Dynamic Hedging of Portfolio Credit Risk in a Markov Copula Model

Bielecki, Tomasz R.; Cousin, Areski; Crépey, Stéphane; Herbertsson, Alexander

doi:10.1007/s10957-013-0318-4

Cited by 54 publications

(37 citation statements)

References 33 publications

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“…This aspect of the Markov copula theory is of fundamental importance for efficient calibration of a model to market data. In [4] and [2] (see also references therein), the strong Markov copula theory was successfully applied to separate calibration of dependence in the pool of 125 obligors (constituting an iTraxx index), from the calibration of univariate characteristics of the individual obligors. We are currently working on using the weak-only Markov copulae for such purpose.…”

Section: 2mentioning

confidence: 99%

Intricacies of dependence between components of multivariate Markov chains: weak Markov consistency and weak Markov copulae

Bielecki

Jakubowski

Niewȩgłowski

2013

Electron. J. Probab.

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In this paper we examine the problem of existence and construction of multivariate Markov chains such that their components are Markov chains with given laws. Specifically, we provide sufficient and necessary conditions, in terms of semimartingale characteristics, for a component of a multivariate Markov chain to be a Markov chain in its own filtration -a property called weak Markov consistency. Accordingly, we introduce and discuss the concept of weak Markov copulae. Finally, we examine relationship between the concepts of weak Markov consistency and weak Markov copulae, and the corresponding strong versions of these concepts.

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Section: 2mentioning

confidence: 99%

Intricacies of dependence between components of multivariate Markov chains: weak Markov consistency and weak Markov copulae

Bielecki

Jakubowski

Niewȩgłowski

2013

Electron. J. Probab.

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“…We point out that many existing models of default contagion are variants of the above described framework. Without being exhaustive, we mention some variants: non absorbing states (Giesecke and Weber [21], [22]); credit migration models with more than two states for each debtor (Davis and Esparragoza-Rodriguez [10], Bielecki et al [4]); the so-called frailty models where the filtration F is (partially) unavailable for pricing and filtering techniques are used (Frey and Schmidt [20]); more than one default is allowed to occur at a time (Bielecki, Cousin, Crépey, Herbertsson [2]). We recommend the survey paper by Bielecki, Crépey, Herbertsson [24] for a more detailed analysis of the Markovian setting.…”

Section: Default Models With Interacting Intensities: the Markovian Amentioning

confidence: 99%

A Default System with Overspilling Contagion

Coculescu

2017

SSRN Journal

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Abstract. In classical contagion models, default systems are Markovian conditionally on the observation of their stochastic environment, with interacting intensities. This necessitates that the environment evolves autonomously and is not influenced by the history of the default events. We extend the classical literature and allow a default system to have a contagious impact on its environment. In our framework, contagion can either be contained within the default system (i.e., direct contagion from a counterparty to another) or spill from the default system over its environment (indirect contagion). This type of model is of interest whenever one wants to capture within a model possible impacts of the defaults of a class of debtors on the more global economy and vice versa. Motivation and aimsDefault events tend to cluster in time, a phenomenon that arises from diverse causes. The literature on dynamic modelling of defaults proposed so far two major mechanisms that produce this effect. First, there is the so-called cyclical correlation, i.e., the dependence of the debtors' financial situation on some common factors. One can naturally think of some macroeconomic factors that impact the default probabilities of many debtors at a time, as the interest rates, the prices of some commodities or the business cycle; a purely statistical approach using abstract or unobserved factors is also possible. Secondly, there is the socalled counterparty risk or direct contagion, i.e., the default of one debtor represents itself a destabilising factor impacting the default rates of surviving debtors (the counterparties of a defaulted debtor). In order to have these two mechanisms of contagion operational simultaneously it is necessary to distinguish within the model the default system from its environment. The role of the random environment is to carry the cyclical dependence.The most classical approach so far was to consider that conditionally on a given realisation of the random environment, the vector of default indicator processes of the different debtors is a time inhomogeneous Markov chain.While the Markovian assumption is convenient, it necessitates that the environment evolves autonomously and is not influenced by the history of the default events. Our aim here is precisely to remove this assumption. This equates to introducing a new source of contagion, that we call overspilling (or indirect) contagion: the one that transmits from the default system to its environment, subsequently having a feedback effect on the system itself. The construction, by its nature is not Markovian, the default probabilities depend not only on the current state of the default system, but also on the circumstances of the occurrence of the past defaults, more precisely the knowledge of their impact on the environment. The theory of enlargements of filtrations reveals to be the right tool to deal with this non Markovian framework.

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“…In the following proposition, we compute the conditional survival probability of ( ) ∈ defined by, for every positive , , Recalling (5), let , , = , , ( , , ).…”

Section: Conditional Survival Distributionmentioning

confidence: 99%

Informationally Dynamized Gaussian Copula

Crépey

Jeanblanc

2013

Int. J. Theor. Appl. Finan.

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In order to dynamize the static Gaussian Copula model of portfolio credit risk, we introduce a model filtration made of a reference Brownian filtration progressively enlarged by the default times. This yields a multidimensional density model of default times, where, as opposed to the classical situation of the Cox model, the reference filtration is not immersed into the enlarged filtration. In mathematical terms this lack of immersion means that martingales in the reference filtration are not martingales in the enlarged filtration. From the point of view of financial interpretation this means defaults contagion, a good feature in the perspective of, say, wrong-way risk modeling of counterparty risk on credit derivatives. Computational tractability is ensured by invariance of multivariate Gaussian distributions through conditioning by some components, the ones corresponding to past defaults. Moreover the model is Markov in an augmented state-space including past default times. After a discussion of different notions of deltas, the model is applied to valuation of counterparty risk on credit derivatives. * The research of the authors benefited from the support of the 'Chaire Risque de crédit', Fédération Bancaire Française. Thanks to Behnaz Zargari, a former PhD student from Evry University, for the computations related to the dynamized Marshall-Olkin model of Section 4 (right panel of Figure 7). Thanks also to Areski Cousin for interesting discussions about the model deltas.

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Dynamic Hedging of Portfolio Credit Risk in a Markov Copula Model

Cited by 54 publications

References 33 publications

Intricacies of dependence between components of multivariate Markov chains: weak Markov consistency and weak Markov copulae

Intricacies of dependence between components of multivariate Markov chains: weak Markov consistency and weak Markov copulae

A Default System with Overspilling Contagion

Informationally Dynamized Gaussian Copula

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