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Integrating Volatility Clustering Into Exponential Lévy Models

Part of: Stochastic processes Stochastic analysis Real functions

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 

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