Interest Rate Swaps and Their Derivatives

An up-to-date look at the evolution of interest rate swaps and derivatives

Interest Rate Swaps and Derivatives bridges the gap between the theory of these instruments and their actual use in day-to-day life. This comprehensive guide covers the main "rates" products, including swaps, options (cap/floors, swaptions), CMS products, and Bermudan callables. It also covers the main valuation techniques for the exotics/structured-notes area, which remains one of the most challenging parts of the market.

• Provides a balance of relevant theory and real-world trading instruments for rate swaps and swap derivatives
• Uses simple settings and illustrations to reveal key results
• Written by an experienced trader who has worked with swaps, options, and exotics

With this book, author Amir Sadr shares his valuable insights with practitioners in the field of interest rate derivatives-from traders and marketers to those in operations.

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 C Girsanov’s Theorem and Change of Numeraire 229

Continuous-Time, Instantaneous-Forwards HJM Framework 230

BGM Result 232

Notes 235

Index 239

AMIR SADR, PHD, has experience as a quant, trader, financial software developer, and academic in fixed income markets. He traded options and exotics at HSBC in New York from 2005 to 2006 and traded at the proprietary desk for Greenwich Capital Markets (GCM) for four years prior to that. Sadr also has experience at Morgan Stanley as a vice president in the derivatives products group where he traded interest rate derivatives and exotics. Since 1996, Sadr has served as an adjunct professor at New York University in the Department of Finance and Accounting.

INTEREST RATE SWAPS AND THEIR DERIVATIVES

Interest rate swaps and their derivatives have become an integral part of the fixed income market, but many of the pricing and risk management issues for these now mainstream products can only be learned on a trading floor. While there are many books on fixed income and interest rate derivatives, they generally suffer from being either too elementary and bond-centric, mentioning swaps in passing, or too technical and focused on exotics and the myriad implementation issues and algorithms used to tackle them.

Rather than focusing on exotics, Interest Rate Swaps and Their Derivatives thoroughly covers the mainstream products—swaps, flow options, Bermudans, semi-exotics—showing the common pricing techniques while also explaining how to generalize the concepts to more nuanced products.

Author Amir Sadr, experienced as a quant, trader, financial software developer, and academic in the fixed income field, begins by presenting plain-vanilla swaps as an extension of fixed rate bonds—revealing how techniques for pricing these instruments are a generalization of similar methods used for pricing bonds and repos, and for the most part involve the concepts of financing cost, discount factors, and projection of forward curves. He then moves on to cover the options markets for flow products, including options on futures, caps and floors, and European swaptions—with detailed attention to the actual trading practice of these products. Sadr explains how, as with any option product, the pricing and risk management of these requires dealing with volatility as the main risk factor—and he shows that one does not need to have a PhD in math to understand options. Sadr presents risk-neutral valuation as the fundamental pricing paradigm for derivatives, and illustrates the core idea of dynamic replication in a simple binomial setting. This unified framework is used to derive industry-standard Black formula for flow products, and is developed into short-rate and full term-structure models for more complex interest rate exotics including Bermudans.

For current or aspiring practitioners in interest rate products, Interest Rate Swaps and Their Derivatives provides a sound working knowledge and appreciation of the main features of these products and their pricing and risk management issues.