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Integrating volatility clustering into exponential Lévy models
Bender Christian and Marquardt Tina
Source: J. Appl. Probab. Volume 46, Number 3 (2009), 609-628.
Abstract
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.
Primary Subjects: 91B28
Secondary Subjects: 60G51, 60H05, 26A33
Keywords: Convoluted Lévy process; financial modeling; subordination; volatility clustering
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Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.jap/1253279842
Digital Object Identifier: doi:10.1239/jap/1253279842
References
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.
Baudoin, F. and Nualart, D. (2003). Equivalence of Volterra processes. Stoch. Process. Appl. 107, 327--350.
Mathematical Reviews (MathSciNet):
MR1999794
Zentralblatt MATH:
1075.60519
Digital Object Identifier: doi:10.1016/S0304-4149(03)00088-7
Bender, C. and Marquardt, T. (2008). Stochastic calculus for convoluted Lévy processes. Bernoulli 14, 499--518.
Bondesson, L. (1982). On simulation from infinitely divisible distributions. Adv. Appl. Prob. 14, 855--869.
Mathematical Reviews (MathSciNet):
MR677560
Zentralblatt MATH:
0494.60013
Digital Object Identifier: doi:10.2307/1427027
JSTOR: links.jstor.org
Brockwell, P. J. (2001). Lévy-driven CARMA processes. Ann. Inst. Statist. Math. 52, 113--124.
Mathematical Reviews (MathSciNet):
MR1820952
Digital Object Identifier: doi:10.1023/A:1017972605872
Brockwell, P. J. and Marquardt, T. (2005). Lévy-driven and fractionally integrated ARMA processes with continuous time parameter. Statistica Sinica 15, 477--494.
Mathematical Reviews (MathSciNet):
MR2190215
Carr, P. and Madan, D. (1998). Option valuation using the fast Fourier transform. J. Comput. Finance 2, 61--73.
Carr, P., Geman, H., Madan, D. B. and Yor, M. (2003). Stochastic volatility for Lévy processes. Math. Finance 13, 345--382.
Mathematical Reviews (MathSciNet):
MR1995283
Digital Object Identifier: doi:10.1111/1467-9965.00020
Cont, R. and Tankov, P. (2004). Financial Modelling with Jump Processes. Chapmann and Hall, Boca Raton, FL.
Mathematical Reviews (MathSciNet):
MR2042661
Zentralblatt MATH:
1052.91043
Eberlein, E. (2001). Application of generalized hyperbolic Lévy motions to finance. In Lévy Processes, eds O. Barndorff-Nielsen et al., Birkhäuser, Boston, pp. 319--336.
Mathematical Reviews (MathSciNet):
MR1833703
Gander, M. P. S. and Stephens, D. A. (2007). Simulation and inference for stochastic volatility models driven by Lévy processes. Biometrika 94, 627--646.
Mathematical Reviews (MathSciNet):
MR2410013
Digital Object Identifier: doi:10.1093/biomet/asm048
Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus (Graduate Texts Math. 113), 2nd edn. Springer, New York.
Mathematical Reviews (MathSciNet):
MR1121940
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.
Madan, D. and Milne, F. (1991). Option pricing with variance gamma martingale components. Math. Finance 1, 39--55.
Mandelbrot, B. B., Calvet, L. and Fisher, A. (1997). The multifractality of the Deutschmark/US Dollar exchange rate. Cowles Foundation of Economics Discussion Papers.
Marquardt, T. (2006). Fractional Lévy processes with an application to long memory moving average processes. Bernoulli 12, 1099--1126.
Mathematical Reviews (MathSciNet):
MR2274856
Digital Object Identifier: doi:10.3150/bj/1165269152
Project Euclid: euclid.bj/1165269152
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.
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.
Mathematical Reviews (MathSciNet):
MR1704556
Digital Object Identifier: doi:10.2307/3318691
Project Euclid: euclid.bj/1171899318
Rosinski, J. (2001). Series representations of Lévy processes from the perspective of point processes. In Lévy Processes, eds O. E. Barndorff-Nielsen et al., Birkhäuser, Boston, pp. 401--415.
Mathematical Reviews (MathSciNet):
MR1833707
Sato, K.-I. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press.
Mathematical Reviews (MathSciNet):
MR1739520
Sato, K.-I. (2006). Addtitive processes and stochastic integrals. Illinois J. Math 50, 825--851.
Mathematical Reviews (MathSciNet):
MR2247848
Zentralblatt MATH:
1103.60051
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