Modelling Single-name and Multi-name Credit Derivatives

Contents

Acknowledgements

About the Author

Introduction

Notation

1 The Credit Derivatives Market

1.1 Introduction

1.2 Credit Derivatives Market Size

1.3 Products

1.4 Market Participants

2 Building the Libor Discount Curve

2.1 Introduction

2.2 The Libor Index

2.3 Money Market Deposits

2.4 Forward Rate Agreements

2.5 Interest Rate Futures

2.6 Interest Rate Swaps

2.7 Bootstrapping the Libor Curve

2.8 Summary

2.9 Technical Appendix

PART I SINGLE-NAME CREDIT DERIVATIVES

3 Single-name Credit Modelling

3.1 Introduction

3.2 Observing Default

3.3 Risk-neutral Pricing Framework

3.4 Structural Models of Default

3.5 Reduced Form Models

3.6 The Hazard Rate Model

3.7 Modelling Default as a Cox Process

3.8 A Gaussian Short Rate and Hazard Rate Model

3.9 Independence and Deterministic Hazard Rates

3.10 The Credit Triangle

3.11 The Credit Risk Premium

3.12 Summary

3.13 Technical Appendix

4 Bonds and Asset Swaps

4.1 Introduction

4.2 Fixed Rate Bonds

4.3 Floating Rate Notes

4.4 The Asset Swap

4.5 The Market Asset Swap

4.6 Summary

5 The Credit Default Swap

5.1 Introduction

5.2 The Mechanics of the CDS Contract

5.3 Mechanics of the Premium Leg

5.4 Mechanics of the Protection Leg

5.5 Bonds and the CDS Spread

5.6 The CDS–Cash basis

5.7 Loan CDS

5.8 Summary

6 A Valuation Model for Credit Default Swaps

6.1 Introduction

6.2 Unwinding a CDS Contract

6.3 Requirements of a CDS Pricing Model

6.4 Modelling a CDS Contract

6.5 Valuing the Premium Leg

6.6 Valuing the Protection Leg

6.7 Upfront Credit Default Swaps

6.8 Digital Default Swaps

6.9 Loan CDS

6.10 Summary

7 Calibrating the CDS Survival Curve

7.1 Introduction

7.2 Desirable Curve Properties

7.3 The Bootstrap

7.4 Interpolation Quantities

7.5 Bootstrapping Algorithm

7.6 Behaviour of the Interpolation Scheme

7.7 Detecting Arbitrage in the Curve

7.8 Example CDS Valuation

7.9 Summary

8 CDS Risk Management

8.1 Introduction

8.2 Market Risks of a CDS Position

8.3 Analytical CDS Sensitivities

8.4 Full Hedging of a CDS Contract

8.5 Hedging the CDS Spread Curve Risk

8.6 Hedging the Libor Curve Risk

8.7 Portfolio Level Hedging

8.8 Counterparty Risk

8.9 Summary

9 Forwards, Swaptions and CMDS

9.1 Introduction

9.2 Forward Starting CDS

9.3 The Default Swaption

9.4 Constant Maturity Default Swaps

9.5 Summary

PART II MULTI-NAME CREDIT DERIVATIVES

10 CDS Portfolio Indices

10.1 Introduction

10.2 Mechanics of the Standard Indices

10.3 CDS Portfolio Index Valuation

10.4 The Index Curve

10.5 Calculating the Intrinsic Spread of an Index

10.6 The Portfolio Swap Adjustment

10.7 Asset-backed and Loan CDS Indices

10.8 Summary

11 Options on CDS Portfolio Indices

11.1 Introduction

11.2 Mechanics

11.3 Valuation of an Index Option

11.4 An Arbitrage-free Pricing Model

11.5 Examples of Pricing

11.6 Risk Management

11.7 Black’s Model Revisited

11.8 Summary

12 An Introduction to Correlation Products

12.1 Introduction

12.2 Default Baskets

12.3 Leveraging the Spread Premia

12.4 Collateralised Debt Obligations

12.5 The Single-tranche Synthetic CDO

12.6 CDOs and Correlation

12.7 The Tranche Survival Curve

12.8 The Standard Index Tranches

12.9 Summary

13 The Gaussian Latent Variable Model

13.1 Introduction

13.2 The Model

13.3 The Multi-name Latent Variable Model

13.4 Conditional Independence

13.5 Simulating Multi-name Default

13.6 Default Induced Spread Dynamics

13.7 Calibrating the Correlation

13.8 Summary

14 Modelling Default Times using Copulas

14.1 Introduction

14.2 Definition and Properties of a Copula

14.3 Measuring Dependence

14.4 Rank Correlation

14.5 Tail Dependence

14.6 Some Important Copulae

14.7 Pricing Credit Derivatives from Default Times

14.8 Standard Error of the Breakeven Spread

14.9 Conclusions

14.10 Technical Appendix

15 Pricing Default Baskets

15.1 Introduction

15.2 Modelling First-to-default Baskets

15.3 Second-to-default and Higher Default Baskets

15.4 Pricing Baskets using Monte Carlo

15.5 Pricing Baskets using a Multi-Factor Model

15.6 Pricing Baskets in the Student-t Copula

15.7 Risk Management of Default Baskets

15.8 Summary

16 Pricing Tranches in the Gaussian Copula Model

16.1 Introduction

16.2 The LHP Model

16.3 Drivers of the Tranche Spread

16.4 Accuracy of the LHP Approximation

16.5 The LHP Model with Tail Dependence

16.6 Conclusion

16.7 Technical Appendix

17 Risk Management of Synthetic Tranches

17.1 Introduction

17.2 Systemic Risks

17.3 The LH+ Model

17.4 Idiosyncratic Risks

17.5 Hedging Tranches

17.6 Conclusion

17.7 Technical Appendix

18 Building the Full Loss Distribution

18.1 Introduction

18.2 Calculating the Tranche Survival Curve

18.3 Building the Conditional Loss Distribution

18.4 Integrating over the Market Factor

18.5 Approximating the Conditional Portfolio Loss Distribution

18.6 A Comparison of Methods

18.7 Perturbing the Loss Distribution

18.8 Summary

19 Implied Correlation

19.1 Introduction

19.2 Implied Correlation

19.3 Compound Correlation

19.4 Disadvantages of Compound Correlation

19.5 No-arbitrage Conditions

19.6 Summary

20 Base Correlation

20.1 Introduction

20.2 Base Correlation

20.3 Building the Base Correlation Curve

20.4 Base Correlation Interpolation

20.5 Interpolating Base Correlation using the ETL

20.6 A Base Correlation Surface

20.7 Risk Management of Index Tranches

20.8 Hedging the Base Correlation Skew

20.9 Base Correlation for Bespoke Tranches

20.10 Risk Management of Bespoke Tranches

20.11 Conclusions

21 Copula Skew Models

21.1 Introduction

21.2 The challenge of Fitting the Skew

21.3 Calibration

21.4 Random Recovery

21.5 The Student-t Copula

21.6 The Double-t Copula

21.7 The Composite Basket Model

21.8 Marshall–Olkin Copula

21.9 Mixing Copula Model

21.10 The Random Factor Loading Model

21.11 The Implied Copula

21.12 Copula Comparison

21.13 Pricing Bespokes

21.14 Summary

22 Advanced Multi-name Credit Derivatives

22.1 Introduction

22.2 Credit CPPI

22.3 Constant Proportion Debt Obligations

22.4 The CDO-squared

22.5 Tranchelets

22.6 Forward Starting Tranches

22.7 Options on Tranches

22.8 Leveraged Super Senior

22.9 Conclusions

23 Dynamic Bottom-up Correlation Models

23.1 Introduction

23.2 A Survey of Dynamic Models

23.3 The Intensity Gamma Model

23.4 The Affine Jump Diffusion Model

23.5 Conclusions

23.6 Technical Appendix

24 Dynamic Top-down Correlation Models

24.1 Introduction

24.2 The Markov Chain Approach

24.3 Markov Chain: Initial Generator

24.4 Markov Chain: Stochastic Generator

24.5 Conclusions

Appendix A Useful Formulae

Bibliography

Index