Marshall–Olkin distributions, subordinators, efficient simulation, and applications to credit risk

Abstract

In the paper we present a novel construction of Marshall–Olkin (MO) multivariate exponential distributions of failure times as distributions of the first-passage times of the coordinates of multidimensional Lévy subordinator processes above independent unit-mean exponential random variables. A time-inhomogeneous version is also given that replaces Lévy subordinators with additive subordinators. An attractive feature of MO distributions for applications, such as to portfolio credit risk, is its singular component that yields positive probabilities of simultaneous defaults of multiple obligors, capturing the default clustering phenomenon. The drawback of the original MO fatal shock construction of MO distributions is that it requires one to simulate 2n-1 independent exponential random variables. In practice, the dimensionality is typically on the order of hundreds or thousands of obligors in a large credit portfolio, rendering the MO fatal shock construction infeasible to simulate. The subordinator construction reduces the problem of simulating a rich subclass of MO distributions to simulating an n-dimensional subordinator. When one works with the class of subordinators constructed from independent one-dimensional subordinators with known transition distributions, such as gamma and inverse Gaussian, or their Sato versions in the additive case, the simulation effort is linear in n. To illustrate, we present a simulation of 100,000 samples of a credit portfolio with 1,000 obligors that takes less than 18 seconds on a PC.

Big Data Credit Report in Credit Risk Management of Consumer Finance

“…It will help implement China's inclusive financial strategy, lower the threshold for financial services, benefit more people, eradicate poverty, and achieve social equity. At the same time, it has a certain reference significance for the application of big data risk control in the industry [3,4]. In terms of inter-pretability, new big data algorithms can evaluate the importance of statistical indicators, and statistical indicators with the highest comprehensive ranking of statistical indicators have better interpretability.…”

Multivariate tempered stable additive subordination for financial models

“…However, it is possible to consider multivariate self-decomposable distributions to define multivariate Sato subordinators. Sato subordinators associated to self-decomposable distributions are easy to construct, introduce time inhomogeneity and perform well on financial data (Sun et al (2017)). Therefore, it is possible to use additive subordination of multivariate Brownian motions to obtain additive processes with inhomogeneous increments to model asset returns.…”

Exchangeable min-id sequences: Characterization, exponent measures and non-decreasing id-processes

“…The relation between the nnnd instances of these subfamilies with the multivariate probability laws of X via de Finetti's Theorem has been explored in several previous articles, which are elegantly unified and extended by the present work. Firstly, if H is a nondecreasing Lévy process (aka Lévy subordinator), the finite-dimensional distributions of the corresponding sequence X are so-called Marshall-Olkin distributions, a result first found in [40], re-discovered and further explored in [58]. Secondly and slightly more general, if H is a non-decreasing additive process (aka additive subordinator), the survival function defined by x → exp(−H x ) is called a neutral-to-the-right prior in non-parametric Bayesian statistics.…”