Credit valuation adjustments with application to credit default swaps

286 Pages·2012·10.08 MB·English

Credit valuation adjustments with
application to credit default swaps
by
Cara Milwidsky
Submitted in partial fulfillment of the requirements for the degree
Magister Scientiae
to the Department of Mathematics and Applied Mathematics

in the Faculty of Natural and Agricultural Sciences

University of Pretoria
Pretoria
August 2011
©© UUnniivveerrssiittyy ooff PPrreettoorriiaa
Declaration
I, Cara Milwidsky, declare that this dissertation, which I hereby submit for the degree
Master of Science at the University of Pretoria, is my own work and has not previously
been submitted by me for a degree at this or any other tertiary institution.
Cara Milwidsky
1 September 2011
Abstract
The credit valuation adjustment (CVA) on an over-the-counter derivative transaction is
the price of the risk associated with the potential default of the counterparties to the
trade. This dissertation provides an introduction to the concept of CVA, beginning with
the required backdrop of counterparty risk and the basics of default risk modelling. Right
and wrong way risks are central themes of the dissertation. A model for the pricing of
both the unilateral and the bilateral CVA on a credit default swap (CDS) is implemented.
Each step of this process is explained thoroughly. Results are reported and discussed for
a range of parameters. The trends observed in the CDS CVA numbers produced by the
model are all justified and the right and wrong way nature of the exposures captured. In
addition, the convergence and stability of the numerical schemes utilised are shown to be
appropriate. A case study, in which the model is applied to a set of market scenarios,
concludes the dissertation. Since the field is far from established, a number of areas are
suggested for further research.
Acknowledgements
I would like to thank my supervisor, Prof. Eben Mar´e, for the valuable advice he has
offered throughout the course of my degree and for sharing the experience he has accumu-
lated over the years. Thank you to T. Schelfaut and P. Mayar for finding an easy way out
of the characteristic function trap and for taking the time to share it with me. To Ivan
Ruscic, thank you for the many hours you have spent discussing ideas and proof reading,
as well as for taking a genuine interest in this dissertation. Thank you to both Ivan Rus-
cic and Glenn Brickhill for giving me the opportunity to implement a CDS CVA model
of my choice. To my friends and family, thank you for the tremendous encouragement,
patience, support and understanding you have demonstrated. And especially to James
for tolerating the piles of papers you so despise.
Cara Milwidsky
1 September 2011
“Courage doesn’t always roar. Sometimes courage is the quiet voice at the
end of the day saying, ‘I will try again tomorrow’.”
Mary Anne Radmacher
Contents
I Introduction, Background and Context 18
1 Introduction 19
1.1 Structure of the Dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.2 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2 Counterparty Credit Risk 23
2.1 Counterparty Credit Exposure . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.2 Potential Future Exposure . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.3 Counterparty Risk Mitigants . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.3.1 Netting Agreements . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.3.2 Credit Support Annex . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.3.3 Early Termination Provisions . . . . . . . . . . . . . . . . . . . . . 30
2.4 Exposure Management & Credit Default Swaps . . . . . . . . . . . . . . . 31
2.5 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3 Credit Valuation Adjustment 34
3.1 An Introduction to CVA . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.1.1 Right and Wrong Way Risk . . . . . . . . . . . . . . . . . . . . . . 36
3.2 Unilateral CVA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.3 The Bilateral Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.3.1 Problems with Hedging DVA . . . . . . . . . . . . . . . . . . . . . 45
3.4 A Simple Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.4.1 A Formula for Unilateral CVA with Independence . . . . . . . . . . 47
3.4.2 Application to a Vanilla Interest Rate Swap . . . . . . . . . . . . . 48
3.4.3 A Formula for Implementation . . . . . . . . . . . . . . . . . . . . . 49
3.4.4 The Bilateral CVA on a Swap . . . . . . . . . . . . . . . . . . . . . 50
3.4.5 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
CONTENTS 6
3.5 Practical Implementation Aspects . . . . . . . . . . . . . . . . . . . . . . . 54
3.5.1 Central Management of CVA . . . . . . . . . . . . . . . . . . . . . 55
3.5.2 System Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.6 Two Relevant Instruments . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.6.1 Contingent Credit Default Swap (CCDS) . . . . . . . . . . . . . . . 58
3.6.2 Credit Extinguisher . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.7 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4 Modelling Default Risk 62
4.1 Structural Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.1.1 The Merton Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.1.2 First Passage Time Models . . . . . . . . . . . . . . . . . . . . . . . 64
4.1.3 Further Remarks on the Structural Approach . . . . . . . . . . . . 66
4.2 Reduced Form Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.2.1 The Role of Poisson Processes . . . . . . . . . . . . . . . . . . . . . 66
4.2.2 Time Homogeneous Poisson Process . . . . . . . . . . . . . . . . . . 67
4.2.3 More General Poisson Processes . . . . . . . . . . . . . . . . . . . . 70
4.3 CDS Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.3.1 General Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.3.2 Market Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.4 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5 Literature Review 79
5.1 Structural Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.2 Intensity Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.2.1 Contagion Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5.2.2 Copula Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.3 Alternative Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.3.1 A Generalized Affine Model . . . . . . . . . . . . . . . . . . . . . . 89
5.3.2 Markov Chain Models . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.4 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
II CDS CVA Model Implementation - the Unilateral Case 93
6 Model Background 94
6.1 Credit Default Swaps in the Spotlight . . . . . . . . . . . . . . . . . . . . . 94
CONTENTS 7
6.2 Model Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
6.3 Preamble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
6.3.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
6.3.2 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
6.3.3 The Stochastic Intensity Process . . . . . . . . . . . . . . . . . . . . 103
6.4 Model Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6.5 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
7 Calibrating Intensities & Simulating Default Times 108
7.1 Calibrating the Intensity Process . . . . . . . . . . . . . . . . . . . . . . . 108
7.1.1 Calibration Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
7.2 Simulating the Default Times . . . . . . . . . . . . . . . . . . . . . . . . . 112
7.2.1 Simulating the Hazard Rates . . . . . . . . . . . . . . . . . . . . . . 112
7.2.2 Integrating intensities . . . . . . . . . . . . . . . . . . . . . . . . . . 117
7.2.3 Determining τ and τ . . . . . . . . . . . . . . . . . . . . . . . . . 117
1 2
7.2.4 Reproducing the Market Survival Curve . . . . . . . . . . . . . . . 123
7.3 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
8 Revaluation of the CDS upon Counterparty Default 126
8.1 Survival Probability Preliminaries . . . . . . . . . . . . . . . . . . . . . . . 127
8.1.1 A Practical Formulation of the Survival Probability . . . . . . . . . 128
8.2 Applying the Fractional Fourier Transform . . . . . . . . . . . . . . . . . . 131
8.2.1 Writing the CDF in the Form of an FRFT Sum . . . . . . . . . . . 132
8.2.2 The Characteristic Function . . . . . . . . . . . . . . . . . . . . . . 134
8.2.3 A Trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
8.2.4 Rescued! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
8.2.5 Parameters Required for the FRFT Implementation . . . . . . . . . 143
8.2.6 A Note on the Limit of the Integrand . . . . . . . . . . . . . . . . . 154
8.3 CDS Valuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
8.3.1 Obtaining the Conditional Copula Function . . . . . . . . . . . . . 155
8.3.2 The Survival Probability . . . . . . . . . . . . . . . . . . . . . . . . 158
8.4 CVA Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
8.5 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
9 Results of the Model Implementation 162
9.1 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
CONTENTS 8
9.2 CVA Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
9.2.1 Reference Entity Riskier than Counterparty . . . . . . . . . . . . . 169
9.2.2 Counterparty Riskier than Reference Entity . . . . . . . . . . . . . 173
9.2.3 A Change in Reference Entity Riskiness . . . . . . . . . . . . . . . 175
9.2.4 Risky Reference Entity . . . . . . . . . . . . . . . . . . . . . . . . . 177
9.2.5 Differing Maturities . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
9.3 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
III A Bilateral CVA Model 183
10 Extending the Model to the Bilateral Case 184
10.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
10.2 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
10.3 Simulating Default Times . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
10.4 Revaluation of the CDS upon Default . . . . . . . . . . . . . . . . . . . . . 190
10.4.1 The Fractional Fast Fourier Transform . . . . . . . . . . . . . . . . 191
10.4.2 The Conditional Copula Function . . . . . . . . . . . . . . . . . . . 192
10.4.3 CDS Valuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
10.4.4 The Bilateral CVA Computation . . . . . . . . . . . . . . . . . . . 197
10.5 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
11 Results of the Bilateral Model Implementation 199
11.1 Investor and Counterparty Equally Risky . . . . . . . . . . . . . . . . . . . 201
11.2 A Less Risky Investor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
11.3 Decreasing the Riskiness of the Counterparty . . . . . . . . . . . . . . . . . 204
11.4 Altering the Reference Entity . . . . . . . . . . . . . . . . . . . . . . . . . 206
11.5 The Effect of a Non Zero Investor-Counterparty Default Time Correlation 208
11.6 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
IV Case Study and Conclusion 212
12 Case Study 213
12.1 Market Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
12.2 Model Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
12.2.1 The Volatility Parameter . . . . . . . . . . . . . . . . . . . . . . . . 216
12.2.2 Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
CONTENTS 9
12.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
12.4 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
13 Conclusion 222
13.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
13.2 Directions for Further Research . . . . . . . . . . . . . . . . . . . . . . . . 226
V Appendices 238
A Proofs of the General Counterparty Risk Pricing Formulae 239
A.1 Proof of the Unilateral Formula . . . . . . . . . . . . . . . . . . . . . . . . 239
A.2 Bilateral Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
B Matters Related to CDS Pricing 244
B.1 Bootstrapping the Par CDS Curve . . . . . . . . . . . . . . . . . . . . . . 244
B.2 Derivation of the CDS Premium and Default Leg Valuation Formulae . . . 248
B.2.1 CDS Premium Leg (Equation (4.27)) . . . . . . . . . . . . . . . . . 248
B.2.2 CDS Protection (Default) Leg (Equation (4.28)) . . . . . . . . . . . 249
B.3 Derivation of the Constant Hazard Rate . . . . . . . . . . . . . . . . . . . 250
C Formulae Related to the CIR Process 252
C.1 CIR Instantaneous Forward Rate . . . . . . . . . . . . . . . . . . . . . . . 252
C.2 CIR Bond Price . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
C.3 CIR Conditional Expected Value . . . . . . . . . . . . . . . . . . . . . . . 253
C.4 CIR Conditional Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
D A Basic Introduction to Copulas 254
D.1 Concepts Required to Define Copulas . . . . . . . . . . . . . . . . . . . . . 254
D.2 The Definition of a Copula . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
D.3 Sklar’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
D.4 The Uniform Distribution of CDFs . . . . . . . . . . . . . . . . . . . . . . 257
D.5 Copula Densities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
D.6 Copula Conditional Distributions . . . . . . . . . . . . . . . . . . . . . . . 258
D.7 ProofoftheStatementthataTrivariateGaussianCopulaInducesBivariate
Marginals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258