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Forwards and European options on CDO tranches. Working paper
, 2006
"... Now that the market for cash and synthetic CDOs is well established, there is increased interest in trading forward contracts and options on CDO tranches. This article develops models for valuing these instruments. The model for valuing European options on CDO tranches has similarities to the standa ..."
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Now that the market for cash and synthetic CDOs is well established, there is increased interest in trading forward contracts and options on CDO tranches. This article develops models for valuing these instruments. The model for valuing European options on CDO tranches has similarities to the standard market model for valuing European swap options and to the model for valuing options on credit default swaps. Once default probabilities, the expected recovery rates and the degree to which defaults tend to cluster have been estimated, it enables traders to calculate option prices from CDO tranche swap spread volatilities and vice versa. 2 FORWARDS AND EUROPEAN OPTIONS ON CDO TRANCHES A credit default swap (CDS) provides protection against default by some particular entity. During the life of the contract the buyer of protection makes periodic payments at some rate, s, times the notional until default or maturity whichever comes first. Typically these payments are made quarterly in arrears. In the event of a default the seller of protection
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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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.
Valuing CDOs of Bespoke Portfolios with Implied MultiFactor Models
, 2007
"... This paper presents a robust and practical CDO valuation framework based on the application of multifactor credit models in conjunction with weighted Monte Carlo techniques used in options pricing. The framework produces arbitragefree prices and can be used to value consistently CDOs of bespoke po ..."
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Cited by 4 (0 self)
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This paper presents a robust and practical CDO valuation framework based on the application of multifactor credit models in conjunction with weighted Monte Carlo techniques used in options pricing. The framework produces arbitragefree prices and can be used to value consistently CDOs of bespoke portfolios, CDOsquared and cash CDOs. Multifactor models allow us to model systematically heterogeneous portfolios, sector and geographical concentrations, deals which refer simultaneously to multiple indices and potentially other risk factors such as recoveries and prepayments. We demonstrate the practical advantages of working through multifactor models, rather than directly on a common hazard rate (or a set of them). The multifactor credit models are defined generally within the mathematical construction of Generalized Linear Mixed Models (GLMMs). The implied copula approach can be seen as a special case of a GLMM, as are other common credit portfolio models. For a given model, the quoted prices of various credit portfolio instruments, such
CDO valuation: term structure, tranche structure, and loss distributions, working paper
"... the Institute LaueLangevin, where some of this work was carried out, for their hospitality; and Julien Houdain and Fortis Investments for providing the market prices used in the examples of this article. 4The support of the Natural Sciences and Engineering Research Council of Canada is acknowledged ..."
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the Institute LaueLangevin, where some of this work was carried out, for their hospitality; and Julien Houdain and Fortis Investments for providing the market prices used in the examples of this article. 4The support of the Natural Sciences and Engineering Research Council of Canada is acknowledged. This article describes a new approach to the riskneutral valuation of CDO tranches, based on a general specification of the loss distribution, and the expected loss at zero default, for the reference portfolio. The new approach can describe tranche termstructures, 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. Also, given a set of arbitragefree market prices, arbitragefree interpolated term structures (plots of tranche price versus maturity for a given tranche) and tranche structures (plots of tranche price versus tranche for a given maturity), as well as implied loss distributions, can be obtained, allowing bespoke tranches to be priced. The marking to market of tranche prices, and the establishment of forwardstart premiums for the index are discussed. An efficient linear programming approach to valuation, essential to the implementation, is also described. The article also makes use of a new, simple yet general, approach to the problem of unequal notionals, and of random, timedependent, riskneutral, recovery rates.
Valuation of Forward Starting CDOs
, 2007
"... A forward starting CDO is a single tranche CDO with a specified premium starting at a specified future time. Pricing and hedging forward starting CDOs has become an active research topic. We present a method for pricing a forward starting CDO by converting it to an equivalent synthetic CDO. The valu ..."
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A forward starting CDO is a single tranche CDO with a specified premium starting at a specified future time. Pricing and hedging forward starting CDOs has become an active research topic. We present a method for pricing a forward starting CDO by converting it to an equivalent synthetic CDO. The value of the forward starting CDO can then be computed by the well developed methods for pricing the equivalent synthetic one. We illustrate our method using the industrystandard Gaussianfactorcopula model. Numerical results demonstrate the accuracy and efficiency of our method.
Lévy Density Based Intensity Modeling of the Correlation Smile
, 2008
"... The jump distribution for the default intensities in a reduced form framework is modeled and calibrated to provide reasonable fits to CDX.NA.IG and iTraxx Europe CDOs, to 5, 7 and 10 year maturities simultaneously. Calibration is carried out using an efficient Monte Carlo simulation algorithm that a ..."
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The jump distribution for the default intensities in a reduced form framework is modeled and calibrated to provide reasonable fits to CDX.NA.IG and iTraxx Europe CDOs, to 5, 7 and 10 year maturities simultaneously. Calibration is carried out using an efficient Monte Carlo simulation algorithm that appears to be suitable for both homogeneous and heterogeneous collections of credit names. The underlying jump process is found to relate closely to a maximally skewed stable Lévy process with index of stability α ∼ 1.5. The market standard for pricing credit derivatives sensitive to default dependency is based on the Gaussian copula. As is wellknown, this method is inadequate to price nonstandard products. The model as such is not able to explain the correlation smile. Better models addressing these issues have been developed by various authors. Some recent work in this direction in a reduced form framework involves modeling the default intensities as in Joshi and Stacey [2005], Chapovsky, Rennie and Tavares [2006], Errais, Giesecke and Goldberg [2006], Balakrishna [2007]; modeling dependency with simultaneous defaults as in
Contents
, 2008
"... We propose a bottomup dynamic credit modelling framework. To achieve a nontrivial coupling, the marginal survival probability processes are multiplied by a common exponential martingale process. Still being a factor coupling, this approach relies on convolution of the conditionally independent ran ..."
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We propose a bottomup dynamic credit modelling framework. To achieve a nontrivial coupling, the marginal survival probability processes are multiplied by a common exponential martingale process. Still being a factor coupling, this approach relies on convolution of the conditionally independent random variables. However, due to the much better analytical tractability, this approach allow getting rid of the traditional recursions as convolution methods, and it does not require tuning the factor quadrature, as the factor integration step is not present. Also the model can be entirely speci…ed only in terms of the local moment surface of the common factor process, with di¤erent moments a¤ecting di¤erent segments of the loss distribution.
Analysis of the Impact of Contagion Flow on High Yield Bond Portfolio
, 2011
"... Portfolios are constructed to increase returns and manage risk. In highrisk investment strategies, central measures of risk must be complemented with tail measures of risk. An unanticipated event impacting securities of one firm can contagiously affect those of other firms through a contagion flow ..."
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Portfolios are constructed to increase returns and manage risk. In highrisk investment strategies, central measures of risk must be complemented with tail measures of risk. An unanticipated event impacting securities of one firm can contagiously affect those of other firms through a contagion flow process. The connections between firms due to a variety of factors can spread the contagion, and potentially impact firms in a network. This can adversely affect the level of tail risk in an investment strategy, especially when a number of these connected firms are included in a portfolio. A model is developed for flow of contagion between firms which will define, characterize and calibrate a contagions impact on default risk of a portfolio of debt instruments. The model assesses the impact of network structure underlying contagion flow and evaluates the contagion related excess risk in a portfolio of highyield debt instruments. 1
Sato Processes in Default Modelling
, 2008
"... Classically, in reduced form default models the instantaneous default intensity is the modelling object and survival probabilities are given by the Laplace transform of At = R t 0 sds. Instead, recent literature has shown a tendency towards specifying the process A directly. We will refer to A as th ..."
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Classically, in reduced form default models the instantaneous default intensity is the modelling object and survival probabilities are given by the Laplace transform of At = R t 0 sds. Instead, recent literature has shown a tendency towards specifying the process A directly. We will refer to A as the cumulative hazard process. We present a new cumulative hazard based framework where survival probabilities are still obtained in closed form but where A belongs to the class of selfsimilar additive processes also termed Sato processes. We analyze two speci…cations for the cumulative hazard process; SatoGamma and SatoIG processes where the unit time distribution A1 is described by a Gamma law and Inverse Gaussian law respectively. The models are calibrated to data on the single names included in the iTraxx Europe index and compared with two OrnsteinUhlenbeck type intensity models. It is shown how the Sato models achieve similar calibration errors with fever parameters, and with more stable parameter estimates in time.
Pricing Portfolio Credit Derivatives Using a Simplified Dynamic Model
, 2008
"... This thesis investigates dynamic methods for pricing portfolio credit derivatives, especially the standardized market for CDO: iTraxx Europe index. Compared with previous static models, i.e., the copula functions, the dynamic models are applicable to much more exotic portfolio credit derivatives. Th ..."
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This thesis investigates dynamic methods for pricing portfolio credit derivatives, especially the standardized market for CDO: iTraxx Europe index. Compared with previous static models, i.e., the copula functions, the dynamic models are applicable to much more exotic portfolio credit derivatives. This thesis uses the concept of the dynamic model from Hull and White (2007). But we modify it by adjusting some parameters. We also find a better way for calibration to give the model more economic sense. The iTraxx Europe index can also be valued analytically using our model. Besides the analytic method, we consider the binomial tree and Monte Carlo method to make pricing more flexible. Finally, the revised dynamic model captures the advantages of the original one and also provides a good fit to CDO quotes.