Vicky Fasen

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.

Extremes of supOU Processes

Barndorff-Nielsen and Shephard [3] investigate supOU processes as volatility models. Empirical volatility has tails heavier than normal, long memory in the sense that the empirical autocorrelation function decreases slower than exponential, and exhibits volatility clusters on high levels. We investigate supOU processes with respect to these stylized facts. The class of supOU processes is vast and can be distinguished by its underlying driving Levy process. Within the classes of convolution equivalent distributions we shall show that extremal clusters and long range dependence only occur for supOU processes, whose underlying driving Levy process has regularly varying increments. The results on the extremal behavior of supOU processes correspond to the results of classical Levy-driven OU processes.

Extremes of regularly varying Lévy-driven mixed moving average processes

In this paper, we study the extremal behavior of stationary mixed moving average processes of the form Y(t)=∫ℝ+×ℝ f(r,t-s) dΛ(r,s), t∈ℝ, where f is a deterministic function and Λ is an infinitely divisible, independently scattered random measure whose underlying driving Lévy process is regularly varying. We give sufficient conditions for the stationarity of Y and compute the tail behavior of certain functionals of Y. The extremal behavior is modeled by marked point processes on a discrete-time skeleton chosen properly by the jump times of the underlying driving Lévy process and the extremes of the kernel function. The sequences of marked point processes converge weakly to a cluster Poisson random measure and reflect extremes of Y at a high level. We also show convergence of the partial maxima to the Fréchet distribution. Our models and results cover short- and long-range dependence regimes.

Limit theory for high frequency sampled MCARMA models

We consider a multivariate continuous-time ARMA (MCARMA) process sampled at a high-frequency time grid {h n , 2h n ,…, nh n }, where h n ↓ 0 and nh n → ∞ as n → ∞, or at a constant time grid where h n = h. For this model, we present the asymptotic behavior of the properly normalized partial sum to a multivariate stable or a multivariate normal random vector depending on the domain of attraction of the driving Lévy process. Furthermore, we derive the asymptotic behavior of the sample variance. In the case of finite second moments of the driving Lévy process the sample variance is a consistent estimator. Moreover, we embed the MCARMA process in a cointegrated model. For this model, we propose a parameter estimator and derive its asymptotic behavior. The results are given for more general processes than MCARMA processes and contain some asymptotic properties of stochastic integrals.

Spectral Estimates for High‐Frequency Sampled Continuous‐Time Autoregressive Moving Average Processes

In this article, we consider a continuous-time autoregressive moving average (CARMA) process driven by either a symmetric α-stable Levy process with α ∈ (0,2) or a symmetric Levy process with finite second moments. In the asymptotic framework of high-frequency data within a long time interval, we establish a consistent estimate for the normalized power transfer function by applying a smoothing filter to the periodogram of the CARMA process. We use this result to propose an estimator for the parameters of the CARMA process and exemplify the estimation procedure by a simulation study.

Quantifying Extreme Risks

Understanding and managing risks caused by extreme events is one of the most demanding problems of our society. We consider this topic from a statistical point of view and present some of the probabilistic and statistical theory, which was developed to model and quantify extreme events. By the very nature of an extreme event there will never be enough data to predict a future risk in the classical statistical sense. However, a rather clever probabilistic theory provides us with model classes relevant for the assessment of extreme events. Moreover, specific statistical methods allow for the prediction of rare events, even outside the range of previous observations. We will present the basic theory and relevant examples from climatology (climate change), insurance (return periods of large claims) and finance (portfolio losses and Value-at-Risk estimation).

TIME SERIES REGRESSION ON INTEGRATED CONTINUOUS-TIME PROCESSES WITH HEAVY AND LIGHT TAILS

The paper presents a cointegration model in continuous time, where the linear combinations of the integrated processes are modeled by a multivariate Ornstein–Uhlenbeck process. The integrated processes are defined as vector-valued Levy processes with an additional noise term. Hence, if we observe the process at discrete time points, we obtain a multiple regression model. As an estimator for the regression parameter we use the least squares estimator. We show that it is a consistent estimator and derive its asymptotic behavior. The limit distribution is a ratio of functionals of Brownian motions and stable Levy processes, whose characteristic triplets have an explicit analytic representation. In particular, we present the Wald and the t -ratio statistic and simulate asymptotic confidence intervals. For the proofs we derive some central limit theorems for multivariate Ornstein–Uhlenbeck processes.

Extremes of Continuous–Time Processes.

In this paper we present a review on the extremal behavior of stationary continuous-time processes with emphasis on generalized Ornstein-Uhlenbeck processes. We restrict our attention to heavy-tailed models like heavy-tailed Ornstein-Uhlenbeck processes or continuous-time GARCH processes. The survey includes the tail behavior of the stationary distribution, the tail behavior of the sample maximum and the asymptotic behavior of sample maxima of our models.

Time consistency of multi-period distortion measures

Abstract 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).

Large Insurance Losses Distributions

Large insurance losses happen infrequently, but they happen. In this paper we present the standard distribution models used in fire, wind–storm, or flood insurance. We also present the Cramer–Lundberg model for the total claim amount and some more recent extensions. The classical insurance risk measure is the ruin probability, and we give a full account of the ruin event in such models. Finally, we present some results for an integrated insurance risk model, where investment risk is also taken into account. Keywords: Cramer-Lundberg model; integrated risk process; integrated tail distribution function; Pollaczek-Khinchine formula; quintuple law; regular variation; renewal measure; risk model; ruin probability; sample path leading to ruin; subexponential distribution