Preface ix
About the Editor xiii
About the Contributors xv
PART ONE
Option Pricing and Volatility Modeling
CHAPTER 1
A Moment Approach to Static Arbitrage 3
Alexandre d’Aspremont
1.1 Introduction 3
1.2 No-Arbitrage Conditions 7
1.3 Example 15
1.4 Conclusion 16
CHAPTER 2
On Black-Scholes Implied Volatility at Extreme Strikes 19
Shalom Benaim, Peter Friz, and Roger Lee
2.1 Introduction 19
2.2 The Moment Formula 20
2.3 Regular Variation and the Tail-Wing Formula 24
2.4 Related Results 27
2.5 Applications 33
2.6 CEV and SABR 35
CHAPTER 3
Dynamic Properties of Smile Models 47
Lorenzo Bergomi
3.1 Introduction 48
3.2 Some Standard Smile Models 50

3.3 A New Class of Models for Smile Dynamics 65
3.4 Pricing Examples 81
3.5 Conclusion 87
CHAPTER 4
A Geometric Approach to the Asymptotics of Implied Volatility 89
Pierre Henry-Labord`ere
4.1 Volatility Asymptotics in Stochastic Volatility Models 91
4.2 Heat Kernel Expansion 92
4.3 Geometry of Complex Curves and Asymptotic
Volatility 100
4.4 λ-SABR Model and Hyperbolic Geometry 106
4.5 SABR Model with β = 0, 1 117
4.6 Conclusions and Future Work 122
4.7 Appendix A: Notions in Differential Geometry 122
4.8 Appendix B: Laplace Integrals in Many Dimensions 125
CHAPTER 5
Pricing, Hedging, and Calibration in Jump-Diffusion Models 129
Peter Tankov and Ekaterina Voltchkova
5.1 Overview of Jump-Diffusion Models 131
5.2 Pricing European Options via Fourier Transform 137
5.3 Integro-differential Equations for Barrier and
American Options 140
5.4 Hedging Jump Risk 147
5.5 Model Calibration 153
PART TWO
Credit Risk
CHAPTER 6
Modeling Credit Risk 163
L. C. G. Rogers
6.1 What Is the Problem? 163
6.2 Hazard Rate Models 166
6.3 Structural Models 175
6.4 Some Nice Ideas 179
6.5 Conclusion 181

CHAPTER 7
An Overview of Factor Modeling for CDO Pricing 185
Jean-Paul Laurent and Areski Cousin
7.1 Pricing of Portfolio Credit Derivatives 185
7.2 Factor Models for the Pricing of CDO Tranches 189
7.3 A Review of Factor Approaches to the Pricing of CDOs 198
7.4 Conclusion 212
CHAPTER 8
Factor Distributions Implied by Quoted CDO Spreads 217
Erik Schlo¨ gl and Lutz Schlo¨ gl
8.1 Introduction 217
8.2 Modeling 220
8.3 Examples 224
8.4 Conclusion 231
8.5 Appendix: Some Useful Results on Hermite
Polynomials under Linear Coordinate Transforms 232
CHAPTER 9
Pricing CDOs with a Smile: The Local Correlation Model 235
Julien Turc and Philippe Very
9.1 The Local Correlation Model 236
9.2 Simplification under the Large Pool Assumption 240
9.3 Building the Local Correlation Function without
the Large Pool Assumption 243
9.4 Pricing and Hedging with Local Correlation 247
CHAPTER 10
Portfolio Credit Risk: Top-Down versus Bottom-Up Approaches 251
Kay Giesecke
10.1 Introduction 251
10.2 Portfolio Credit Models 251
10.3 Information and Specification 253
10.4 Default Distribution 259
10.5 Calibration 264
10.6 Conclusion 265
CHAPTER 11
Forward Equations for Portfolio Credit Derivatives 269
Rama Cont and Ioana Savescu
11.1 Portfolio Credit Derivatives 270
11.2 Top-Down Models for CDO Pricing 273

11.3 Effective Default Intensity 275
11.4 A Forward Equation for CDO Pricing 278
11.5 Recovering Forward Default Intensities from
Tranche Spreads 282
11.6 Conclusion 288
Index 295