Asymptotic results for sample autocovariance functions and extremes of integrated generalized Ornstein-Uhlenbeck processes

We consider a positive stationary generalized Ornstein--Uhlenbeck process V_t=\mathrm{e}^{-\xi_t}\biggl(\int_0^t\mathrm{e}^{\xi_{s-}}\ ,\mathrm{d}\eta_s+V_0\biggr)\qquadfor t\geq0, and the increments of the integrated generalized Ornstein--Uhlenbeck process $I_k=\int_{k-1}^k\sqrt{V_{t-}} \mathrm{d}L_t$, $k\in\mathbb{N}$, where $(\xi_t,\eta_t,L_t)_{t\geq0}$ is a three-dimensional L\'{e}vy process independent of the starting random variable $V_0$. The genOU model is a continuous-time version of a stochastic recurrence equation. Hence, our models include, in particular, continuous-time versions of $\operatorname {ARCH}(1)$ and $\operatorname {GARCH}(1,1)$ processes. In this paper we investigate the asymptotic behavior of extremes and the sample autocovariance function of $(V_t)_{t\geq0}$ and $(I_k)_{k\in\mathbb{N}}$. Furthermore, we present a central limit result for $(I_k)_{k\in\mathbb{N}}$. Regular variation and point process convergence play a crucial role in establishing the statistics of $(V_t)_{t\geq0}$ and $(I_k)_{k\in\mathbb{N}}$. The theory can be applied to the $\operatorname {COGARCH}(1,1)$ and the Nelson diffusion model.Comment: Published in at http://dx.doi.org/10.3150/08-BEJ174 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Extremal behavior of stochastic volatility models

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 Levy driven volatility processes as, for instance, by Levy driven Ornstein-Uhlenbeck models. They can capture heavy tails and volatility jumps. Also volatility clusters can be found in such models, provided the driving Levy 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 Levy process it exhibits regularly varying tails, volatility upwards jumps and clusters on high levels

Time consistency of multi-period distortion measures

Dynamic risk measures play an important role for the acceptance or non-acceptance of risks in a bank portfolio. Dynamic consistency and weaker versions like conditional and sequential consistency guarantee that acceptability decisions remain consistent in time. An important set of static risk measures are so-called distortion measures. We extend these risk measures to a dynamic setting within the framework of the notions of consistency as above. As a prominent example, we present the Tail-Value-at-Risk (TVaR

Derivative pricing under the possibility of long memory in the supOU stochastic volatility model

We consider the supOU stochastic volatility model which is able to exhibit long-range dependence. For this model we give conditions for the discounted stock price to be a martingale, calculate the characteristic function, give a strip where it is analytic and discuss the use of Fourier pricing techniques. Finally, we present a concrete specification with polynomially decaying autocorrelations and calibrate it to observed market prices of plain vanilla options

Dependence Estimation for High Frequency Sampled Multivariate CARMA Models

The paper considers high frequency sampled multivariate continuous-time ARMA (MCARMA) models, and derives the asymptotic behavior of the sample autocovariance function to a normal random matrix. Moreover, we obtain the asymptotic behavior of the cross-covariances between different components of the model. We will see that the limit distribution of the sample autocovariance function has a similar structure in the continuous-time and in the discrete-time model. As special case we consider a CARMA (one-dimensional MCARMA) process. For a CARMA process we prove Bartlett's formula for the sample autocorrelation function. Bartlett's formula has the same form in both models, only the sums in the discrete-time model are exchanged by integrals in the continuous-time model. Finally, we present limit results for multivariate MA processes as well which are not known in this generality in the multivariate setting yet