PART ONE
The Relative Pricing of Fixed Income Securities with Fixed Cash Flows 1
CHAPTER 1
Bond Prices, Discount Factors, and Arbitrage 3
The Time Value of Money 3
Treasury Bond Quotations 4
Discount Factors 6
The Law of One Price 8
Arbitrage and the Law of One Price 10
Treasury STRIPS 12
APPENDIX 1A
Deriving the Replicating Portfolio 17
APPENDIX 1B
APPLICATION: Treasury Triplets and High Coupon Bonds 18
CHAPTER 2
Bond Prices, Spot Rates, and Forward Rates 23
Semiannual Compounding 23
Spot Rates 25
Forward Rates 28
Maturity and Bond Price 32
Maturity and Bond Return 34
Treasury STRIPS, Continued 37
APPENDIX 2A
The Relation between Spot and Forward Rates and the Slope of the
Term Structure 38
CHAPTER 3
Yield-to-Maturity 41
Definition and Interpretation 41
Yield-to-Maturity and Spot Rates 46
Yield-to-Maturity and Relative Value: The Coupon Effect 50
Yield-to-Maturity and Realized Return 51
CHAPTER 4
Generalizations and Curve Fitting 53
Accrued Interest 53
Compounding Conventions 56
Yield and Compounding Conventions 59
Bad Days 60
Introduction to Curve Fitting 61
Piecewise Cubics 69
APPLICATION: Fitting the Term Structure in the U.S. Treasury Market
on February 15, 2001 71
TRADING CASE STUDY: A 7s-8s-9s Butterfly 77
APPENDIX 4A
Continuous Compounding 83
APPENDIX 4B
A Simple Cubic Spline 85
PART TWO
Measures of Price Sensitivity and Hedging 87
CHAPTER 5
One-Factor Measures of Price Sensitivity 89
DV01 91
A Hedging Example, Part I: Hedging a Call Option 95
vi CONTENTS
Duration 98
Convexity 101
A Hedging Example, Part II: A Short Convexity Position 103
Estimating Price Changes and Returns with DV01, Duration, and Convexity 105
Convexity in the Investment and Asset-Liability Management Contexts 108
Measuring the Price Sensitivity of Portfolios 109
A Hedging Example, Part III: The Negative Convexity of Callable Bonds 111
CHAPTER 6
Measures of Price Sensitivity Based on Parallel Yield Shifts 115
Yield-Based DV01 115
Modified and Macaulay Duration 119
Zero Coupon Bonds and a Reinterpretation of Duration 120
Par Bonds and Perpetuities 122
Duration, DV01, Maturity, and Coupon: A Graphical Analysis 124
Duration, DV01, and Yield 127
Yield-Based Convexity 127
Yield-Based Convexity of Zero Coupon Bonds 128
The Barbell versus the Bullet 129
CHAPTER 7
Key Rate and Bucket Exposures 133
Key Rate Shifts 134
Key Rate 01s and Key Rate Durations 135
Hedging with Key Rate Exposures 137
Choosing Key Rates 140
Bucket Shifts and Exposures 142
Immunization 147
Multi-Factor Exposures and Risk Management 147
CHAPTER 8
Regression-Based Hedging 149
Volatility-Weighted Hedging 150
One-Variable Regression-Based Hedging 153
Two-Variable Regression-Based Hedging 158
TRADING CASE STUDY: The Pricing of the 20-Year U.S. Treasury Sector 161
A Comment on Level Regressions 166
Contents vii
PART THREE
Term Structure Models 169
CHAPTER 9
The Science of Term Structure Models 171
Rate and Price Trees 171
Arbitrage Pricing of Derivatives 174
Risk-Neutral Pricing 177
Arbitrage Pricing in a Multi-Period Setting 179
Example: Pricing a CMT Swap 185
Reducing the Time Step 187
Fixed Income versus Equity Derivatives 190
CHAPTER 10
The Short-Rate Process and the Shape of the Term Structure 193
Expectations 194
Volatility and Convexity 196
Risk Premium 201
A Mathematical Description of Expectations, Convexity, and
Risk Premium 206
APPLICATION: Expectations, Convexity, and Risk Premium in the
U.S. Treasury Market on February 15, 2001 212
APPENDIX 10A
Proofs of Equations (10.19) and (10.25) 214
CHAPTER 11
The Art of Term Structure Models: Drift 219
Normally Distributed Rates, Zero Drift: Model 1 219
Drift and Risk Premium: Model 2 225
Time-Dependent Drift: The Ho-Lee Model 228
Desirability of Fitting to the Term Structure 229
Mean Reversion: The Vasicek (1977) Model 232
CHAPTER 12
The Art of Term Structure Models: Volatility and Distribution 245
Time-Dependent Volatility: Model 3 245
Volatility as a Function of the Short Rate: The Cox-Ingersoll-Ross
and Lognormal Models 248
viii CONTENTS
Tree for the Original Salomon Brothers Model 251
A Lognormal Model with Mean Reversion: The Black-Karasinski Model 253
Selected List of One-Factor Term Structure Models 255
APPENDIX 12A
Closed-Form Solutions for Spot Rates 257
CHAPTER 13
Multi-Factor Term Structure Models 259
Motivation from Principal Components 259
A Two-Factor Model 263
Tree Implementation 265
Properties of the Two-Factor Model 269
Other Two-Factor and Multi-Factor Modeling Approaches 274
APPENDIX 13A
Closed-Form Solution for Spot Rates in the Two-Factor Model 275
CHAPTER 14
Trading with Term Structure Models 277
Example Revisited: Pricing a CMT Swap 278
Option-Adjusted Spread 278
Profit and Loss (P&L) Attribution 280
P&L Attributions for a Position in the CMT Swap 283
TRADING CASE STUDY: Trading 2s-5s-10s in Swaps with a
Two-Factor Model 286
Fitting Model Parameters 295
Hedging to the Model versus Hedging to the Market 297
PART FOUR
Selected Securities 301
CHAPTER 15
Repo 303
Repurchase Agreements and Cash Management 303
Repurchase Agreements and Financing Long Positions 305
Reverse Repurchase Agreements and Short Positions 308
Carry 311
General Collateral and Specials 314
Contents ix
Special Repo Rates and the Auction Cycle 316
Liquidity Premiums of Recent Issues 319
APPLICATION: Valuing a Bond Trading Special in Repo 321
APPLICATION: Disruption in the Specials Market after September 11, 2001 323
CHAPTER 16
Forward Contracts 325
Definitions 325
Forward Price of a Deposit or a Zero Coupon Bond 326
Using Forwards to Hedge Borrowing Costs or Loan Proceeds 328
Forward Price of a Coupon Bond 329
Forward Yield and Forward DV01 331
Forward Prices with Intermediate Coupon Payments 332
Value of a Forward Contract 335
Forward Prices in a Term Structure Model 336
CHAPTER 17
Eurodollar and Fed Funds Futures 339
LIBOR and Eurodollar Futures 339
Hedging with Eurodollar Futures 343
Tails: A Closer Look at Hedging with Futures 344
Futures on Prices in a Term Structure Model 347
Futures on Rates in a Term Structure Model 349
The Futures-Forward Difference 350
TED Spreads 355
APPLICATION: Trading TED Spreads 359
Fed Funds 362
Fed Funds Futures 364
APPLICATION: Fed Funds Contracts and Predicted Fed Action 366
APPENDIX 17A
Hedging to Dates Not Matching Fed Funds and Eurodollar
Futures Expirations 369
CHAPTER 18
Interest Rate Swaps 371
Swap Cash Flows 371
Valuation of Swaps 373
x CONTENTS
Floating Rate Notes 374
Valuation of Swaps, Continued 376
Note on the Measurement of Fixed and Floating Interest Rate Risk 378
Swap Spreads 378
Major Uses of Interest Rate Swaps 381
Asset Swap Spreads and Asset Swaps 382
TRADING CASE STUDY: 30-Year FNMA Asset Swap Spreads 386
On the Credit Risk of Swap Agreements 388
APPENDIX 18A
TRADING CASE STUDY: Five-Year On-the-Run/Off-the-Run Spread
of Spreads 390
CHAPTER 19
Fixed Income Options 397
Definitions and Review 397
Pricing American and Bermudan Bond Options in a Term
Structure Model 400
APPLICATION: FNMA 6.25s of July 19, 2011, and the Pricing of
Callable Bonds 405
Graphical Analysis of Callable Bond Pricing 408
A Note on Yield-to-Call 411
Swaptions, Caps, and Floors 413
Quoting Prices with Volatility Measures in Fixed Income Options Markets 416
Smile and Skew 420
CHAPTER 20
Note and Bond Futures 423
Mechanics 423
Cost of Delivery and the Determination of the Final Settlement Price 426
Motivations for a Delivery Basket and Conversion Factors 428
Imperfection of Conversion Factors and the Delivery Option at Expiration 431
Gross and Net Basis 435
Quality Option before Delivery 438
Some Notes on Pricing the Quality Option in Term Structure Models 441
Measures of Rate Sensitivity 443
Timing Option 444
End-of-Month Option 445
TRADING CASE STUDY: November ’08 Basis into TYM0 446
Contents xi
CHAPTER 21
Mortgage-Backed Securities 455
Basic Mortgage Mathematics 455
Prepayment Option 459
Overview of Mortgage Pricing Models 464
Implementing Prepayment Models 467
Price-Rate Curve of a Mortgage Pass-Through 471
APPLICATION: Mortgage Hedging and the Directionality of Swap Spreads 473
Mortgage Derivatives, IOs, and POs 475
EXERCISES 479
REFERENCES AND SUGGESTIONS FOR FURTHER READING 497
INDEX 501