Markov multi-variate survival indicators for default simulation as a new characterization of the Marshall–Olkin law

Abstract

A new characterization of the Marshall-Olkin distribution is provided: all subvectors of the associated survival indicators are continuous-time Markov chains. This property is crucial to overcome practical limitations for the modeling of highdimensional default times (rebalancing, iterative simulation, consistent sub-portfolios).Classification Codes: AMS 60E07, 62H05, 62H20, 62H99; JEL C15, C16.

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

“…This article analyzes further desirable theoretical and practical conditions on the resulting simulation process and as a result focuses on the subclasses of Marshall-Olkin distributions as well as a multivariate extension of the bivariate Freund distribution. In particular, the practically important requirement of having the Markov property also for sub-vectors of indicators leads to a new characterization of the Marshall-Olkin law that has been first discussed in [8] and is recalled here in the context of the present paper. Our general aim is to increase awareness of the fact that the stepwise simulation of default indicators (approach (ii) above) is a hard task in general, and in particular that the practical implementation is not feasible without huge efforts (both theoretical and computational), and that sizable errors and undesired effects may occur by iterating under the wrong conditions.…”

“…4 the "mixed default/survival" problem is discussed, for looping default models, Freund distributions, and multivariate phase-type distributions. A special focus lies on the Marshall-Olkin class, leveraging its new characterization in terms of Markov property of vectors and subvectors of indicators, as in [8], and different simulation strategies as well as a convenient construction through Lévy-frailty models. The final section concludes the article.…”

“…In [8] it was shown that τ has a Marshall-Olkin distribution if and only if for every non-empty subset I the marginal survival-indicator process Z I (t) := (1 {τi>t} , i ∈ I) is time-homogeneous Markovian. The following theorem shows that every continuous-time, time-homogeneous Markovian survivalindicator process can be constructed using a finite sequence of MarshallOlkin distributed random vectors.…”

“…This section starts with summarizing the findings and results of [8], in which it is emphasized that for practical applications even the assumption of a continuous-time, time-homogeneous Markovian survival-indicator process has serious drawbacks if the corresponding default-times vector τ does not have a Marshall-Olkin distribution. The findings are:…”

“…4.2 has the advantage of being a De Finetti model for extendible sequences, which renders the approach independent of the dimension d. This solves not only the problem described in (b), but also provides an alternative simulation strategy, see [8] for a detailed account. The alternative simulation strategy has the advantage that its runtime scales linearly with increasing dimension, which makes it particularly interesting for large d. The approach comes with the drawback that a simulation bias is introduced as we can only sample the random walk corresponding to some embedding of Λ on a discrete time-grid.…”