Extreme Times For Volatility Processes PDF Download

Empirical volatility changes in time and exhibits tails, which are heavier than those of normal distributions. 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: subexponential Lévy driven MA processes in the maximum domain of attraction of the Gumbel distribution, regularly varying mixed MA processes, Ornstein-Uhlenbeck processes with exponentially decreasing tails and COGARCH processes. The basic volatility models of this thesis are subexponential Lévy driven MA processes $Y(t)=\int_{-\infty}^{\infty}f(t-s)\, dL(s)$ for $t\in \R$ where f is a deterministic function and L is a Lévy process. In Chapter 1 we study the extremal behavior of subexponential MA processes in the maximum domain of attraction of the Gumbel distribution and in Chapter 2 of the Fréchet distribution. The behavior is quite different in these different regimes. For both classes we give sufficient conditions for the kernel function f, such that a stationary version of the MA process Y exists, which preserves the infinitely divisibility of L. We calculate the tail behavior of the stationary distribution, which is again subexponential and in the same maximum domain of attraction as the driving Lévy process L. Hence they capture heavy tails and volatility jumps. Our investigation on the extremal behavior of Y is based on a discrete-time skeleton of Y chosen to incorporate those times, where large jumps of the Lévy process L and extremes of the kernel function f occur. Adding marks to this discrete-time skeleton, we obtain, by the weak limit of marked point processes, complete information about the extremal behavior. A complementary result guarantees the convergence of running maxima. Both models have volatility clusters. Regularly varying MA processes have long high level excursion in contrast to subexp.