Christian Y. Robert

A Sliding Blocks Estimator for the Extremal Index

In extreme value statistics for stationary sequences, blocks estimators are usually constructed by using disjoint blocks because exceedances over high thresholds of different blocks can be assumed asymptotically independent. In this paper we focus on the estimation of the extremal index which measures the degree of clustering of extremes. We consider disjoint and sliding blocks estimators and compare their asymptotic properties. In particular we show that the sliding blocks estimator is more efficient than the disjoint version and has a smaller asymptotic bias. Moreover we propose a method to reduce its bias when considering sufficiently large block sizes.

STOCHASTIC UNIT ROOT MODELS

This paper develops a dynamic switching model, with a random walk and a stationary regime, where the time spent in the random walk regime is endogeneously predetermined. More precisely, we assume that the process is recursively defined by Yt = I¼ + Yt−1 + Iµt, with stochastic probability I€rw(Yt−1), Yt = I¼ + Iµt, with stochastic probability 1 − I€rw(Yt−1), where (Iµt) is a strong white noise and I€rw is a nondecreasing function. Then, the dynamics of the process (Yt), its marginal distribution, and the distribution of the time spent in the unit root regime depend on the pattern of random walk intensity I€rw and on the noise distribution F. Moreover, we study the links between the endogeneous switching regime and the degree of persistence of the process (Yt).

Extreme dependence of multivariate catastrophic losses

Natural catastrophes cause insurance losses in several different lines of business. An approach to modelling the dependence in loss severities is to assume that they are related to the intensity of the natural disaster. In this paper we introduce a factor model and investigate the extreme dependence. We derive a specific extreme dependence structure when considering an heavy-tailed intensity. Estimation procedures are presented and their moderate sample properties are compared in a simulation study. We also motivate our approach by an illustrative example from storm insurance.

On the Microstructural Hedging Error

We consider the issue of hedging a European derivative security in the presence of microstructure noise. In a market where the efficient price of the asset is driven by a stochastic volatility process, we assume an agent wants to use a (possibly misspecified) local volatility-type replication strategy. Focusing on microstructure noise effects, our goal is to evaluate the error between the theoretical, but practically unfeasible, strategy and its market adapted versions. The microstructural hedging error is in particular due to transaction price discreteness and endogenous trading times. Thus, we consider a transaction price model that accommodates such inherent properties of ultrahigh frequency data with the assumption of a continuous semimartingale efficient price. In this framework, we study two hedging strategies derived from the local volatility-type hedging strategy: (i) the hedging portfolio is rebalanced every time that the transaction price moves; (ii) the hedging portfolio is rebalanced only once the transaction price has varied by more than a selected value. To assess these strategies, we use an asymptotic approach where the number of rebalancing transactions goes to infinity. For the first strategy, we show that, because of microstructure noise effects, the hedging error does not vanish. However, an optimal strategy of the second type enables us to reduce it significantly.

SPACE-TIME MAX-STABLE MODELS WITH SPECTRAL SEPARABILITY

Abstract Natural disasters may have considerable impact on society as well as on the (re-)insurance industry. Max-stable processes are ideally suited for the modelling of the spatial extent of such extreme events, but it is often assumed that there is no temporal dependence. Only a few papers have introduced spatiotemporal max-stable models, extending the Smith, Schlather and Brown‒Resnick spatial processes. These models suffer from two major drawbacks: time plays a similar role to space and the temporal dynamics are not explicit. In order to overcome these defects, we introduce spatiotemporal max-stable models where we partly decouple the influence of time and space in their spectral representations. We introduce both continuous- and discrete-time versions. We then consider particular Markovian cases with a max-autoregressive representation and discuss their properties. Finally, we briefly propose an inference methodology which is tested through a simulation study.

Transparency matters: Price formation in the presence of order preferencing

We present a model of market-making in which dealers differ by their current inventory positions and by their preferencing agreements. Under preferencing, dealers receive captive orders that they guarantee to execute at the best price. We show that preferencing raises the inventory holding costs of preferenced dealers. In turn, competitors post less aggressive quotes. Since price-competition is softened, expected spreads widen. The entry of unpreferenced dealers, or the ability to route preferenced orders to best-quoting dealers, as internalization does restore price competitiveness. We also show that a greater transparency may negatively affect expected spreads, depending on the scale of preferencing.

Testing the type of a semi-martingale: Itō against multifractal

We consider high frequency observations of a semi-martingale. From these data, we build simple test statistics allowing to distinguish between the two following situations: i) the data generating process is an Ito semi-martingale; ii) the data generating process is a Multifractal Random Walk. We also investigate the finite sample behavior of the test statistics on some simulated data.

Estimating the multivariate extremal index function

The multivariate extremal index function relates the asymptotic distribution of the vector of pointwise maxima of a multivariate stationary sequence to that of the independent sequence from the same stationary distribution. It also measures the degree of clustering of extremes in the multivariate process. In this paper, we construct nonparametric estimators of this function and prove their asymptotic normality under long-range dependence and moment conditions. The results are illustrated by means of a simulation study.

Stochastic stability of some state-dependent growth-collapse processes

In this paper we consider a discrete-time process which grows according to a random walk with nonnegative increments between crash times at which it collapses to 0. We assume that the probability of crashing depends on the level of the process. We study the stochastic stability of this growth-collapse process. Special emphasis is given to the case in which the probability of crashing tends to 0 as the level of the process increases. In particular, we show that the process may exhibit long-range dependence and that the crash sizes may have a power law distribution.