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Integrating Volatility Clustering Into Exponential Lévy Models

Part of: Stochastic processes Real functions Stochastic analysis

Published online by Cambridge University Press:  14 July 2016

Christian Bender  and
Tina Marquardt
Christian Bender*
Affiliation:
Technische Universität Braunschweig
Tina Marquardt*
Affiliation:
Munich University of Technology
*
∗Current address: Department of Mathematics, Saarland University, PO Box 151150, D-66041 Saarbrücken, Germany. Email address: bender@math.uni-sb.de
∗∗Postal address: Center of Mathematical Sciences, Munich University of Technology, D-85747 Garching, Germany. Email address: marquard@ma.tum.de
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Abstract

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We introduce a class of stock models that interpolates between exponential Lévy models based on Brownian subordination and certain stochastic volatility models with Lévy-driven volatility, such as the Barndorff-Nielsen–Shephard model. The driving process in our model is a Brownian motion subordinated to a business time which is obtained by convolution of a Lévy subordinator with a deterministic kernel. We motivate several choices of the kernel that lead to volatility clusters while maintaining the sudden extreme movements of the stock. Moreover, we discuss some statistical and path properties of the models, prove absence of arbitrage and incompleteness, and explain how to price vanilla options by simulation and fast Fourier transform methods.

Keywords

Convoluted Lévy processfinancial modelingsubordinationvolatility clustering

MSC classification

Secondary: 60G51: Processes with independent increments; L'evy processes 60H05: Stochastic integrals 26A33: Fractional derivatives and integrals
Type
Research Article
Information
Journal of Applied Probability , Volume 46 , Issue 3 , September 2009 , pp. 609 - 628
Copyright
Copyright © Applied Probability Trust 2009 

References

[1] Barndorff-Nielsen, O. E. and Shephard, N. (2001). Non-Gaussian Ornstein–Uhlenbeck-based models and some of their uses in financial economics.” J. Roy. Statist. Soc. B 63, 167–241.Google Scholar
[2] Baudoin, F. and Nualart, D. (2003). Equivalence of Volterra processes. Stoch. Process. Appl. 107, 327–350.Google Scholar
[3] Bender, C. and Marquardt, T. (2008). Stochastic calculus for convoluted Lévy processes. Bernoulli 14, 499–518.CrossRefGoogle Scholar
[4] Bondesson, L. (1982). On simulation from infinitely divisible distributions. Adv. Appl. Prob. 14, 855–869.Google Scholar
[5] Brockwell, P. J. (2001). “Lévy-driven CARMA processes. Ann. Inst. Statist. Math. 52, 113–124.Google Scholar
[6] Brockwell, P. J. and Marquardt, T. (2005). Lévy-driven and fractionally integrated ARMA processes with continuous time parameter. Statistica Sinica 15, 477–494.Google Scholar
[7] Carr, P. and Madan, D. (1998). Option valuation using the fast Fourier transform. J. Comput. Finance 2, 61–73.Google Scholar
[8] Carr, P., Geman, H., Madan, D. B. and Yor, M. (2003). Stochastic volatility for Lévy processes. Math. Finance 13, 345–382.CrossRefGoogle Scholar
[9] Cont, R. and Tankov, P. (2004). Financial {Modelling with Jump Processes}. Chapmann and Hall, Boca Raton, FL.Google Scholar
[10] Eberlein, E. (2001). Application of generalized hyperbolic Lévy motions to finance. In Lévy Processes, eds Barndorff-Nielsen, O. et al., Birkhäuser, Boston, pp. 319–336.CrossRefGoogle Scholar
[11] Gander, M. P. S. and Stephens, D. A. (2007). Simulation and inference for stochastic volatility models driven by Lévy processes. Biometrika 94, 627–646.Google Scholar
[12] Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus (Graduate Texts Math. 113), 2nd edn. Springer, New York.Google Scholar
[13] Lewis, A. (2001). A simple option formula for general Jump-diffusion and other exponential Lévy processes. Tech. Rep. Available at www.optioncity.net/publications.htm.Google Scholar
[14] Madan, D. and Milne, F. (1991). Option pricing with variance gamma martingale components. Math. Finance 1, 39–55.Google Scholar
[15] Mandelbrot, B. B., Calvet, L. and Fisher, A. (1997). The multifractality of the Deutschmark/{US D}ollar exchange rate. Cowles Foundation of Economics Discussion Papers.Google Scholar
[16] Marquardt, T. (2006). Fractional Lévy processes with an application to long memory moving average processes. Bernoulli 12, 1099–1126.Google Scholar
[17] Muzy, J., Delour, J. and Bacry, E. (2000). Modeling fluctuations of financial time series: from cascade processes to stochastic volatility models. Europ. J. Phys. B 17, 537–548.CrossRefGoogle Scholar
[18] Norros, I., Valkeila, E. and Virtamo, J. (1999). An elementary approach to a Girsanov formula and other analytical results on fractional Brownian motions. Bernoulli 5, 571–587.Google Scholar
[19] Rosinski, J. (2001). Series representations of {Lévy} processes from the perspective of point processes. In Lévy Processes, eds Barndorff-Nielsen, O. E. et al., Birkhäuser, Boston, pp. 401–415.Google Scholar
[20] Sato, K.-I. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press.Google Scholar
[21] Sato, K.-I. (2006). Addtitive processes and stochastic integrals. Illinois J. Math 50, 825–851.Google Scholar