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Levy Processes in Credit Risk

Levy Processes in Credit Risk

Wim Schoutens, Jessica Cariboni

ISBN: 978-0-470-68506-8

Jun 2010

200 pages



This book is an introductory guide to using Lévy processes for credit risk modelling. It covers all types of credit derivatives: from the single name vanillas such as Credit Default Swaps (CDSs) right through to structured credit risk products such as Collateralized Debt Obligations (CDOs), Constant Proportion Portfolio Insurances (CPPIs) and Constant Proportion Debt Obligations (CPDOs) as well as new advanced rating models for Asset Backed Securities (ABSs).

Jumps and extreme events are crucial stylized features, essential in the modelling of the very volatile credit markets - the recent turmoil in the credit markets has once again illustrated the need for more refined models.

Readers will learn how the classical models (driven by Brownian motions and Black-Scholes settings) can be significantly improved by using the more flexible class of Lévy processes. By doing this, extreme event and jumps can be introduced into the models to give more reliable pricing and a better assessment of the risks.

The book brings in high-tech financial engineering models for the detailed modelling of credit risk instruments, setting up the theoretical framework behind the application of Lévy Processes to Credit Risk Modelling before moving on to the practical implementation. Complex credit derivatives structures such as CDOs, ABSs, CPPIs, CPDOs are analysed and illustrated with market data.




1 An Introduction to Credit Risk.

1.1 Credit Risk.

1.1.1 Historical and Risk-Neutral Probabilities.

1.1.2 Bond Prices and Default Probability.

1.2 Credit Risk Modelling.

1.3 Credit Derivatives.

1.4 Modelling Assumptions.

1.4.1 Probability Space and Filtrations.

1.4.2 The Risk-Free Asset.

2 An Introduction to Lévy Processes.

2.1 Brownian Motion.

2.2 Lévy Processes.

2.3 Examples of Lévy Processes.

2.3.1 Poisson Process.

2.3.2 Compound Poisson Process.

2.3.3 The Gamma Process.

2.3.4 Inverse Gaussian Process.

2.3.5 The CMY Process.

2.3.6 The Variance Gamma Process.

2.4 Ornstein–Uhlenbeck Processes.

2.4.1 The Gamma-OU Process.

2.4.2 The Inverse Gaussian-OU Process.


3 Single-Name Credit Derivatives.

3.1 Credit Default Swaps.

3.1.1 Credit Default Swaps Pricing.

3.1.2 Calibration Assumptions.

3.2 Credit Default Swap Forwards.

3.2.1 Credit Default Swap Forward Pricing.

3.3 Constant Maturity Credit Default Swaps.

3.3.1 Constant Maturity Credit Default Swaps Pricing.

3.4 Options on CDS.

4 Firm-Value Lévy Models.

4.1 The Merton Model.

4.2 The Black–Cox Model with Constant Barrier.

4.3 The Lévy First-Passage Model.

4.4 The Variance Gamma Model.

4.4.1 Sensitivity to the Parameters.

4.4.2 Calibration on CDS Term Structure Curve.

4.5 One-Sided Lévy Default Model.

4.5.1 Wiener–Hopf Factorization and Default Probabilities.

4.5.2 Illustration of the Pricing of Credit Default Swaps.

4.6 Dynamic Spread Generator.

4.6.1 Generating Spread Paths.

4.6.2 Pricing of Options on CDSs.

4.6.3 Black’s Formulas and Implied Volatility.

Appendix: Solution of the PDIE.

5 IntensityLévy Models.

5.1 Intensity Models for Credit Risk.

5.1.1 Jarrow–Turnbull Model.

5.1.2 Cox Models.

5.2 The Intensity-OU Model.

5.3 Calibration of the Model on CDS Term Structures.


6 Multivariate Credit Products.

6.1 CDOs.

6.2 Credit Indices.

7 Collateralized Debt Obligations.

7.1 Introduction.

7.2 The Gaussian One-Factor Model.

7.3 Generic One-Factor Lévy Model.

7.4 Examples of Lévy Models.

7.5 Lévy Base Correlation.

7.5.1 The Concept of Base Correlation.

7.5.2 Pricing Non-Standard Tranches.

7.5.3 Correlation Mapping for Bespoke CDOs.

7.6 Delta-Hedging CDO tranches.

7.6.1 Hedging with the CDS Index.

7.6.2 Delta-Hedging with a Single-Name CDS.

7.6.3 Mezz-Equity hedging.

8 Multivariate Index Modelling.

8.1 Black’s Model.

8.2 VG Credit Spread Model.

8.3 Pricing Swaptions using FFT.

8.4 Multivariate VG Model.


9 Credit CPPIs and CPDOs.

9.1 Introduction.

9.2 CPPIs.

9.3 Gap Risk.

9.4 CPDOs.

10 Asset-Backed Securities.

10.1 Introduction.

10.2 Default Models.

10.2.1 Generalized Logistic Default Model.

10.2.2 Lévy Portfolio Default Model.

10.2.3 Normal One-Factor Default Model.

10.2.4 Generic One-Factor Lévy Default Model.

10.3 Prepayment Models.

10.3.1 Constant Prepayment Model.

10.3.2 Lévy Portfolio Prepayment Model.

10.3.3 Normal One-Factor Prepayment Model.

10.4 Numerical Results.



""This text introduces into the use of Levy processes in credit risk modeling. After a general overview of credit risk and standard credit derivatives, the authors provide a short introduction into Levy processes in general. This material is then used to study single-name credit derivatives. Following this, the authors introduce into firm-value Levy models, including the Merton model, Black-Cox model, Levy first passage model, variance gamma model and the one sided Levy default model. The problem of calibration is discussed. After that, the authors introduce intensity Levy models such as the Jarrow and Turnbull model, the Cox model and the intensity-OU model. Multivariate credit products, collateralized debt obligations and multivariate index modeling are discussed in the following. In the final part of their book, the authors study credit CPPIs and CPDOs as well as asset-backed securities."" (Zentralblatt MATH, 2010)