Affine Point Processes and Portfolio Credit Risk

Abstract

Abstract. This paper analyzes a family of multivariate point process models of correlated event timing whose arrival intensity is driven by an affine jump diffusion. The components of an affine point process are self-and cross-exciting and facilitate the description of complex event dependence structures. ODEs characterize the transform of an affine point process and the probability distribution of an integer-valued affine point process. The moments of an affine point process take a closed form. This guarantees a high degree of computational tractability in applications. We illustrate this in the context of portfolio credit risk, where the correlation of corporate defaults is the main issue. We consider the valuation of securities exposed to correlated default risk and demonstrate the significance of our results through market calibration experiments. We show that a simple model variant can capture the default clustering implied by index and tranche market prices during September 2008, a month that witnessed significant volatility.

Multivariate dynamic intensity peaks‐over‐threshold models

“…Traditional concepts to describe the tail of a loss distribution are value‐at‐risk (VaR) and expected shortfall (ES); see, for example, McNeil and Frey (), Cotter and Dowd (), or Chavez‐Demoulin, Embrechts, and Sardy (). On the other hand, point process methods allow the dynamic behavior of (extreme) events to be captured and are typically applied in the context of portfolio credit risk, market microstructure analysis, contagion analysis, or jump‐diffusion models; see, for example, Engle and Russell (), Bauwens and Hautsch (), Errais, Giesecke, and Goldberg (), Bacry and Muzy (), or Aït‐Sahalia, Cacho‐Diaz, and Laeven (). Moreover, point process theory provides an elegant formulation for the characterization of the limiting distribution of extreme value distributions, see Pickands () or Smith (), and therefore builds a natural complementary framework to extreme value analysis.…”

Modeling financial contagion using mutually exciting jump processes

“…2 where US news and market events lead successive market moves elsewhere around the world. Self-exciting models have recently been employed to model joint defaults in portfolios of credit derivatives (see Errais, Giesecke, and Goldberg, 2010). It takes some time for the transmission, if at all, to take place.…”

Cumulants of Hawkes point processes

“…In his work, Adamopoulos [27], for example, attempts to derive the probability generating functional of the Hawkes process, but manages only to represent it implicitly, as a solution of an intractable functional equation. Errais et al [10], using the elegant theory of affine jump processes, show that the moments of Hawkes processes can be computed by solving a system of nonlinear ODEs. Once again, however, explicit formulas turn out to be unobtainable by analytic means.…”