Abstract
Lévy processes are developed in the more general framework of semimartingale theory with a focus on purely discontinuous processes. The fundamental exponential Lévy model is given, which allows us to describe stock prices or indices in a more realistic way than classical diffusion models. A number of standard examples including generalized hyperbolic and CGMY Lévy processes are considered in detail.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bachelier, L. (1900): Théorie de la spéculation. PhD thesis, Annales Scientifiques de l’École Normale Superieure II I–17, 21–86. English Translation in: Cootner, P. (Ed.) (1964): Random Character of Stock Market Prices, 17–78. Massachusetts Institute of Technology, Cambridge.
Barndorff-Nielsen, O. E. (1978): Hyperbolic distributions and distributions on hyperbolae. Scandinavian Journal of Statistics 5, 151–157.
Barndorff-Nielsen, O. E. (1998): Processes of normal inverse Gaussian type. Finance and Stochastics, 2, 41–68.
Barndorff-Nielsen, O. E, Mikosch, T. and Resnick, S. I. (Eds.) (2001): Lévy Processes. Theory and Applications. Birkhäuser, Basel.
Bertoin, J. (1996): Lévy processes. Cambridge University Press, Cambridge.
Björk, T. (2008): An overview of interest rate theory. In: Andersen, T.G , Davis, R.A., Kreiss, J.-P. and Mikosch, T. (Eds.): Handbook of Financial Time Series, 614–651. Springer, New York.
Black, F. and Scholes, M. (1973): The pricing of options and corporate liabilities. Journal of Political Economy, 81 637–654.
Breiman, L. (1968): Probability. Addison-Wesley, Reading.
Carr, P., Geman, H., Madan, D. and Yor, M. (2002): The fine structure of asset returns: An empirical investigation. Journal of Business 75, 305–332.
Cont, R. and Tankov, P. (2004): Financial Modelling with Jump Processes. Chapman & Hall/CRC, New York.
Eberlein, E. (2001): Application of generalized hyperbolic Lévy motions to finance. In:Barndorff-Nielsen, O. E (Eds.): Lévy Processes. Theory and Applications, 319–336. Birkhäuser, Basel.
Eberlein, E. and Keller, U. (1995): Hyperbolic distributions in finance. Bernoulli, 1, 281–299.
Eberlein, E. and Kluge, W. (2006): Exact pricing formulae for caps and swaptions in a Lévy term structure model. Journal of Computational Finance 9, 99–125.
Eberlein, E. and Kluge, W. (2007): Calibration of Lévy term structure models. In:Fu, M., Jarrow, R. A., Yen, J.-Y. and Elliot, R. J.(Eds.): Advances in Mathematical Finance: In Honor of Dilip B. Madan, 155–180. Birkhäuser, Basel.
Eberlein, E. and Koval, N. (2006): A cross-currency Lévy market model. Quantitative Finance 6, 465–480.
Eberlein, E. and Özkan, F. (2003a): The defaultable Lévy term structure: Ratings and restructuring. Mathematical Finance 13, 277–300.
Eberlein, E. and Özkan, F. (2003b): Time consistency of Lévy models. Quantitative Finance 3, 40–50.
Eberlein, E. and Özkan, F. (2005): The Lévy LIBOR model. Finance and Stochastics 9, 327–348.
Eberlein, E. and Prause, K. (2002): The generalized hyperbolic model: Financial derivatives and risk measures. In: Mathematical Finance—Bachelier Congress, 2000 (Paris), 245–267. Springer, New York.
Eberlein, E. and Raible, S. (1999): Term structure models driven by general Lévy processes. Mathematical Finance 9, 31–53.
Eberlein, E. and von Hammerstein, E. A. (2004): Generalized hyperbolic and inverse gaussian distributions: Limiting cases and approximation of processes. In: Dalang, R. C., Dozzi, M., and Russo, F. (Eds.): Seminar on Stochastic Analysis, Random Fields and Applications IV, Progress in Probability 58, 221–264. Birkhäuser, Basel.
Eberlein, E., Jacod, J. and Raible, S. (2005): Lévy term structure models: No-arbitrage and completeness. Finance and Stochastics 9, 67–88.
Fama, E. (1965): The behaviour of stock market prices. Journal of Business 38, 34–105.
Jacod, J. and Shiryaev, A. N. (1987): Limit Theorems for Stochastic Processes. Springer, New York.
Kou, S. G. (2002): A jump diffusion model for option pricing. Management Science 48, 1086–1101.
Madan, D. and Milne, F. (1991): Option pricing with V.G. martingale component. Mathematical Finance 1, 39–55.
Madan, D. and Seneta, E. (1990): The variance gamma (V.G.) model for share market returns. Journal of Business 63, 511–524.
Morton, R. C. (1976): Option pricing when underlying stock returns are discontinuous. Journal Financ. Econ. 3, 125–144.
Protter, P. E. (2004): Stochastic Integration and Differential Equations. (2nd ed.). Volume 21 of Applications of Mathematics. Springer, New York.
Rachev, S. and Mittnik, S. (2000): Stable Paretian Models in Finance. Wiley, New York.
Raible, S. (2000): Lévy processes in finance: Theory, numerics, and empirical facts. PhD thesis, University of Freiburg.
Samorodnitsky, G. and Taqqu, M. (1999): Stable Non-Gaussian Random Processes: Stochastic Models With Infinite Variance. Chapman & Hall/CRC, London.
Samuelson, P. (1965): Rational theory of warrant pricing. Industrial Management Review 6:13–32.
Sato, K.-I. (1999): Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge.
Schoutens, W. (2003): Lévy Processes in Finance: Pricing Financial Derivatives. Wiley, New York.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Eberlein, E. (2009). Jump–Type Lévy Processes. In: Mikosch, T., Kreiß, JP., Davis, R., Andersen, T. (eds) Handbook of Financial Time Series. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-71297-8_19
Download citation
DOI: https://doi.org/10.1007/978-3-540-71297-8_19
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-71296-1
Online ISBN: 978-3-540-71297-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)