Consistent Iterated Simulation of Multi-Variate Default Times: A Markovian Indicators Characterization

24 Pages Posted: 5 Jun 2013 Last revised: 2 May 2014

See all articles by Damiano Brigo

Damiano Brigo

Imperial College London - Department of Mathematics

Jan-Frederik Mai

Technische Universität München (TUM) - HVB Institute for Mathematical Finance

Matthias A. Scherer

Technische Universität München (TUM)

Date Written: May 1, 2014

Abstract

We investigate under which conditions a single simulation of joint default times at a final time horizon can be decomposed into a set of simulations of joint defaults on subsequent adjacent sub-periods leading to that final horizon. Besides the theoretical interest, this is also a practical problem as part of the industry has been working under the misleading assumption that the two approaches are equivalent for practical purposes. As a reasonable trade-off between realistic stylized facts, practical demands, and mathematical tractability, we propose models leading to a Markovian multi-variate survival -- indicator process, and we investigate two instances of static models for the vector of default times from the statistical literature that fall into this class. On the one hand, the "looping default'' case is known to be equipped with this property, and we point out that it coincides with the classical "Freund distribution'' in the bivariate case. On the other hand, if all sub-vectors of the survival indicator process are Markovian, this constitutes a new characterization of the Marshall-Olkin distribution, and hence of multi-variate lack-of-memory. A paramount property of the resulting model is stability of the type of multi-variate distribution with respect to elimination or insertion of a new marginal component with marginal distribution from the same family. The practical implications of this "nested margining'' property are enormous.

To implement this distribution we present an efficient and unbiased simulation algorithm based on the Levy-frailty construction. We highlight different pitfalls in the simulation of dependent default times and examine, within a numerical case study, the effect of inadequate simulation practices.

Keywords: Stepwise default simulation, default modeling, credit modeling, default dependence, default correlation, default simulation, arrival times, credit risk, Marshall-Olkin distribution, nested margining, Freund distribution, looping default models

JEL Classification: C15, C16

Suggested Citation

Brigo, Damiano and Mai, Jan-Frederik and Scherer, Matthias A., Consistent Iterated Simulation of Multi-Variate Default Times: A Markovian Indicators Characterization (May 1, 2014). Available at SSRN: https://ssrn.com/abstract=2274369 or http://dx.doi.org/10.2139/ssrn.2274369

Damiano Brigo (Contact Author)

Imperial College London - Department of Mathematics ( email )

South Kensington Campus
London SW7 2AZ, SW7 2AZ
United Kingdom

HOME PAGE: http://www.imperial.ac.uk/people/damiano.brigo

Jan-Frederik Mai

Technische Universität München (TUM) - HVB Institute for Mathematical Finance ( email )

Matthias A. Scherer

Technische Universität München (TUM) ( email )

Arcisstrasse 21
Munich, 80333
Germany

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