Consistent Iterated Simulation of Multivariate Defaults: Markov Indicators, Lack of Memory, Extreme-Value Copulas, and the Marshall–Olkin Distribution

Abstract

A current market-practice to incorporate multivariate defaults in global riskfactor simulations is the iteration of (multiplicative) i.i.d. survival indicator increments along a given time-grid, where the indicator distribution is based on a copula ansatz. The underlying assumption is that the behavior of the resulting iterated default distribution is similar to the one-shot distribution. It is shown that in most cases this assumption is not fulfilled and furthermore numerical analysis is presented that shows sizable differences in probabilities assigned to both "survival-of-all" and "mixed default/survival" events. Moreover, the classes of distributions for which probabilities from the "terminal one-shot" and "terminal iterated" distribution coincide are derived for problems considering "survival-of-all" events as well as "mixed default/survival" events. For the former problem, distributions must fulfill a lack-of-memory type property, which is, e.g., fulfilled by min-stable multivariate exponential distributions. These correspond in a copula-framework to exponential margins coupled via extreme-value copulas. For the latter problem, while looping default inspired multivariate Freund distributions and more generally multivariate phase-type

Hypothesis testing for tail dependence parameters on the boundary of the parameter space

“…In Cui and Zhang (2018) and Zhao and Zhang (2018), max-linear models are used to model a latent factor structure for financial returns. Finally, the Marshall-Olkin model (Embrechts et al, 2003;Segers, 2012) is a submodel that is frequently used in practice, see for instance Burtschell et al (2009), Su and Furman (2017) or Brigo et al (2018).…”