| 1. |
- Bielecki, Tomasz R., et al.
(författare)
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A Bottom-Up Dynamic Model of Portfolio Credit Risk. Part I: Markov Copula Perspective
- 2014
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Ingår i: Recent Advances in Financial Engineering 2012. - New Jersey London Hong Kong : World Scientific. - 9789814571630 ; , s. 25-49
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Bokkapitel (refereegranskat)abstract
- We consider a bottom-up Markovian copula model of portfolio credit risk where instantaneous contagion is possible in the form of simultaneous defaults. Due to the Markovian copula nature of the model, calibration of marginals and dependence parameters can be performed separately using a two-steps procedure, much like in a standard static copula set-up. In this sense this model solves the bottom-up top-down puzzle which the CDO industry had been trying to do for a long time. It can be applied to any dynamic credit issue like consistent valuation and hedging of CDSs, CDOs and counterparty risk on credit portfolios.
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| 2. |
- Bielecki, Tomasz R., et al.
(författare)
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A Bottom-Up Dynamic Model of Portfolio Credit Risk. Part II: Common-Shock Interpretation, Calibration and Hedging Issues
- 2014
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Ingår i: Recent Advances in Financial Engineering 2012. - New Jersey London Hong Kong : World Scientific. - 9789814571630 ; , s. 51-73
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Bokkapitel (refereegranskat)abstract
- In this paper, we prove that the conditional dependence structure of default times in the Markov model of [4] belongs to the class of Marshall- Olkin copulas. This allows us to derive a factor representation in terms of “common-shocks”, the latter beeing able to trigger simultaneous defaults in some pre-specified groups of obligors. This representation depends on the current default state of the credit portfolio so that fast convolution pricing schemes can be exploited for pricing and hedging credit portfolio derivatives. As emphasized in [4], the innovative breakthrough of this dynamic bottom-up model is a suitable decoupling property between the dependence structure and the default marginals as in [10] (like in static copula models but here in a full-flesh dynamic “Markov copula” model). Given the fast deterministic pricing schemes of the present paper, the model can then be jointly calibrated to single-name and portfolio data in two steps, as opposed to a global joint optimization procedures involving all the model parameters at the same time which would be untractable numerically. We illustrate this numerically by results of calibration against market data from CDO tranches as well as individual CDS spreads. We also discuss hedging sensitivities computed in the models thus calibrated.
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| 3. |
- Bielecki, Tomasz R., et al.
(författare)
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A Bottom-Up Dynamic Model of Portfolio Credit Risk with Stochastic Intensities and Random Recoveries
- 2014
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Ingår i: Communications in Statistics - Theory and Methods. - : Informa UK Limited. - 0361-0926 .- 1532-415X. ; 43:7, s. 1362-1389
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Tidskriftsartikel (refereegranskat)abstract
- In Bielecki et al. (2014a), the authors introduced a Markov copula model of portfolio credit risk where pricing and hedging can be done in a sound theoretical and practical way. Further theoretical backgrounds and practical details are developed in Bielecki et al. (2014b,c) where numerical illustrations assumed deterministic intensities and constant recoveries. In the present paper, we show how to incorporate stochastic default intensities and random recoveries in the bottom-up modeling framework of Bielecki et al. (2014a) while preserving numerical tractability. These two features are of primary importance for applications like CVA computations on credit derivatives (Assefa et al., 2011; Bielecki et al., 2012), as CVA is sensitive to the stochastic nature of credit spreads and random recoveries allow to achieve satisfactory calibration even for “badly behaved” data sets. This article is thus a complement to Bielecki et al. (2014a), Bielecki et al. (2014b) and Bielecki et al. (2014c).
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| 4. |
- Bielecki, Tomasz R., et al.
(författare)
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A Markov Copula Model of Portfolio Credit Risk with Stochastic Intensities and Random Recoveries
- 2012
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- In [4], the authors introduced a Markov copula model of portfolio credit risk. This model solves the top-down versus bottom-up puzzle in achieving efficient joint calibration to single-name CDS and to multi-name CDO tranches data. In [4], we studied a general model, that allows for stochastic default intensities and for random recoveries, and we conducted empirical study of our model using both deterministic and stochastic default intensities, as well as deterministic and random recoveries only. Since, in case of some “badly behaved” data sets a satisfactory calibration accuracy can only be achieved through the use of random recoveries, and, since for important applications, such as CVA computations for credit derivatives, the use of stochastic intensities is advocated by practitioners, efficient implementation of our model that would account for these two issues is very important. However, the details behind the implementation of the loss distribution in the case with random recoveries were not provided in [4]. Neither were the details on the stochastic default intensities given there. This paper is thus a complement to [4], with a focus on a detailed description of the methodology that we used so to implement these two model features: random recoveries and stochastic intensities.
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| 5. |
- Bielecki, Tomasz R., et al.
(författare)
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Dynamic Hedging of Portfolio Credit Risk in a Markov Copula Model
- 2014
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Ingår i: Journal of Optimization Theory and Applications. - : Springer Science and Business Media LLC. - 0022-3239 .- 1573-2878. ; 161:1, s. 90-102
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Tidskriftsartikel (refereegranskat)abstract
- We devise a bottom-up dynamic model of portfolio credit risk where instantaneous contagion is represented by the possibility of simultaneous defaults. Due to a Markovian copula nature of the model, calibration of marginals and dependence parameters can be performed separately using a two-step procedure, much like in a standard static copula setup. In this sense this solves the bottom-up top-down puzzle which the CDO industry had been trying to do for a long time. This model can be used for any dynamic portfolio credit risk issue, such as dynamic hedging of CDOs by CDSs, or CVA computations on credit portfolios.
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| 6. |
- Bielecki, T.R., et al.
(författare)
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Dynamic Modeling of Portfolio Credit Risk with Common Shocks
- 2011
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- We consider a bottom-up Markovian model of portfolio credit risk where dependence among credit names stems from the possibility of simultaneous defaults. A common shocks interpretation of the model is possible so that efficient convolution recursion procedures are available for pricing and hedging CDO tranches, conditionally on any given state of the Markov model. Calibration of marginals and dependence parameters can be performed separately using a two-steps procedure, much like in a standard static copula set-up. As a result this model allows us to hedge CDO tranches using single- name CDS-s in a theoretically sound and practically convenient way. To illustrate this we calibrate the model against market data on CDO tranches and the underlying single- name CDS-s. We then study the loss distributions as well as the min-variance hedging strategies in the calibrated portfolios.
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| 7. |
- Bielecki, T.R., et al.
(författare)
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In search of a grand unifying theory
- 2013
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Ingår i: Creditflux Newsletter. - 1475-0716. ; :July 2013
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Tidskriftsartikel (övrigt vetenskapligt/konstnärligt)
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| 8. |
- Herbertsson, Alexander, 1977
(författare)
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CDS index options in Markov chain models
- 2019
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- We study CDS index options in a credit risk model where the defaults times have intensities which are driven by a finite-state Markov chain representing the underlying economy. In this setting we derive compact computationally tractable formulas for the CDS index spread and the price of a CDS index option. In particular, the evaluation of the CDS index option is handled by translating the Cox-framework into a bivariate Markov chain. Due to the potentially very large, but extremely sparse matrices obtained in this reformulating, special treatment is needed to efficiently compute the matrix exponential arising from the Kolmogorov Equation. We provide details of these computational methods as well as numerical results. The finite-state Markov chain model is calibrated to data with perfect fits, and several numerical studies are performed. In particular we show that under same exogenous circumstances, the CDS index options prices in the Markov chain framework can be close to or sometimes larger than prices in models which assume that the CDS index spreads follows a log-normal process. We also study the different default risk components in the option prices generated by the Markov model, an investigation which is difficult to do in models where the CDS index spreads follows a log-normal process.
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| 9. |
- Herbertsson, Alexander, 1977, et al.
(författare)
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CDS INDEX OPTIONS UNDER INCOMPLETE INFORMATION
- 2016
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- We derive practical formulas for CDS index spreads in a credit risk model under incomplete information. The factor process driving the default intensities is not directly observable, and the filtering model of Frey & Schmidt (2012) is used as our setup. In this framework we find a computationally tractable expressions for the payoff of a CDS index option which naturally includes the so-called armageddon correction. A lower bound for the price of the CDS index option is derived and we provide explicit conditions on the strike spread for which this inequality becomes an equality. The bound is computationally feasible and do not depend the noise parameters in the filtering model. We outline how to explicitly compute the quantities involved in the lower bound for the price of the credit index option as well as implement and calibrate this model to market data. A numerical study is performed where we show that the lower bound in our model can be several hundred percent bigger compared with models which assume that the CDS index spreads follows a log-normal process. Also a systematic study is performed in order to understand the impact of various model parameters on CDS index options (and on the index itself).
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| 10. |
- Herbertsson, Alexander, 1977
(författare)
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Default Contagion in Large Homogeneous Portfolios
- 2007
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- We study default contagion in large homogeneous credit portfolios. Using data from the iTraxx Europe series, two synthetic CDO portfolios are calibrated against their tranche spreads, index CDS spreads and average CDS spreads, all with five year maturity. After the calibrations, which render perfect fits, we investigate the implied expected ordered defaults times, implied default correlations, and implied multivariate default and survival distributions, both for ordered and unordered default times. Many of the numerical results differ substantially from the corresponding quantities in a smaller inhomogeneous CDS portfolio. Furthermore, the studies indicate that market CDO spreads imply extreme default clustering in upper tranches. The default contagion is introduced by letting individual intensities jump when other defaults occur, but be constant between defaults. The model is translated into a Markov jump process. Expressions for the investigated quantities are derived by using matrix-analytic methods.
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