Session 22. Multivariate stochastic modelling in finance, insurance and risk management

On the construction of high-dimensional models for dependent default times

Jan-Frederik Mai, Technische Universität München, Germany
The talk is based on joint work with German Bernhart, Damiano Brigo and Matthias Scherer
Several financial applications require a mathematical model for \((X_1,\ldots,X_d)\), where component \(X_k\) denotes the random future time point, when the \(k\)-th asset in a portfolio of size \(d \gg 2\) defaults. This high-dimensional modeling task is difficult in general because typical applications require the model to be low-parametric and efficient to simulate on a standard PC, while at the same time being realistic enough to capture common stylized facts. The use of a min-stable multivariate exponential (MSMVE) distribution in the sense of [Esary, Marshall (1974)] can be seen as a natural candidate in many cases.

(a) We show how to define low-parametric MSMVEs from parametric families of Bernstein functions, and how to simulate them efficiently. The key here is a novel stochastic representation for one-factor MSMVEs based on strong IDT processes, worked out in [Mai, Scherer (2014), Bernhart et al. (2014)].

(b) We derive a novel characterization for the subfamily of Marshall--Olkin distributions in terms of Markovianity, which sets this subfamily apart as a class of probability laws for dependent default times satisfying many practically desirable axioms, see [Brigo et al. (2014)].

References

  [Bernhart et al. (2014)] G. Bernhart, J.-F. Mai, M. Scherer, Constructing min-stable multivariate exponential distributions from Bernstein functions, working paper (2014).

  [Brigo et al. (2014)] D. Brigo, J.-F. Mai, M. Scherer, On the Markovianity of multi- variate default indicators and its impact on simulation for economics and finance, working paper (2014).

  [Esary, Marshall (1974)] J.D. Esary, A.W. Marshall, Multivariate distributions with exponential minimums , Annals of Statistics 2 (1974) pp. 84-98.

  [Mai, Scherer (2014)] J.-F. Mai, M. Scherer, Characterization of extendible distribu- tions with exponential minima via processes that are infinitely divisible with respect to time , Extremes 17 (2014) pp. 77-95.

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