Bond Pricing and Yield Curve Modeling: A Structural Approach

Frontmatter......Page 2
Dedication......Page 6
Contents......Page 8
Acknowledgements......Page 24
Symbols and Abbreviations......Page 26
Part I The Foundations......Page 30
1.1 My Goal in Writing This Book......Page 32
1.2 What My Account Leaves Out......Page 34
1.3 Affine Models......Page 35
1.4 A Simple Taxonomy......Page 37
1.5.1 Latent versus Observable Variables......Page 39
1.5.2 The Spanning Problem......Page 44
1.5.3 The Constraint Problem......Page 45
1.6 Why Do We Need No-Arbitrage Models After All?......Page 48
1.7 Stamp Collecting and Shallow versus Deep Explanations......Page 49
1.8 The Ideal Reader and Plan of the Book......Page 50
2.2.1 Arbitrage......Page 53
2.2.2 Pseudo-Arbitrage......Page 54
2.2.3 Sharpe Ratios......Page 56
2.2.4 Bond Prices and Yields......Page 57
2.2.5 Duration and Convexity......Page 60
2.2.6 Forward Rates......Page 61
2.3 Log Prices and Log Returns......Page 62
2.4 Dimensional Analysis......Page 63
2.5.1 Definition......Page 65
2.5.2 Transformations of Vectors......Page 66
2.5.3 Orthogonal Matrices......Page 67
2.5.4 Row Vectors......Page 68
2.5.5 Exponential of a Matrix......Page 69
2.6.1 The Ornstein–Uhlenbeck Process......Page 70
2.6.2 The AR(1) Process......Page 71
2.6.3 Parallels between AR(1) Processes and the Ornstein–Uhlenbeck Process......Page 72
2.7.1 Ito’s Lemma......Page 73
2.7.2 Stochastic-Calculus Rules for d pt dxt......Page 74
2.7.3 Expectations of Ito Integrals......Page 75
2.7.4 The Ito Isometry......Page 76
2.7.5 Risk-less Portfolios......Page 77
3.1 The Purpose of This Chapter......Page 78
3.2 The Monetary Channels......Page 79
3.3 A Modelling Framework......Page 81
3.4 The Monetary Actions: A Simple Model......Page 85
3.5.1 General Considerations......Page 87
3.5.2 Assessing the Quality of the Calibration Process......Page 89
3.5.3 State Variables versus Model Parameters......Page 90
4.1 The Purpose of This Chapter......Page 92
4.2.1 Inflation Risk......Page 93
4.2.2 Real-Rate Risk......Page 94
4.2.3 Putting the Pieces Together......Page 95
4.3 Real-World and Risk-Neutral Probabilities: The Market Price of Risk......Page 97
4.3.1 Introducing the P and Q Measures......Page 98
4.3.2 Introducing the Market Price of Risk......Page 101
4.4 An Important First Result: Bond Prices as Q-Expectations......Page 105
4.5.1 The General Case......Page 106
4.5.2 The Affine Case......Page 107
4.6 Nominal Rates, Inflation and Real Rates: Definitions......Page 108
5.2.1 An Account of What Happened......Page 110
5.2.2 Possible Explanations of What Happened......Page 114
5.3 How Can We Estimate Risk Premia?......Page 116
5.4 Different Types of Risk Premia......Page 117
5.5.1 Decomposition of the Risk Premium......Page 120
5.5.2 ‘Which’ Liquidity Are TIPS-Investors Compensated For?......Page 122
5.6 What Is and What Is Not a True Risk Premium......Page 123
5.7 Does It Matter if a Risk Premium Is ‘Really’ a Risk Premium?......Page 125
6.2.1 The Axis Rotation......Page 127
6.3 How Many Principal Components Do We Need for Yields?......Page 132
6.4 First Conclusions......Page 133
6.5 Some Mathematical Results......Page 134
7.2.1 Descriptive Features......Page 137
7.2.2 Mean-Reverting Properties – Each PC in Isolation......Page 141
7.2.3 The Joint Mean-Reverting Behaviour of Principal Components......Page 145
7.3 Real Rates and Break-Even Inflation......Page 151
7.4 Correlation between Nominal, Inflation and Real Principal Components......Page 157
Part II The Building Blocks: A First Look......Page 162
8.2 Linking Expectations with No-Arbitrage......Page 166
8.2.1 A One-Factor World......Page 167
8.2.2 Moving to Many Factors......Page 170
8.3 An Example: A Mean-Reverting Process for the Short Rate......Page 172
8.4 Expectations and Survey Data......Page 174
9.2 Where Does Convexity Come from?......Page 176
9.3 The Links between Convexity and Jensen’s Inequality......Page 178
9.3.1 A Special but Important Case: Gaussian Random Variables......Page 179
9.4 What Does Convexity Depend On?......Page 181
9.5 Why Convexity Is Different......Page 183
9.6 Why Isn’t Convexity ‘Always Good’?......Page 185
9.7 Who Sets the Price of Convexity? A Bit of Story-Telling......Page 186
10.1 The Purpose of This Chapter......Page 189
10.3.1 Distributional Properties of the Vasicek Model......Page 190
10.3.3 The Duration in the Vasicek Model......Page 194
10.4 Rate Expectations and the Shape of the Vasicek Yield Curve......Page 197
10.5.1 An Expression for Convexity......Page 199
10.5.2 Convexity and the Volatility of Yields......Page 200
10.5.3 How Big Should One Expect the Convexity Effect to Be?......Page 201
10.5.4 What Is the ‘Right’ Reversion Speed?......Page 202
10.6 The Risk Premium in the Vasicek Model......Page 204
10.7 The Functional Form of the Market Price of Risk......Page 205
10.8 The Link between the Market Price of Risk and the Sharpe Ratio......Page 207
10.9 Appendix 10A: Proof that rt = .........Page 210
Part III The Conditions of No-Arbitrage......Page 212
11.1 The Purpose of This Chapter......Page 214
11.2 Type-I Arbitrage......Page 215
11.3 Bounds to the Price-Correction Term: Type-I Arbitrage......Page 217
11.4 Bounds to the Price-Correction Term: Type-II Arbitrage......Page 218
11.5 A Useful Rewriting......Page 221
11.6 Extension to Many Factors......Page 222
11.7 The Task of the Long-Term Bond Investor......Page 224
12.2 Constructing a Risk-Less Portfolio: The Market Price of Risk Again......Page 225
12.3 Interpretations of the Market Price of Risk......Page 228
12.4 Excess Returns......Page 229
12.5 What the Market Price of Risk Can Depend On......Page 230
12.6 Appendix 12A: The Market Price of Risk and Excess Return with Many Factors......Page 231
13.1 The Purpose of This Chapter......Page 235
13.2 A Bird’s Eye View of the ‘Traditional’ and ‘Modern’ Approaches......Page 236
13.3 Pricing Assets: The Building-Blocks Approach......Page 237
13.4.1 Prices as Expectations in the Risk-Neutral Measure – Again......Page 240
13.4.2 The Equivalence of the State-Price Deflator and the Stochastic Discount Factor......Page 242
13.5 The Process for the State-Price Deflator......Page 243
13.7 Deriving the Drift of the State-Price Deflator......Page 245
13.8 The Short Rate Again......Page 247
13.9 Deriving the Volatility of the State-Price Deflator......Page 248
13.9.1 Evaluation of the Three Terms......Page 249
13.9.2 The Link between the Volatility of the State-Price Deflator and the Market Price of Risk......Page 250
13.9.3 Where Does the Volatility of Bonds Come from?......Page 251
13.9.4 Summary of Results......Page 252
14.2 The Expression for the Real State-Price Deflator......Page 253
14.3 The Process for the Real State-Price Deflator......Page 255
14.4 The Link between Break-Even Inflation and Inflation Expectations......Page 258
14.4.1 Inflation Expectation Under P......Page 259
14.4.2 Inflation Expectation under Q......Page 261
14.4.3 Inflation Expectation under T......Page 262
14.4.4 Inflation Expectations under Different Measures......Page 263
14.5 The Risk Premium as a Covariance......Page 264
14.6 Moving to an Affine World......Page 267
14.7 The Market Price of Inflation Risk – Affine Models......Page 268
15.1 The Purpose of This Chapter......Page 270
15.2 First Derivation of the SDF......Page 271
15.3 From the SDF to Risk Premia......Page 274
15.4 Real versus Nominal Prices......Page 277
15.5 Idiosyncratic Risk......Page 278
15.6 The Links between the SDF and the Risk-Less Rate......Page 279
15.6.1 The No-Uncertainty Case......Page 280
15.6.2 Reintroducing Uncertainty......Page 281
15.6.3 But Does It Work?......Page 282
15.7 SDFs in Continuous and Discrete Time......Page 285
15.8 A More General Result for the Sharpe Ratio......Page 286
Part IV Solving the Models......Page 290
16.1 Purpose of This Chapter......Page 292
16.2.1 The PDE Satisfied by Bond Prices......Page 293
16.3 A Special Case: Affine Term Structures......Page 295
16.4 The Vasicek Case......Page 298
16.5 Affinity of the Vasicek Model under P and under Q......Page 300
16.6.1 Yields......Page 301
16.6.3 Forward Rates......Page 302
16.6.4 Calibrating to the Volatility Structure: Factorization......Page 305
16.6.5 Fitting to the Yield Curve......Page 306
16.7 Why Do We Care about a Humped Volatility Curve?......Page 310
16.8 How to Lengthen the Short Blanket......Page 311
17.2 Affine Models with Many State Variables......Page 314
17.2.1 The N = 2 Case......Page 316
17.2.2 An Expression for the Variance for Generic N......Page 317
17.2.3 Stability......Page 319
17.2.4 Changing Variables......Page 320
17.3 Multivariable Exponentially Affine Models......Page 321
17.4.1 Yields and Forward Rates......Page 322
17.4.2 Distributional Properties......Page 323
17.5 Appendix 17A: Derivation of the Variance of a One-Dimensional Mean-Reverting Process......Page 324
17.6 Appendix 17B: Derivation of the Variance of a Multidimensional Mean-Reverting Process......Page 325
17.7 Appendix 17C: Stability of the Mean-Reverting System......Page 326
18.1 The Purpose of This Chapter......Page 328
18.2 What Is an Affine Model?......Page 329
18.3 The General Strategy......Page 330
18.4 Summary of the Equations Derived in Appendix 18A......Page 332
18.5.1 Expression for the Yield......Page 333
18.5.2 Expression for the Yield Covariance Matrix and the Yield Volatilities......Page 334
18.5.3 Expression for the Volatility of the Instantaneous Forward Rates......Page 335
18.6 Derivation of the Mean and Variance of the State Variables......Page 336
18.7.1 Simple Vasicek......Page 338
18.7.2 The Doubly-Mean-Reverting Vasicek Model......Page 339
18.7.3 The Trebly-Mean-Reverting Vasicek Model......Page 340
18.7.4 The Stochastic-Market-Price-of-Risk Model......Page 341
18.8 Appendix 18A: Solving for .........Page 342
18.8.1 Solving the ODE for .........Page 343
18.8.2 Solving the ODE for A(τ )......Page 345
18.9.1 The Meaning of eA......Page 352
18.10 Explicit Calculation of the Formal Solution .........Page 353
18.10.1 The Up-and-Down Theorem......Page 354
18.10.2 Commutation Relationships for A and f (A)......Page 355
18.10.4 Integral of eAt......Page 356
18.10.5 Evaluation of the Integral .........Page 357
19.1 The Purpose of This Chapter......Page 358
19.2 Motivation: Why the Shadow Rate Matters......Page 359
19.3 How the Shadow Rate Affects the Whole Yield Curve......Page 361
19.4.1 The Setting......Page 362
19.4.2 An Approximate Solution......Page 363
19.5.2 The Effect of the Shadow Rate on Long Yields......Page 368
19.6 A Broader View of the Topic......Page 372
Part V The Value of Convexity......Page 378
20.2 Break-Even Volatility – The Vasicek Setting......Page 380
20.3 Problems with the Vasicek Version of the Break-Even Volatility......Page 384
20.4 Generalizing to Many Factors......Page 386
20.4.1 Calculating the Terms.........Page 389
20.4.2 Expressing the Convexity in Terms of Yield Volatilities......Page 391
20.5 What to Do with This Expression for the Convexity......Page 392
20.5.1 An Important Aside......Page 393
20.6 An Intuitive Aside: Simplifying the Analysis......Page 394
20.7 A Graphical Interpretation......Page 397
20.8 Appendix 20A......Page 398
21.1 The Purpose of This Chapter......Page 400
21.2 Equivalent Affine Models......Page 402
21.3 The Expression for Convexity in an Affine Setting......Page 403
21.4 An Expression for the Theoretical Convexity of the Portfolio......Page 406
21.5.1 Theoretical Portfolio Convexity as a Function of Forward Rates......Page 409
21.5.2 The Portfolio Time Decay as a Function of ‘Carry’ and ‘Roll-Down’......Page 410
21.6 What These Results Imply......Page 412
21.7 Linking the Term 21 Tr[STDS] with Yield Volatilities......Page 413
21.8 Making the Weights (Almost) Model Independent......Page 415
21.9 How General Are the Results?......Page 417
21.10 Model-Based or Empirical?......Page 418
22.1 The Purpose of This Chapter......Page 420
22.2 The Strategy: A Reminder......Page 422
22.3.1 Determining the Optimal Weights......Page 424
22.3.2 Estimating the Yield Volatilities......Page 425
22.4.1 Is the Yield Curve Fairly Curved?......Page 427
22.4.2 Why Are the Strategies Not Always Profitable?......Page 431
22.4.3 Is the Strength of the Signal Correlated with the Money Made?......Page 434
22.4.4 Explaining the Residuals – Residual Exposure?......Page 435
22.4.5 Explaining the Residuals – Wrong Volatility Estimate?......Page 438
22.5 Conclusions......Page 440
Part VI Excess Returns......Page 442
23.2 The (Local) Expectation Hypothesis......Page 444
23.3 What One Really Tests for When One Tests the (L)EH......Page 446
23.4.1 General Exact Results......Page 448
23.4.2 Special Cases......Page 449
23.4.3 Approximate Results for the τ = n = 1 Case......Page 450
23.6 Excess Returns with Real Rates......Page 451
23.7 Excess Returns: Links with Carry, Roll-Down and Related Market Lore......Page 454
23.8 Why ‘Carry’ and ‘Roll-Down’ Matter......Page 458
24.1 The Purpose of This Chapter......Page 460
24.2.2 The ‘Forwards-Come-True’ Condition......Page 461
24.3.1 Market Yields versus Expected Yields......Page 462
24.4 The Link between Excess Returns and Term Premia......Page 465
24.5 The Link between Term Premia and Expected Excess Returns......Page 467
24.6 Reconciling Results......Page 469
24.7 Expected versus Realized Excess Returns......Page 470
24.8 Forwards-Come-True versus Yields-Don’t-Move: Roll-Down Again......Page 475
24.9 When to Invest......Page 477
25.1 The Purpose of This Chapter......Page 478
25.2.1 The Empirical Questions......Page 479
25.2.2 A Very Important Caveat on Spanning......Page 481
25.2.3 The Methodological Dilemma......Page 483
25.3 Unconditional Results: Nominal Bonds......Page 484
25.4.1 1- to 10-Year Returns......Page 486
25.4.2 Effectiveness of Various Regressors......Page 488
25.4.3 5-Year Returns: Comparison with Cochrane–Piazzesi (2005)......Page 490
25.5 Where Has the Volatility Gone?......Page 491
25.6 Regression Results: Real Bonds......Page 492
25.8 The Real Excess Returns......Page 493
25.9 Extracting the Real-Rate Risk Premium......Page 497
25.10 Estimating the Inflation Premium in Nominal Bonds......Page 499
25.10.1 Isolating the Liquidity Component......Page 500
26.1 The Purpose of This Chapter......Page 502
26.2 The Early Work......Page 503
26.3 Cochrane and Piazzesi (2005)......Page 504
26.5 Robustness of the Tent Shape: Tents versus Bats......Page 507
26.6 The Link Between the Tent and the Bat Factors: Constrained Regressions......Page 511
26.6.1 Constrained Regression: The Investigation Methodology......Page 512
26.6.2 Constrained Regression: Results......Page 514
26.6.3 Constrained Regression: First Conclusions......Page 516
26.7.1 Tent versus Slope Shape Similarity: The Investigation Methodology......Page 517
26.7.2 Tent versus Slope Shape Similarity: Results......Page 519
26.8.1 Tent versus Slope Robustness: Methodology......Page 520
26.8.2 Tent versus Slope Robustness: Results......Page 522
26.8.3 Tent versus Slope Robustness: Conclusions......Page 524
27.1 The Purpose of This Chapter......Page 526
27.2.1 Features and Highlights of the Radwanski Results......Page 527
27.2.2 The Methodology and Results......Page 528
27.2.3 Comments and Conclusions......Page 532
27.3.1 Main Results......Page 533
27.3.2 The Spanning of Yield Curve Factors Revisited: Implication for Affine Models......Page 536
27.4 Yield-Curve Spanning: Why One May Need Five Factors After All......Page 537
27.4.1 The Essentially Affine Description......Page 538
27.4.3 Augmenting the State Vector......Page 540
27.4.4 Switching Back to an ‘Augmented’ Set of Yields as State Variables......Page 541
27.4.6 Spanning in Principle versus Spanning in Practice......Page 542
27.5.1 The Set-Up and Main Features......Page 543
27.5.2 The Investigation Methodology and Results......Page 544
27.5.3 The Link with Forward-Rate-Based RPFs......Page 548
27.5.4 Intrinsic Limitations of Forward-Rate-Based Factors......Page 549
27.5.6 Re-Interpretation of the Cieslak–Povala RPF: Conditional Slope and Level......Page 551
27.6 Related Work......Page 554
28.1 The Purpose of This Chapter......Page 556
28.2 What Does Not Qualify as an Explanation......Page 557
28.3 Excess Returns, the Slope and the Real Economy......Page 558
28.4 The Data-Generating, Subjective and Risk-Neutral Measures......Page 560
28.5 Does It Matter?......Page 562
28.6 Why Is the Slope Significant? A Heterogeneous-Expectations Model......Page 563
28.7 Why Is the Slope Significant? An Over-reaction Model......Page 567
28.8.1 The Actions of the Central Bank......Page 568
28.8.3 The Bond Price Formation......Page 570
28.8.5 The Simulations......Page 571
28.8.6 Summary of Results......Page 575
29.2 What Is the Spanning Problem?......Page 576
29.3 The Empirical Spanning Problem......Page 577
29.4 The Theoretical Spanning Problem......Page 580
29.5 The Modelling Choices to Handle the Spanning Problem......Page 581
Part VII What the Models Tell Us......Page 586
30.1 The Purpose of This Chapter......Page 588
30.2 The Doubly Mean-Reverting Vasicek Model......Page 589
30.3 Bond Prices and Properties of the Solution......Page 590
30.4 The Volatility of the Instantaneous Forward Rate......Page 591
30.5 The Building Blocks......Page 593
30.6 Initial Conclusions......Page 597
30.7.1 Calibrating the Model to the Volatility Structure......Page 598
30.7.2 Calibrating the Model to the Yield Curve......Page 600
30.8 The Value of Convexity......Page 602
30.9 What Happened to the P-Measure?......Page 603
31.1 The Purpose of This Chapter......Page 604
31.2 Empirical Findings about Inflation......Page 605
31.3 The No-Arbitrage Relationships......Page 606
31.3.1 What the No-Arbitrage Relationships Really Imply......Page 608
31.4 The Assumptions About the Process for the State Variables......Page 610
31.5 Inflation Expectations and Risk Premia......Page 611
31.6 Adding Liquidity......Page 612
31.7 The Parameter Estimation Procedure and Results......Page 613
31.7.1 The Difficulty of Parameter Estimation......Page 615
31.8.1 Full-Sample Analysis......Page 617
31.8.2 Prediction of Nominal and Real Excess Returns......Page 618
31.8.3 Analysis of the May–September 2013 Events......Page 623
31.9 Conclusions......Page 625
31.10 Related Work......Page 628
32.1 The Purpose of This Chapter......Page 631
32.2 Turning a Snapshot Model into a Dynamic Model......Page 632
32.3 Turning a Dynamic Model into a No-Arbitrage Affine Model......Page 635
32.4 Are the Variables Really Principal Components?......Page 639
32.5.1 On-the-Run, Off-the-Run Bonds......Page 641
32.5.2 The Modelling Approach......Page 642
32.5.3 The Results......Page 644
32.5.4 Conclusions......Page 645
33.1 The Purpose of This Chapter......Page 647
33.2 Why PC-Based Models Are Special (Again)......Page 649
33.3 Specified-Variable Models Revisited......Page 651
33.3.1 Parameter Constraints for PCA Prespecified Models......Page 653
33.4 Our Strategy to Link the P- and Q-Measures......Page 654
33.5.2 The Geometry (Kinematics) of the Problem......Page 655
33.5.3 The Dynamics of the Problem......Page 656
33.5.4 Solution......Page 657
33.5.5 Necessary Conditions for Identifiability......Page 658
33.6.1 Impossibility of Identification When K Is Diagonal......Page 660
33.6.2 What Does It Mean to Require that the Factors −→xt Should Be Principal Components?......Page 661
33.6.3 Constraints on K for Identifiability......Page 662
33.6.4 What the Q-measure Reversion-Speed Matrix Affects......Page 664
33.7 Moving from the Q- to the P-Measure......Page 668
33.8 Estimating the Parameters of .........Page 670
33.9.1 Cross-Sectional Fit to Yields......Page 672
33.10 Calibration Results......Page 673
33.11 Generalizable Results on Term Premia from a PC-Based Affine Model......Page 678
33.12 The Existential Dilemma......Page 683
33.13.1 Preliminaries......Page 686
33.13.2 Some Ancillary Results......Page 687
33.13.3 The Derivation of the Main Result......Page 688
33.13.4 The Conditions on the Vector .........Page 689
33.14 Appendix 33B: Switching Regressors......Page 690
34.1 The Purpose of This Chapter......Page 692
34.2 The Strategy Behind the Adrian–Crump–Moench Model......Page 693
34.3 A High-Level Description of the Model......Page 694
34.4 State-Price Deflators: Generalizing the Results......Page 696
34.5 Establishing an Expression for the Excess Returns......Page 700
34.6 The Estimation Procedure......Page 705
34.7 Establishing a Link with the Affine Model: The Discount Factor......Page 706
34.8 Some Observations......Page 708
34.9.1 Full-Sample Analysis......Page 709
34.9.2 Analysis of the May–September 2013 Events......Page 712
34.10 Conclusions......Page 716
35.1 The Purpose of This Chapter......Page 717
35.2 Why Do We Need Another Affine Model?......Page 718
35.3 Another Justification for a Stochastic-Market-Price-of-Risk Model......Page 720
35.4 The Model......Page 722
35.5 In Which Measure(s) Are We Working?......Page 723
35.6 The Qualitative Behaviour of the Model......Page 725
35.7 Calibration of the Model......Page 727
35.8 Calibration Results......Page 729
35.9 Comments on the Solution......Page 734
35.10 Term Premia in the Stochastic-Market-Price-of-Risk Model......Page 737
36.1.1 The Road Followed......Page 743
36.1.2 The Case for the Prosecution: Models As Regurgitators......Page 744
36.1.3 The Case for the Defence: Models as Enforcers of Parsimony......Page 745
36.1.4 The Case for the Defence: Models as Enforcers of Cross-Sectional Restrictions......Page 747
36.1.5 The Case for the Defence: Models as Revealers of Forward-Looking Informations......Page 748
36.1.6 The Case for the Defence: Models as Integrators......Page 749
36.1.7 The Case for the Defence: Models as Enhancers of Understanding......Page 750
References......Page 754
Index......Page 766