| 1. |
- Bielecki, Tomasz R., et al.
(author)
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A Markov Copula Model of Portfolio Credit Risk with Stochastic Intensities and Random Recoveries
- 2012
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Reports (other academic/artistic)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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| 2. |
- Bielecki, T.R., et al.
(author)
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Dynamic Modeling of Portfolio Credit Risk with Common Shocks
- 2011
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Reports (other academic/artistic)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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| 3. |
- Herbertsson, Alexander, 1977
(author)
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CDS index options in Markov chain models
- 2019
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Reports (other academic/artistic)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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| 4. |
- Herbertsson, Alexander, 1977, et al.
(author)
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CDS INDEX OPTIONS UNDER INCOMPLETE INFORMATION
- 2016
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Reports (other academic/artistic)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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| 5. |
- Herbertsson, Alexander, 1977
(author)
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Default Contagion in Large Homogeneous Portfolios
- 2007
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Reports (other academic/artistic)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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| 6. |
- Herbertsson, Alexander, 1977
(author)
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Modelling Default Contagion Using Multivariate Phase-Type Distributions
- 2007
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Reports (other academic/artistic)abstract
- We model dynamic credit portfolio dependence by using default contagion in an intensity-based framework. Two different portfolios (with 10 obligors), one in the European auto sector, the other in the European financial sector, are calibrated against their market CDS spreads and the corresponding CDS-correlations. After the calibration, which are perfect for the banking portfolio, and good for the auto case, we study several quantities of importance in active credit portfolio management. For example, implied multivariate default and survival distributions, multivariate conditional survival distributions, implied default correlations, expected default times and expected ordered defaults times. The default contagion is modelled by letting individual intensities jump when other defaults occur, but be constant between defaults. This model is translated into a Markov jump process, a so called multivariate phase-type distribution, which represents the default status in the credit portfolio. Matrix-analytic methods are then used to derive expressions for the quantities studied in the calibrated portfolios.
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| 7. |
- Herbertsson, Alexander, 1977, et al.
(author)
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Pricing basket default swaps in a tractable shot-noise model
- 2009
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Reports (other academic/artistic)abstract
- We value CDS spreads and kth-to-default swap spreads in a tractable shot noise model. The default dependence is modelled by letting the individual jumps of the default intensity be driven by a common latent factor. The arrival of the jumps is driven by a Poisson process. By using conditional independence and properties of the shot noise processes we derive tractable closed-form expressions for the default distribution and the ordered survival distributions in a homogeneous portfolio. These quantities are then used to price and study CDS spreads and kth-to-default swap spreads as function of the model parameters. We study the kth-to-default spreads as function of the CDS spread, as well as other parameters in the model. All calibrations lead to perfect fits.
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| 8. |
- Herbertsson, Alexander, 1977
(author)
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Pricing Synthetic CDO Tranches in a Model with Default Contagion Using the Matrix-Analytic Approach
- 2007
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Reports (other academic/artistic)abstract
- We value synthetic CDO tranche spreads, index CDS spreads, kth-to-default swap spreads and tranchelets in an intensity-based 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 allow us to use a matrix-analytic approach to derive computationally tractable closed-form 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 and FtD spreads, all taken from the iTraxx Europe series. After the calibration, which render perfect fits, we compute spreads for tranchelets and kth-to-default swap spreads for different subportfolios of the main portfolio. We also investigate implied tranche-losses and the implied loss distribution in the calibrated portfolios.
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| 9. |
- Herbertsson, Alexander, 1977
(author)
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Risk management of stock portfolios with jumps at exogenous default events
- 2023
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Other publication (other academic/artistic)abstract
- In this paper we study equity risk management of stock portfolios where the individual stock prices have downward jumps at the defaults of an exogenous group of defaultable entities. The default times can come from any type of credit portfolio model. In this setting we derive computational tractable formulas for several stock-related quantizes, such as loss distributions of equity portfolios and apply it to Value-at-Risk computations. We start with individual stock prices and then extend the setting to a portfolio framework. In the portfolio case our studies considers both small-time expansions of the loss-distribution for a heterogeneous portfolio via a linearization of the loss, but also for general time points when the stock portfolio is large and homogeneous and where we use a conditional version of the law of large numbers. Most of the derived formulas will heavily rely on the ability to efficiently compute the number of defaults distribution of the entities in the exogenous group of corporates negative affecting the stock prices in our equity portfolio. If the stock prices are unaffected by the exogenous defaults then our framework collapses into the traditional Black-Scholes model under the real probability measure. Finally, we give several numerical applications. For example, in a setting where the jumps in the stock prices are at default times which are generated by a one-factor Gaussian copula model, we study the time evolution of Value-at-Risk (i.e. VaR as function of time) for stock portfolios, both for a 20-day period and for a two-year period. We also perform similar numerical VaR-studies in a setting where the individual default intensities follow a CIR process. Our results are compared with the corresponding VaR-values in the Black-Scholes case with same drift and volatilises as in the jump models. Not surprisingly, we show that the VaR-values in stock portfolios with downward jumps at defaults of external entities, will have substantially higher VaR-values compared to the corresponding Black-Scholes cases. The numerical computations of the number of default distribution will in all our studies use fast and efficient saddlepoint methods.
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| 10. |
- Herbertsson, Alexander, 1977
(author)
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Saddlepoint approximations for credit portfolio distributions with applications in equity risk management
- 2023
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Other publication (other academic/artistic)abstract
- We study saddlepoint approximations to the tail-distribution for credit portfolio losses in continuous time intensity based models under conditional independent homogeneous settings. In such models, conditional on the filtration generated by the individual default intensity up to time t, the conditional number of defaults distribution (in the portfolio) will be a binomial distribution that is a function of a factor Z_t which typically is the integrated default intensity up to time t. This will lead to an explicit closed-form solution of the saddlepoint equation for each point used in the number of defaults distribution when conditioning on the factor Z_t, and we hence do not have to solve the saddlepoint equation numerically. The ordo-complexity of our algorithm computing the whole distribution for the number of defaults will be linear in the portfolio size, which is a dramatic improvement compared to e.g. recursive methods which have a quadratic ordo-complexity in the portfolio size. The individual default intensities can be arbitrary as long as they are conditionally independent given the factor Z_t in a homogeneous portfolio. We also outline how our method for computing the number of defaults distribution can be extend to heterogeneous portfolios. Furthermore, we show that all our results can be extended to hold for any factor copula model. We give several numerical applications and in particular, in a setting where the individual default intensities follow a CIR process we study both the tail distribution and the number of defaults distribution. We then repeat similar numerical studies in a one-factor Gaussian copula model. We also numerically benchmark our saddlepoint method to other computational methods. Finally, we apply of our saddlepoint method to efficiently investigate Value-at-Risk for equity portfolios where the individual stock prices have simultaneous downward jumps at the defaults of an exogenous group of defaultable entities driven by a one-factor Gaussian copula model were we focus on Value-at-Risk as function of the default correlation parameter in the one-factor Gaussian copula model.
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