Multivariate dynamic intensity peaks‐over‐threshold models

Abstract

Summary We propose a multivariate dynamic intensity peaks‐over‐threshold model to capture extremes in multivariate return processes. The random occurrence of extremes is modeled by a multivariate dynamic intensity model, while temporal clustering of their size is captured by an autoregressive multiplicative error model. Applying the model to daily returns of three major stock indexes yields strong empirical support for a temporal clustering of both the occurrence and the size of extremes. Backtesting value‐at‐risk and expected shortfall forecasts shows that the consideration of clustering effects and of feedback between the magnitudes and the intensity of extremes results in better forecasts of risk.

Looking at Extremes without Going to Extremes: A New Self-Exciting Probability Model for Extreme Losses in Financial Markets

“…However, there is a plethora of VaR models in the literature—therefore, there are no two or three candidate specifications against which the SEP-POT model should be benchmarked and compared. Only among the point process-based POT models there have been variants put forward, including the ACD-POT model (which is based on the dynamic specifications of time, i.e., duration, that elapses between consecutive extreme losses [ 6 , 7 , 8 ]) or the ACI-POT model (with its multivariate extensions) that provides an explicit autoregressive specification for the intensity function [ 13 ]. All these dynamic versions of POT models exploit both strands of the literature: the point process theory and the EVT, accounting for the clustering of extreme losses and the heavy-tailness of the loss distribution.…”

“…The more recent dynamic versions of the classical POT model found in several studies (i.e., [ 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 ]), are directly motivated by the behavior of the non-homogeneous Poisson point process, where the intensity rate of threshold exceedances, , can vary over time due to the temporal bursts in volatility. According to such a point process approach to POT models, the first factor on the left-hand side of Equation ( 3 ) (i.e., the conditional probability of a threshold exceedance over day ) can be derived based on the (time varying) conditional intensity function as follows: because the probability of no events in (i.e., ) can be given as [ 21 ].…”

“…A new avenue for forecasting VaR was opened up when the point-process approach to POT models was released in [ 4 ]. This methodology was later extended in several publications [ 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 ]. The benefit of this model is that it does not require prefiltering returns using GARCH-family estimates while at the same time it can capture the clustering effects of extreme losses and maintain the merits of the extreme value theory.…”

“…The point-process POT model approximates the time-varying conditional probability of an extreme loss over a given day with the help of a conditional intensity function that characterizes the arrival rate of such extreme events. The intensity function can either be formulated in the spirit of the self-exciting Hawkes process [ 4 , 5 , 10 , 11 , 12 ] (which is extensively used in geophysics and seismology), in the form of the observation-driven autoregressive conditional intensity (ACI) model [ 13 ], or using the autoregressive conditional duration (ACD) models [ 6 , 7 , 8 ] (the last two methodologies were very popular in the area of market microstructure and the modeling of financial ultra-high-frequency data [ 15 , 16 , 17 ]). In all cases, the timing of extreme losses depends on the timing of extreme losses observed in the past.…”

Modelling Short- and Long-Term Dependencies of Clustered High-Threshold Exceedances in Significant Wave Heights

“…A related approach is the so-called self-exciting POT (SEPOT) suggested by McNeil et al [21], where one Hawkes process influences the intensity of both event times and marks. The various conditional POT models were reviewed in [3,[22][23][24], and extended to a multivariate framework by Grothe et al [25] and Hautsch and Herrera [26]. Conditional POT models are based on similar ideas as the epidemic-type aftershock sequence (ETAS) model proposed by Ogata [27], which is designed to describe the occurrence of earthquakes based on previous ones and was further explored for example by Helmstetter and Sornette [28].…”