007) - available from www.math.jussieu.fr/~tankov
  • P. Tankov, Pricing and hedging gap risk, working paper, available from www.math.jussieu.fr/~tankov
Secondary references:
  • J. Kallsen, F. Hubalek, and L. Krawczyk, Variance-optimal hedging for processes with stationary independent increments, The Annals of Applied Probability, 16 (2006), pp. 853-885.
  • P. Tankov and E. Voltchkova, Asymptotic analysis of hedging errors in models with jumps, Stochastic Processes and their Applications (to appear), available from www.math.jussieu.fr/~tankov

Ernst Eberlein

Ernst Eberlein is Professor of Stochastics and Mathematical Finance at the University of Freiburg. He is one of the founders of the Freiburg Center for Data Analysis and Modeling (FDM). He is also an elected member of the International Statistical Institute and at present Executive Secretary of the Bachelier Finance Society, the international association of researchers and practitioners working in mathematical finance. Ernst Eberlein is a frequent speaker at conferences on financial models and risk management. He pioneered the introduction of Lévy models in finance. His current research interests and his consulting activities focus on realistic modeling of financial markets, risk management, as well as pricing of derivative products.




Peter Tankov

Peter Tankov is a leading expert on financial applications of jump processes. He is currently Associate professor at Ecole Polytechnique, France, and has previously worked in the University of Paris VII and the INRIA research institute after obtaining his PhD from Ecole Polytechnique. He is the author of the book Financial Modelling with Jump Processes and of many publications in international journals on subjects ranging from risk management and model calibration to stochastic control and commodity price modeling. He is frequent speaker at international conferences and professional and academic training courses on various aspects of quantitative finance.