Dynamic single portfolio credit models
From FinMath
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[edit] Top Down Models
The phrase "top down" is used by Giesecke and Goldberg (2005) and Errais and Giesecke (2007). Many dynamic single portfolio credit models comprise a global or aggregate default intensity or loss process (or integral thereof) for a single group of names, with optional subsequent selection of idiosyncratic defaults. Top down models find application in the modelling of basket options, levered vehicles with spread or default triggers, forward starting CDOs and other securities where aggregate dynamics are sufficient statistics.
[edit] Explicit Contagion Models
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[edit] Structured Models
In "Dynamic credit portfolio modelling in structural models with jumps", Kiesel and Scherer 2007 present a structural jump-diffusion portfolio model with a bottom-up flavour. Dependence is introduced through a market factor and also common jumps in firm values.
[edit] Transform Methods
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[edit] Markovian contagion models
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[edit] Binomial trees
In "Dynamic models of portfolio credit risk: a simplified approach", Hull and White 2007 propose a model in which hazard rates follow a deterministic process subject to periodic impulses. The recombining nature allows for easier calibration to market data.
The recombining feature is also proposed by Erik Schlogl and Tao Peng 2007. Here dependence is also driven by a common factor but a more detailed conditional probability structure emerges from a global iterative calibration procedure. Global calibration of parameters can succeed in instances where bootstrap methods back the user into a corner.
Binomial trees are also exploited in "Dynamic Conditioning and Credit Correlation Baskets", Albanese and Vidler 2008.
[edit] Recent Papers
- Lauren, Cousin and Fermanian (2007). Hedging default risks of CDOs in Markovian Contagion Models
- Giesecke and Kim (pdf) implement exact simulation of default times using an affine point process for the loss.
- Frey and Backhaus (pdf) construct a Markov chain for CDO to measure hedging efficiency.
[edit] Symmetry considerations
The pricing of basket default swaps for a particular fixed portfolio (c.f. many) depends only superficially on the timing and ordering of defaults. The distribution of defaults by time slice is adequate, assuming similar recoveries, and otherwise the distribution of losses by time slice will do nicely (that is, in the absence of any knowledge of the joint distribution of losses by time slice other than the marginals).
A second conceptual reduction applies also to path dependent products. The pathwise pricing of several important financial securities are invariant under permutations of reference obligations. For example, the premiums and credit event payments owed to parties in a basket default swap contract with artificial reserve fund might be influenced by the amount and timing of losses, but not the precise choice of reference entity defaulting. Further, if recovery rates are assumed equal (or more generally, exchangeable), then the present values of contingent cashflows depend on the timing and number of defaults but not the precise selection of reference entity defaulting. Trivially then, any two risk neutral distributions applied to the pricing of a basket default swap and differing only by a permutation of coordinates will assign the same present value to both legs of the swap (the same off the run valuation and so forth). It further follows that the price assigned to a symmetric multi-name security such as a basket default swap is equal to the price assigned by a model (i.e. joint distribution on defaults) is equal to the price assigned by a similar model equal to the average of the original model over all possible permutations of coordinates. This latter model is by construction exchangeable, and it is then obvious that the class of exchangeable models is sufficient. Taking this line of reasoning a step further, we observe by a generalization of de Finetti's theorem (Kerns and Szekely - 2006) that signed mixtures of i.i.d. models are also adequate.
These considerations suggest that the space of single portfolio credit models is, in some loose sense, small.
[edit] Recent Papers
- Dynamic models of portfolio credit risk: a simplified approach (pdf) Hull and White, May 2007
