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Fasen, Vicky
Klüppelberg, Claudia
Lindner, Alexander M.
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Discussion paper // Sonderforschungsbereich 386 der Ludwig-Maximilians-Universität München 427
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 nonclassical 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.
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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Working Paper

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