Interest Rate Swaps and Their Derivatives: A Practitioner's Guide

Preface ix

“Rates” Market ix

Background ix

Book Structure xi

Acknowledgments xvii

About the Author xix

List of Symbols and Abbreviations xxi

PART ONE Cash, Repo, and Swap Markets 1

CHAPTER 1 Bonds: It’s All About Discounting 3

Time Value of Money: Future Value, Present Value 3

Price-Yield Formula 5

PV01, PVBP, Convexity 11

Repo, Reverse Repo 16

Forward Price/Yield, Carry, Roll-Down 19

CHAPTER 2 Swaps: It’s Still About Discounting 25

Discount Factor Curve, Zero Curve 26

Forward Rate Curve 27

Par-Swap Curve 31

Construction of the Swap/Libor Curve 34

CHAPTER 3 Interest Rate Swaps in Practice 43

Market Instruments 43

Swap Trading—Rates or Spreads 48

Swap Spreads 51

Risk, PV01, Gamma Ladder 56

Calendar Rules, Date Minutiae 59

CHAPTER 4 Separating Forward Curve from Discount Curve 67

Forward Curves for Assets 67

Implied Forward Rates 69

Float/Float Swaps 70

Libor/Libor Basis Swaps 73

Overnight Indexed Swaps (OIS) 75

PART TWO Interest-Rate Flow Options 77

CHAPTER 5 Derivatives Pricing: Risk-Neutral Valuation 79

European-Style Contingent Claims 80

One-Step Binomial Model 80

From One Time-Step to Two 84

From Two Time-Steps to . . . 90

Relative Prices 91

Risk-Neutral Valuation: All Relative Prices Must be Martingales 92

Interest-Rate Options Are Inherently Difficult to Value 93

From Binomial Model to Equivalent Martingale Measures 94

CHAPTER 6 Black’s World 97

A Little Bit of Randomness 97

Modeling Asset Changes 103

Black-Scholes-Merton/Black Formulae 104

Greeks 112

Digitals 116

Call Is All You Need 117

Calendar/Business Days, Event Vols 120

CHAPTER 7 European-Style Interest-Rate Derivatives 123

Market Practice 124

Interest-Rate Option Trades 124

Caplets/Floorlets: Options on Forward Rates 125

European-Style Swaptions 129

Skews, Smiles 137

CMS Products 140

Bond Options 147

PART THREE Interest-Rate Exotics 149

CHAPTER 8 Short-Rate Models 151

A Quick Tour 152

Dynamics to Implementation 153

Lattice/Tree Implementation 154

BDT Lattice Model 156

Hull-White, Black-Karasinski Models 168

Simulation Implementation 169

CHAPTER 9 Bermudan-Style Options 175

Bellman’s Equation—Backward Induction 176

Bermudan Swaptions 177

Bermudan Cancelable Swaps, Callable/Puttable Bonds 180

Bermudan-Style Options in Simulation Implementation 183

CHAPTER 10 Full Term-Structure Interest-Rate Models 185

Shifting Focus from Short Rate to Full Curve: Ho-Lee Model 186

Heath-Jarrow-Morton (HJM) Full Term-Structure Framework 186

Discrete-Time, Discrete-Tenor HJM Framework 188

Forward-Forward Volatility 191

Multifactor Models 197

HJM Framework Typically Leads to Nonrecombining Trees 199

CHAPTER 11 Forward-Measure Lens 201

Numeraires Are Arbitrary 201

Forward Measures 206

BGM/Jamshidian Results 208

Different Measures for Different Rates 210

“Classic” or “New Improved”: Pick Your Poison! 212

CHAPTER 12 In Search of “The” Model 215

Migration to Full-Term Structure Models 215

Implementation Era 216

Model versus Market: Liquidity and Concentration Risk 216

Complexity Risk 217

Remaining Challenges 218

APPENDIX A Taylor Series Expansion 219

Function of One Variable 219

Function of Several Variables 220

Ito’s Lemma: Taylor Series for Diffusions 220

APPENDIX B Mean-Reverting Processes 223

Normal Dynamics 224

Log-Normal Dynamics 226

APPENDIX CGirsanov’s Theorem and Change of Numeraire 229

Continuous-Time, Instantaneous-Forwards

HJM Framework 230

BGM Result 232

Notes 235

Index 239