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34
BSLP: Markovian bivariate spreadloss model for portfolio credit derivatives
, 2007
"... BSLP is a twodimensional dynamic model of interacting portfoliolevel loss and loss intensity processes. It is constructed as a Markovian, shortrate intensity model, which facilitates fast lattice methods for pricing various portfolio credit derivatives such as tranche options, forwardstarting tr ..."
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Cited by 48 (0 self)
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BSLP is a twodimensional dynamic model of interacting portfoliolevel loss and loss intensity processes. It is constructed as a Markovian, shortrate intensity model, which facilitates fast lattice methods for pricing various portfolio credit derivatives such as tranche options, forwardstarting tranches, leveraged supersenior tranches etc. A semiparametric model specification is used to achieve near perfect calibration to any set of consistent portfolio tranche quotes. The onedimensional local intensity model obtained in the zero volatility limit of the stochastic intensity is useful in its own right for pricing nonstandard index tranches by arbitragefree interpolation. Opinions expressed in this paper are those of the authors, and do not necessarily reflect the view of
Calibration of CDO tranches with the dynamical generalizedPoisson loss model
 RISK
, 2006
"... In the first part we consider a dynamical model for the number of defaults of a pool of names. The model is based on the notion of generalized Poisson process, allowing for more than one default in small time intervals, contrary to many alternative approaches to loss modeling. We illustrate how to d ..."
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Cited by 36 (5 self)
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In the first part we consider a dynamical model for the number of defaults of a pool of names. The model is based on the notion of generalized Poisson process, allowing for more than one default in small time intervals, contrary to many alternative approaches to loss modeling. We illustrate how to define the pool default intensity and discuss recovery assumptions. The models are tractable, pricing and simulation are straightforward, and consistent calibration to quoted index CDO tranches and tranchelets for several maturities is feasible, as we illustrate with numerical examples. In the second part we model directly the pool loss and we introduce extensions based on piecewisegamma, scenariobased or CIR random intensities, leading to richer spread dynamics, investigating calibration improvements and stability.
Twodimensional Markovian model for dynamics of aggregate credit loss
 ADVANCES IN ECONOMETRICS
, 2007
"... We propose a new model for the dynamics of the aggregate credit portfolio loss. The model is Markovian in two dimensions with the state variables being the total accumulated loss Lt and the stochastic default intensity λt. The dynamics of the default intensity are governed by the equation dλt = κ(ρ( ..."
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Cited by 25 (1 self)
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We propose a new model for the dynamics of the aggregate credit portfolio loss. The model is Markovian in two dimensions with the state variables being the total accumulated loss Lt and the stochastic default intensity λt. The dynamics of the default intensity are governed by the equation dλt = κ(ρ(Lt, t) − λt)dt + σ √ λtdWt. The function ρ depends both on time t and accumulated loss Lt, providing sufficient freedom to calibrate the model to a generic distribution of loss. We develop a computationally efficient method for model calibration to the market of synthetic single tranche CDOs. The method is based on the Markovian projection technique which reduces the full model to a onestep Markov chain having the same marginal distributions of loss. We show that once the intensity function of the effective Markov chain consistent with the loss distribution implied by the tranches is found, the function ρ can be recovered with a very moderate computational effort. Because our model is Markovian and has low dimensionality, it offers a convenient framework for the pricing of dynamic credit instruments, such as options on indices and tranches, by backward induction. We calibrate the model to a set of recent market quotes on CDX index tranches and apply it to the pricing of tranche options.
Dynamic models of portfolio credit risk: A simplified approach. Working paper
 Joshi, M S and A M Stacey
, 2006
"... We propose a simple dynamic model that is an attractive alternative to the (static) Gaussian copula model. The model assumes that the hazard rate of a company has a deterministic drift with periodic impulses. The impulse size plays a similar role to default correlation in the Gaussian copula model. ..."
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Cited by 25 (2 self)
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We propose a simple dynamic model that is an attractive alternative to the (static) Gaussian copula model. The model assumes that the hazard rate of a company has a deterministic drift with periodic impulses. The impulse size plays a similar role to default correlation in the Gaussian copula model. The model is analytically tractable and can be represented as a binomial tree. It can be calibrated so that it exactly matches the term structure of CDS spreads and provides a good fit to CDO quotes of all maturities. Empirical research shows that as the default environment worsens default correlation increases. Consistent with this research we find that in order to fit market data it is necessary to assume that as the default environment worsens impulse size increases. We present both a homogeneous and heterogeneous version of the model and provide results on the use of the calibrated model to value forward CDOs, CDO options, and leveraged super senior transactions. *We are grateful to Moody’s Investors Services for providing financial support for this research.
Dynamic CDO Term Structure Modelling
, 2008
"... This paper provides a unifying approach for valuing contingent claims on a portfolio of credits, such as collateralized debt obligations (CDOs). We introduce the defaultable (T, x)bonds, which pay one if the aggregated loss process in the underlying pool of the CDO has not exceeded x at maturity T, ..."
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Cited by 12 (5 self)
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This paper provides a unifying approach for valuing contingent claims on a portfolio of credits, such as collateralized debt obligations (CDOs). We introduce the defaultable (T, x)bonds, which pay one if the aggregated loss process in the underlying pool of the CDO has not exceeded x at maturity T, and zero else. Necessary and sufficient conditions on the stochastic term structure movements for the absence of arbitrage are given. Background market risk as well as feedback contagion effects of the loss process are taken into account. Moreover, we show that any exogenous specification of the volatility and contagion parameters actually yields a unique consistent loss process and thus an arbitragefree family of (T, x)bond prices. For the sake of analytical and computational efficiency we then develop a tractable class of doubly stochastic affine term structure models. Key words: affine term structure, collateralized debt obligations, loss process, single tranche CDO, term structure of forward spreads 1
Credit risk models IV: Understanding and pricing CDOs. www.abelelizalde.com
, 2005
"... Some investors in the Collateralized Debt Obligations (CDOs) market have been publicly accused of not fully understanding the risks and dynamics of these products. They won’t have an excuse any more. This report explains the mechanics of CDOs: their implied cash flows, the variables affecting those ..."
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Cited by 12 (0 self)
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Some investors in the Collateralized Debt Obligations (CDOs) market have been publicly accused of not fully understanding the risks and dynamics of these products. They won’t have an excuse any more. This report explains the mechanics of CDOs: their implied cash flows, the variables affecting those cash flows, their pricing, the sensitivity of CDO prices to those variables, the functioning of the markets where they are traded, their different types, the conventions used for trading CDOs,... We built our description of CDOs pricing upon the Vasicek asymptotic single factor model because of its simplicity and the insights it provides regarding the pricing of CDOs. Additionally, we provide an extensive and updated review of the literature which extends the Vasicek model by relaxing its, somehow restrictive, assumptions in order to build more
HJM: A Unified Approach to Dynamic Models for Fixed Income, Credit and Equity Markets
"... Summary. The purpose of this paper is to highlight some of the key elements of the HJM approach as originally introduced in the framework of fixed income market models, to explain how the very same philosophy was implemented in the case of credit portfolio derivatives and to show how it can be exten ..."
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Cited by 11 (4 self)
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Summary. The purpose of this paper is to highlight some of the key elements of the HJM approach as originally introduced in the framework of fixed income market models, to explain how the very same philosophy was implemented in the case of credit portfolio derivatives and to show how it can be extended to and used in the case of equity market models. In each case we show how the HJM approach naturally yields a consistency condition and a noarbitrage conditions in the spirit of the original work of Heath, Jarrow and Morton. Even though the actual computations and the derivation of the drift condition in the case of equity models seems to be new, the paper is intended as a survey of existing results, and as such, it is mostly pedagogical in nature. 1
UP AND DOWN CREDIT RISK
, 2008
"... This paper discusses the main modeling approaches that have been developed so far for handling portfolio credit derivatives. In particular the so called top, top down and bottom up approaches are considered. We first provide an overview of these approaches. Then we give some mathematical insights to ..."
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Cited by 10 (7 self)
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This paper discusses the main modeling approaches that have been developed so far for handling portfolio credit derivatives. In particular the so called top, top down and bottom up approaches are considered. We first provide an overview of these approaches. Then we give some mathematical insights to the fact that information, namely, the choice of a relevant model filtration, is the major modeling issue. In this regard, we examine the notion of thinning that was recently advocated for the purpose of hedging a multiname derivative by singlename derivatives. We then give a further analysis of the various approaches using simple models, discussing in each case the issue of possibility of hedging. Finally we explain by means of numerical simulations (semistatic hedging experiments) why and when the portfolio loss process may not
CDO Models  Towards the Next Generation: Incomplete Markets and Term Structure
, 2006
"... This article describes a new approach to the riskneutral valuation of CDO tranches, based on a general specification of the tranche loss distributions and the index default distribution. The new model is a termstructure model, and the generality with which the basic distributions are specified all ..."
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Cited by 7 (0 self)
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This article describes a new approach to the riskneutral valuation of CDO tranches, based on a general specification of the tranche loss distributions and the index default distribution. The new model is a termstructure model, and the generality with which the basic distributions are specified allows it to be perfectly calibrated to any set of market prices (for any number of tranches and maturities) that is arbitragefree. The use of the new model is illustrated by testing market prices for the standardized iTraxx index tranches (for all marketed tranches and maturities) to see if they are arbitragefree. Other examples include the determination of the arbitragefree range of prices allowed for an unmarketed standardized tranche and the determination of the cost of exiting a tranche position. For the latter example, both arbitragefree price ranges, and a preferred price, are obtained. Prices for unmarketed maturities and unmarketed nonstandard tranches are also obtained by an interpolation and extrapolation procedure. Because the model is an incompletemarket model characterized by many more parameters than market prices, it was essential to develop an efficient optimization approach to valuation. The article also makes use of a new approach to the problem of unequal recovery rates and notionals.
A Dynamic Correlation Modelling Framework with Consistent Recovery.” defaultrisk.com
"... This paper describes a flexible and tractable bottomup dynamic correlation modelling framework with a consistent stochastic recovery specification. In this modelling framework, only the joint distributions of default indicators are determined from the calibration to the index tranches; and the join ..."
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Cited by 6 (2 self)
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This paper describes a flexible and tractable bottomup dynamic correlation modelling framework with a consistent stochastic recovery specification. In this modelling framework, only the joint distributions of default indicators are determined from the calibration to the index tranches; and the joint distribution of default time and spread dynamics can be changed independently from the CDO tranche pricing by applying one of the existing topdown methods to the common factor process. Numerical results showed that the proposed modelling method achieved good calibration to the index tranches across multiple maturities under the current market conditions. This modelling framework offers a practical approach to price and risk manage the exotic correlation products.