Dynamic multiple portfolio credit models
From FinMath
Risk neutral models for default intended for simultaneous valuation of tranche prices from multiple overlapping portfolios, and plausible internally consistent dynamics thereof. Separate pages exist for single portfolio models and latent variable dynamic models which warrant a category unto themselves. See also an overview of portfolio credit models.
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[edit] Overview
Characterized by simultaneous modeling of credit default swap and tranche pricing dynamics for a plurality of potentially overlapping reference portfolios, dynamic multiple portfolio credit models are desirable from practitioner, academic and ascetic standpoint. In the context of risk neutral pricing they are, arguably, something of an unobtainable holy grail. In principal a vast number of institutional credit products could be priced simultaneously. In practice, most progress has been made on the periphery of this problem, on special cases, and in contexts where calibration to market prices is not essential. Some of the reasons for this lack of progress are explained on this page.
Notwithstanding the pragmatics of this problem formulations of dynamic multiple portfolio credit models have been used in industry at least as early as 1999 and probably before, but often yeilded to simpler static counterparts in day to day business. A starting point is the class of intensity model (also called "reduced form" in this context) whose principal implementation exploits conditional independence of defaults given correlated hazard rates (so called doubly stochastic or Cox processes). This class has been criticised because even extreme covariation in hazard rates can translate to relatively benign default correlation inconsistent with tranche pricing, and a study of US corporations from 1979 to 2004 (Duffie et al 2007) reject the independence hypothesis, though the "residual" correlation is not large. Several authors have taken up the defense (Yu 2005), Mortensen (2005)).
A popular framework involves the introduction of latent variables whose posterior distributions are inferred at subsequent time steps, with some care required avoiding inconsistency and computational intractability. A rather large number of attempts fall into the rubrik of latent variable portfolio credit models (a.k.a. inferred hidden factor models). Challenges to the task at hand are presented in sections to follow described below may or may not be interpreted as criticism of the latent variable approach. Notwithstanding some philosophical and practical distinctions, mathematical equivalences within the former class are frequent (Andersen - pdf).
It may be argued that credit markets have provided surprisingly little incentive to pursue the construction of dynamic credit models and incumbent calibration procedures. Portfolio credit products tend to be treated in isolation (and amenable to "top down" approaches and other single portfolio dynamic credit models for forward starting CDOs, leveraged super senior transactions) or exhibit sufficient - or perceived - insensitivity to dynamics and intertemporal loss distribution behaviour. Symmetries and weak path dependence of the more liquid portfolio credit products such as basket default swaps is certainly a factor. Valuations of these products leaves flaws in popular static portfolio credit models largely unscrutinized in the competitive market sense, however nauseating their mathematical flaws might seem.
Countering this argument is the emergence of credit derivative product companies (essentially infinitely lived CDO or CDO squared vehicles), growing volumes in bespoke correlation flow trading, greater sophistication in the analysis of cash products with non-trivial path dependence, and increased sophistication in counterparty risk management and other applications requiring plausible evolution of modelled quantities and non-trivial overlap in reference portfolios.
We consider on this page modelling attempts which address directly the fundamental tension between cross portfolio consistency, tractable dynamics and market satisfying levels of default dependence.
[edit] Intensity Models
"Conditionally non-systematic default risk" was coined by David Lando in his paper "On Cox Processes and Credit Risky Securities" (Derivatives Research, Vol 2 1998). Lando's formalism assumes a firm's state and default process are conditionally independent (though and extension by Guo, Jarrow and Menn 2007 does not). The instantaneous probability of default is determined by some finite dimensional stochastic process Xt. The latter process summarizes the viability of the firm whose time of default τ may be written as the first time the integrated intensity process crosses an exponential barrier:
where E1 is unit exponential random. The probability of survival to time t is
and any (e.g. affine) process admitting closed form solution to the right hand side integral is a convenient choice for bond and default swap valuation. Cox processes are also called doubly stochastic Poisson process because the default of an asset at time t is random and occurs with intensity λt which is also stochastic. The simplest intensity process exhibiting mean reversion is the Ornstein-Uhlenbeck process - known in the interest rate context as the Vasicek model. Gaussian processes such as this admit negative values whereas others in the affine class (pioneered by Duffie, Singleton et al) do not. Jumps in the hazard rate process have been considered by many authors ( Zhou 2001 and more recently Cariboni and Schoutens - 2007, Brigo and El-Bachir 2006, Garcia, Goossens, Schoutens 2007, Matovu 2007 and Ahangarani 2007). The right hand side also suggests direct modelling of the integrated process or approximation thereof, even if the joint hazard rate and integrated hazard rate is not gaussian. Fast convolution and transform methods may be borrowed from the interest rate literature (Duffie, Pan Singleton 2000). Integrated hazard rate processes may be approximated by lower dimensional processes (as with the Karhunen-Loeve expansion). Further defense of the practicality of intensity models is provided by Eckner ("Computational Techniques for Basic Affine Models of Portfolio Credit Risk 2007) who demonstrates good fit to CDX tranches prices.
Despite these efforts and the significant time since early expositions (Jarrow and Turnbull 1995, Duffie 1998 and others), computational complexity continues to hinder the adoption of intensity models and reduce their popularity. Comments to this effect have been made by several authors including Hull and White (2004).
As in the case of single portfolio models and latent variable models, judicious choice of conditionally independent variables (factors) provides some tradeoff between tractability and fidelity. A recent paper by Ronnegard and Roszbach however, highlights other dangers when too much emphasis is placed on systemic risk factors (2007). They write "clearly, restrictive assumptions about the dependency between firms, such as assuming common factors and independent disturbances, paired with an inability to match peaks in default rates, form an unattractive and undesirable property for portfolio credit risk models."
Running counter to this advice, large collective jumps in hazard rates have been suggested and might be used with appropriate care in a risk neutral context. To the extent that spread time series exist they certainly do exhibit significant co-movement - though perhaps not quite to the extent charicatured by extreme systemwide common jumps.
The conditional independence assumption is brought into question by near-simultaneous chapter 11 filings. However, Lando's "pure" approach has simplicity and dynamic consistency on it side and a resurgence of interest is possible. To to appreciate the advantages of intensity models, some discussion of alternatives is supplied.
Rudimentary implementations of affine jump diffusion models are provided in Code:sde_util.
[edit] Combining Intensity and Copula Models
Motivated in part by senior tranche prices, "bastardizations" of intensity models been employed in industry since 1999 or earlier usually can be classified as latent variable models. In one extension a Copula may be used to introduce correlation in the respective exponential variables E1,...,En, thereby increasing the correlation in default times. This setup nests both intensity models and Copula models. The reduction to the former is obvious and the latter follows from elementary properties of Copula functions. The connection may also be seen from the perspective of the following computational "shortcut" in the default time simulation described above. As a thought experiment, replace the burdensome pathwise hazard rate integrals with a deterministic proxies 
satisfying
![\exp\left( \int_0^t \bar{\lambda}^i_s ds \right) = E \left[ \exp \left( -\int_0^t \lambda^i(X_s) ds \right) \right]](/w/upload/math/8/2/c/82cf506b8e052693c0bd18cde4d66724.png)
Here marginals are preserved (easily checked) but to no surprise, we have merely returned to the static Copula framework. Copulas are preserved by monotonic transformations of marginals - in this case the transformations taking Ei into default times τi. Adding additional correlation in this manner is something of a kludge, given the suspect dynamics of Copula models. However, the inconsistencies are at least reduced, since the Copula function can be calibrated to residual correlation only.
Hybrid intensity/copula models were introduced into the academic literature by Schonbucher and Schubert (2001) who provide a comprehensive treatment including a proof that under the filtration containing only the individual obligator and Xt, the quantity 
is indeed the default intensity for the i'th asset. Formulas for conditional survival under the full filtration are also derived for Archimedian Copulas, and in general before the first default occurs. Duffie et al 2005 provide an example of calibrating the residual correlation in a hybrid copula/intensity model.
[edit] Motivation: the challenge of cross portfolio consistency
The introduction of additional correlation by means of a Copula function is - for common copulas - equivalent to the introduction of latent factors F1,..,Fm. Case in point, the one time draws of the auxiliary exponential random variables Ei are conditionally independent given F1,...,Fm. In theory, a self-consistent dynamic model for the defaults and risk neutral hazard rates can be defined by sensible choice of filtration - for instance by including all defaults in the set of observed variables at time t as well as some random functions of hazard rates and latent variables. I principal all quantities of interest can subsequently be computed, if need be by computationally intensive Bayesian inference techniques (Metropolis Hastings sampling, mean field approximations and coupling from the past.
Unfortunately the introduction of variables requiring filtering poses conceptual as well as computational challenges, as the "universal" portfolio is not fixed, well defined, and certainly is not referenced to liquid tranches. The inclusion of new portfolios and assets into the modelled system should increase the filtration by at minimal the defaults - lest simulations of names referenced to two different portfolio be inconsistent. The same problem plagues explicit contagion models (violating Lando's independence assumption with the introduction of specific rules for intensity increases following defaults of assets), since the filtration in use is tied to the particular subset of names considered which might expand at any time. It is of desireable to use a model for a particular collection of names in ignorance of all possible future choices of portfolios, reference obligations and unrelated trades. Though one may take the philosophical position that no trade or name is truly unrelated, the overnight valuations of a collection of CDOs should in all reasonableness not depend on whether or not an extraneous unreferenced name has been added to the database. Such behaviour violates a number of inoffensive assumptions in the architecture of commercial risk systems, not to mention regulatory requirements.
Arguably, considerations such as these motivate the categorization of explicit contagion models as dynamic single portfolio credit models, since simulation of paths for one portfolio should in principle be drawn from a one time global simulation of all names - but in practice cannot be.
[edit] Other approaches
(Please add to this and other sections!)
In "Factor Models for Credit Correlation", 2007 Inglis and Lipton propose a single factor dynamic model outside the affine family. Motivated by the structural approach, the authors employ a logit function linking the integral of a (functional of a) jump diffusion to default. Motivated by a "softening" of the structural model, this approach admits closed form conditional default probabilities despite considerably less benign hazard rates than the affine framework. Though a single factor model is unlikely to mimic global dynamics of the credit universe, Inglis and Lipton demonstrate good fits when mapping calibrated CDX parameters to iTrax and vice versa.
[edit] Regime switching and transition models
Schuermann (2007) provides an overview of the use of credit transition models and the estimation of rating migration matrices. Ratings migration models have been used extensively by rating agencies not only for forecasting individual bond downgrades but also in the analysis of debt obligations and structured finance vehicles, most recently derivative product companies. Evidently, mathematics describing rating transition models can also be used in more generic frameworks. Typically, a finite set of states or regimes is specified in conjunction with a continuous time Markov process and, overlaid, dynamics for other variables conditioned on the state of the Markov chain.
The Markov assumption applied to ratings transitions in isolation is convenient but deserving of scrutiny. Additional economic variables might be introduced to capture business cycle variation and serial correlation in rating changes. Evidence for the latter is presented in Christensen et al 2004.
The special case involving only two states was suggested by Hansen and Poulsen (2000): "A simple regime switching interest rate model" and generalized by Cotton 2001 to allow for regime dependent volatility. However, closed form solutions in the latter case are known only in the case of rapid regime switches and equal unconditional regime densities, assumptions are not a priori desirable.
[edit] Amnesic Models
One attempt around this problem of cross portfolio consistency involves the use of rapidly varying processes whose Bayesian updates are very close to their invariant distributions. The filtered distributions of these fluctuating latent variables are roughly independent of the filtration time t, thereby minimizing the cross portfolio dynamic inconsistencies described above. When used in conjunction with asymptotic techniques yields computationally feasible marginals, default swap pricing and so forth. Bursts of volatility encourage large joint excursions in hazard rates and significantly increase short term default correlation.
Tractability of multiscale intensity models relies on a separation of time scales (volatility fluctuating either more rapidly or less rapidly than the hazard rate) which allows a partial uncoupling of the Feynman-Kac partial differential equations for the moment generating function. Martingale techniques can also be applied, but only if the same modeling ansatz is applied. The use of multiscale models for hazard rates is considered by Cotton (2001) and by Papageorgiou and Sircar (2007) who present new portfolio level analytic results with name grouping.
[edit] Operator models
Albanese, Chen, Dallessandro and Vidler (2007) formulate a model in which the status of a company is represented by a continuous variable ξ with ξ = 0 corresponding to default and ξ = 1 corresponding to a divine state of perfect credit worthiness. Discretized dynamics of the credit variable ξ are made tractable by methods in linear algebra, beginning with the diagonalization of its infinitesimal generator. Some further discussion here.
[edit] Structural models
See also debt equity models. As with the single name problem, structural multi-name models have been criticized for failure to generate short term default probabilities of sensible magnitude. Stochastic volatility techniques have been used in this context by Fouque, Wignall and Zhou, "Modeling Correlated Defaults: First Passage Model under Stochastic Volatility" (2007). The authors extend analytic results to the multi-asset setting, including a correction to the binomial formula.
[edit] Contagion models
A common critique of explicit contagion models is failure to address individual CDS pricing. Herbertsson suggests a solution where hazard rates respond to defaults but are constant between defaults ("Modelling Default Contagion Using Multivariate Phase-Type Distribution" (2007). This makes it possible to adapt some results from multivariate phase-type distributions, re-interpreting intensity based models as markov jump processes. Though the events of 2007 bring the key assumption into question (wild moves in spreads absent corporate defaults), the reduction to matrix analytics is convenient. Herbertsson illustrates name by name calibration and exhibits a formula for the CDS spreads.
[edit] See also
Static multiple portfolio credit models, Latent variable portfolio credit models, Credit transition models.
