Extremal behavior of stochastic volatility models

  • Vicky Fasen
  • Claudia Klüppelberg
  • Alexander Lindner
Conference paper

DOI: 10.1007/0-387-28359-5_4

Cite this paper as:
Fasen V., Klüppelberg C., Lindner A. (2006) Extremal behavior of stochastic volatility models. In: Shiryaev A.N., Grossinho M.R., Oliveira P.E., Esquível M.L. (eds) Stochastic Finance. Springer, Boston, MA

Summary

Empirical volatility changes in time and exhibits tails, which are heavier than normal. Moreover, empirical volatility has — sometimes quite substantial — upwards jumps and clusters on high levels. We investigate classical and non-classical stochastic volatility models with respect to their extreme behavior. We show that classical stochastic volatility models driven by Brownian motion can model heavy tails, but obviously they are not able to model volatility jumps. Such phenomena can be modelled by Lévy driven volatility processes as, for instance, by Lévy driven Ornstein-Uhlenbeck models. They can capture heavy tails and volatility jumps. Also volatility clusters can be found in such models, provided the driving Lévy process has regularly varying tails. This results then in a volatility model with similarly heavy tails. As the last class of stochastic volatility models, we investigate a continuous time GARCH(1,1) model. Driven by an arbitrary Lévy process it exhibits regularly varying tails, volatility upwards jumps and clusters on high levels.

Key words

COGARCH extreme value theory generalized Cox-Ingersoll-Ross model Lévy process Ornstein-Uhlenbeck process Poisson approximation regular variation stochastic volatility model subexponential distribution tail behavior volatility cluster 

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Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • Vicky Fasen
    • 1
  • Claudia Klüppelberg
    • 1
  • Alexander Lindner
    • 1
  1. 1.Center for Mathematical SciencesMunich University of TechnologyGarchingGermany

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