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27
Pricing synthetic CDO tranches in a model with default contagion using the matrixanalytic approach
 CONTAGION IN PORTFOLIO CREDIT RISK 25
, 2007
"... We value synthetic CDO tranche spreads, index CDS spreads, k thtodefault swap spreads and tranchelets in an intensitybased credit risk model with default contagion. The default dependence is modelled by letting individual intensities jump when other defaults occur. The model is reinterpreted as ..."
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Cited by 17 (5 self)
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We value synthetic CDO tranche spreads, index CDS spreads, k thtodefault swap spreads and tranchelets in an intensitybased credit risk model with default contagion. The default dependence is modelled by letting individual intensities jump when other defaults occur. The model is reinterpreted as a Markov jump process. This allows us to use a matrixanalytic approach to derive computationally tractable closedform expressions for the credit derivatives that we want to study. Special attention is given to homogenous portfolios. For a fixed maturity of five years, such a portfolio is calibrated against CDO tranche spreads, index CDS spread and the average CDS spread, all taken from the iTraxx Europe series. After the calibration, which renders perfect fits, we compute spreads for tranchelets and k thtodefault swap spreads for different subportfolios of the main portfolio. Studies of the implied tranchelosses and the implied loss distribution in the calibrated portfolios are also performed. We implement two different numerical methods for determining the distribution of the Markovprocess. These are applied in separate calibrations in order to verify that the matrixanalytic method is independent of the numerical approach used to find the law of the process. Monte Carlo simulations are also performed to check the correctness of the numerical implementations.
Dynamic hedging of synthetic CDO tranches with spread risk and default contagion
, 2007
"... We study the hedging of synthetic CDO tranches in a dynamic portfolio credit risk model which incorporates spread risk and default contagion. The model is constructed and studied via Markovchain techniques. We discuss the immunization of a CDO tranche against spread and event risk in the Markovch ..."
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Cited by 16 (6 self)
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We study the hedging of synthetic CDO tranches in a dynamic portfolio credit risk model which incorporates spread risk and default contagion. The model is constructed and studied via Markovchain techniques. We discuss the immunization of a CDO tranche against spread and event risk in the Markovchain model and compare the results with hedge ratios obtained in the standard Gauss copula model. Moreover, we derive modelbased dynamic hedging strategies using the concept of risk minimization. Numerical experiments are used to illustrate some of the properties of the riskminimizing hedging strategies.
Multiscale intensity models and name grouping for valuation of multiname credit derivatives
 Appl. Math. Finance
"... Abstract. The pricing of collateralized debt obligations and other basket credit derivatives is contingent upon (i) a realistic modeling of the firms ’ default times and the correlation between them, and (ii) efficient computational methods for computing the portfolio loss distribution from the indi ..."
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Cited by 15 (4 self)
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Abstract. The pricing of collateralized debt obligations and other basket credit derivatives is contingent upon (i) a realistic modeling of the firms ’ default times and the correlation between them, and (ii) efficient computational methods for computing the portfolio loss distribution from the individual firms ’ default time distributions. Factor models, a widelyused class of pricing models, are computationally tractable despite the large dimension of the pricing problem, thus satisfying issue (ii), but to have any hope of calibrating CDO data, numerically intense versions of these models are required. We revisit the intensitybased modeling setup for basket credit derivatives and, with the aforementioned issues in mind, we propose improvements (a) via incorporating fast meanreverting stochastic volatility in the default intensity processes, and (b) by considering homogeneous groups within the original set of firms. This can be thought of as a hybrid of topdown and bottomup approaches. We present a calibration example, including data in the midst of the 2008 financial credit crisis, and discuss the relative performance of the framework. 1.
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 14 (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.
Hedging Default Risks of CDOs in Markovian Contagion Models
 ISFA ACTUARIAL SCHOOL, UNIVERSITÉ DE LYON
, 2008
"... We describe a hedging strategy of CDO tranches based upon dynamic trading of the corresponding credit default swap index. We rely upon a homogeneous Markovian contagion framework, where only single defaults occur. In our framework, a CDO tranche can be perfectly replicated by dynamically trading the ..."
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Cited by 12 (2 self)
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We describe a hedging strategy of CDO tranches based upon dynamic trading of the corresponding credit default swap index. We rely upon a homogeneous Markovian contagion framework, where only single defaults occur. In our framework, a CDO tranche can be perfectly replicated by dynamically trading the credit default swap index and a riskfree asset. Default intensities of the names only depend upon the number of defaults and are calibrated onto an input loss surface. Numerical implementation can be carried out fairly easily thanks to a recombining tree describing the dynamics of the aggregate loss. Both continuous time market and its discrete approximation are complete. The computed credit deltas can be seen as a credit default hedge and may also be used as a benchmark to be compared with the market credit deltas. Though the model is quite simple, it provides some meaningful results which are discussed in detail. We study the robustness of the hedging strategies with respect to recovery rate and examine how input loss distributions drive the credit deltas. Using market inputs, we find that the deltas of the equity tranche are lower than those computed in the standard base correlation framework. This is related to the dynamics of dependence between defaults. We can think of our model as a “sticky implied tree” while the hedge ratios computed by market participants correspond to “sticky
2009, Pricing Credit Derivatives under Incomplete Information: a NonlinearFiltering Approach
"... Abstract This paper considers a general reduced form pricing model for credit derivatives where default intensities are driven by some factor process X. The process X is not directly observable for investors in secondary markets; rather, their information set consists of the default history and of n ..."
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Cited by 8 (5 self)
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Abstract This paper considers a general reduced form pricing model for credit derivatives where default intensities are driven by some factor process X. The process X is not directly observable for investors in secondary markets; rather, their information set consists of the default history and of noisy price observation for traded credit products. In this context the pricing of credit derivatives leads to a challenging nonlinear filtering problem. We provide recursive updating rules for the filter, derive a finite dimensional filter for the case where X follows a finite state Markov chain and propose a novel particle filtering algorithm. A numerical case study illustrates the properties of the proposed algorithms.
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. The stochastic recovery specification only models the first two moments of the spot recovery rate as the higher moments of the recovery rate have almos ..."
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Cited by 5 (2 self)
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This paper describes a flexible and tractable bottomup dynamic correlation modelling framework with a consistent stochastic recovery specification. The stochastic recovery specification only models the first two moments of the spot recovery rate as the higher moments of the recovery rate have almost no contribution to the loss distribution and CDO tranche pricing. Observing that only the joint distribution of default indicators is needed to build the portfolio loss distribution, we argue that the default indicator copula should be used instead of the default time copula for the purpose of CDO tranche calibration and pricing. We then defined a generic class of default indicator copula with the “time locality ” property, which makes it easy to calibrate to index tranche prices across multiple maturities. This correlation modelling framework has the unique advantage that the joint distribution of default time and other dynamic properties of the model can be changed independently from the loss distribution and tranche prices. After calibrating the model to index tranche prices, existing topdown methods can be applied to the common factor process to construct very flexible systemic dynamics without changing the already calibrated tranche prices. This modelling framework therefore combines the best features of the bottomup and topdown models: it is fully consistent with all the single name market information
A simple dynamic model for pricing and hedging heterogenous CDOs
, 2008
"... We present a simple bottomup dynamic credit model that can be calibrated simultaneously to the market quotes on CDO tranches and individual CDSs constituting the credit portfolio. The model is most suitable for the purpose of evaluating the hedge ratios of CDO tranches with respect to the underlyin ..."
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Cited by 5 (0 self)
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We present a simple bottomup dynamic credit model that can be calibrated simultaneously to the market quotes on CDO tranches and individual CDSs constituting the credit portfolio. The model is most suitable for the purpose of evaluating the hedge ratios of CDO tranches with respect to the underlying credit names. Default intensities of individual assets are modeled as deterministic functions of time and the total number of defaults accumulated in the portfolio. To overcome numerical difficulties, we suggest a semianalytic approximation that is justified by the large number of portfolio members. We calibrate the model to the recent market quotes on CDO tranches and individual CDSs and find the hedge ratios of tranches. Results are compared with those obtained within the static Gaussian Copula model.
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 5 (5 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
Advanced credit portfolio modeling and CDO pricing
"... Credit risk represents by far the biggest risk in the activities of a traditional bank. In particular, during recession periods financial institutions loose enormous amounts as a consequence of bad loans and default events. Traditionally the risk arising from a loan contract could not be transferred ..."
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Cited by 1 (0 self)
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Credit risk represents by far the biggest risk in the activities of a traditional bank. In particular, during recession periods financial institutions loose enormous amounts as a consequence of bad loans and default events. Traditionally the risk arising from a loan contract could not be transferred and remained