Dynamic Term Structure Modeling The Fixed Income Valuation Course & CD-ROM

by Nawalkha, Sanjay K.; Soto, Gloria M.; Beliaeva, Natalia A.

• ISBN13: 9780471737148
• ISBN10: 0471737143
• Edition: 1st
• Format: Hardcover
• Copyright: 2007-06-04
• Publisher: Wiley

Summary

Praise for Dynamic Term Structure Modeling "This book offers the most comprehensive coverage of term-structure models I have seen so far, encompassing equilibrium and no-arbitrage models in a new framework, along with the major solution techniques using trees, PDE methods, Fourier methods, and approximations. It is an essential reference for academics and practitioners alike." -Sanjiv Ranjan Das Professor of Finance, Santa Clara University, California, coeditor, Journal of Derivatives "Bravo! This is an exhaustive analysis of the yield curve dynamics. It is clear, pedagogically impressive, well presented, and to the point." -Nassim Nicholas Taleb author, Dynamic Hedging and The Black Swan "Nawalkha, Beliaeva, and Soto have put together a comprehensive, up-to-date textbook on modern dynamic term structure modeling. It is both accessible and rigorous and should be of tremendous interest to anyone who wants to learn about state-of-the-art fixed income modeling. It provides many numerical examples that will be valuable to readers interested in the practical implementations of these models." -Pierre Collin-Dufresne Associate Professor of Finance, UC Berkeley "The book provides a comprehensive description of the continuous time interest rate models. It serves an important part of the trilogy, useful for financial engineers to grasp the theoretical underpinnings and the practical implementation." -Thomas S. Y. Ho, PHD President, Thomas Ho Company, Ltd, coauthor, The Oxford Guide to Financial Modeling

Author Biography

Sanjay K. Nawalkha, PhD, is an Associate Professor of Finance at the Isenberg School of Management, University of Massachusetts Amherst, where he teaches graduate courses in finance theory and fixed income. He has published extensively in academic and practitioner journals, and is the President and founder of Nawalkha and Associates—a fixed income training and consulting firm.

Natalia A.Beliaeva, PhD, is an Assistant Professor of Finance at the Sawyer Business School, Suffolk University, Boston. She also holds a master's degree in computer science (artificial intelligence) from the University of Massachusetts Amherst. Dr. Beliaeva's expertise is in the area of applied numerical methods for pricing fixed income derivatives.

Gloria M.Soto, PhD, is a Professor of Applied Economics and Finance at the University of Murcia, Spain, where she teaches courses in financial markets and institutions and applied economics. Dr. Soto has published extensively in both Spanish and international journals in finance and economics, especially in the areas of interest rate risk management and related fixed income topics.

Table of Contents

List of Figures	p. xxxi
List of Tables	p. xxxv
A Simple Introduction to Continuous-Time Stochastic Processes	p. 1
Continuous-Time Diffusion Processes	p. 3
Wiener Process	p. 3
Ito Process	p. 5
Ito's Lemma	p. 7
Simple Rules of Stochastic Differentiation and Integration	p. 9
Obtaining Unconditional Mean and Variance of Stochastic Integrals under Gaussian Processes	p. 9
Examples of Gaussian Stochastic Integrals	p. 11
Mixed Jump-Diffusion Processes	p. 14
The Jump-Diffusion Process	p. 14
Ito's Lemma for the Jump-Diffusion Process	p. 15
Arbitrage-Free Valuation	p. 17
Arbitrage-Free Valuation: Some Basic Results	p. 18
A Simple Relationship between Zero-Coupon Bond Prices and Arrow Debreu Prices	p. 20
The Bayes Rule for Conditional Probabilities of Events	p. 20
The Relationship between Current and Future AD Prices	p. 21
The Relationship between Cross-Sectional AD Prices and Intertemporal Term Structure Dynamics	p. 22
Existence of the Risk-Neutral Probability Measure	p. 23
Stochastic Discount Factor	p. 28
Radon-Nikodym Derivative	p. 30
Arbitrage-Free Valuation in Continuous Time	p. 31
Change of Probability Measure under a Continuous Probability Density	p. 32
The Girsanov Theorem and the Radon-Nikodym Derivative	p. 34
Equivalent Martingale Measures	p. 35
Stochastic Discount Factor and Risk Premiums	p. 43
The Feynman-Kac Theorem	p. 43
Valuing Interest Rate and Credit Derivatives: Basic Pricing Frameworks	p. 49
Eurodollar and Other Time Deposit Futures	p. 54
Valuing Futures on a Time Deposit	p. 58
Convexity Bias	p. 60
Treasury Bill Futures	p. 61
Valuing T-Bill Futures	p. 62
Convexity Bias	p. 63
Treasury Bond Futures	p. 64
Conversion Factor	p. 65
Cheapest-to-Deliver Bond	p. 67
Options Embedded in T-Bond Futures	p. 68
Valuing T-Bond Futures	p. 68
Treasury Note Futures	p. 72
Forward Rate Agreements	p. 73
Interest Rate Swaps	p. 74
Day-Count Conventions	p. 76
The Financial Intermediary	p. 77
Motivations for Interest Rate Swaps	p. 78
Pricing Interest Rate Swaps	p. 82
Interest Rate Swaptions	p. 85
Caps and Floors	p. 88
Caplet	p. 90
Floorlet	p. 91
Collarlet	p. 92
Caps, Floors, and Collars	p. 92
Black Implied Volatilities for Caps and Swaptions	p. 93
Black Implied Volatilities: Swaptions	p. 95
Black Implied Volatilities: Caplet	p. 96
Black Implied Volatilities: Caps	p. 97
Black Implied Volatilities: Difference Caps	p. 98
Pricing Credit Derivatives Using the Reduced-Form Approach	p. 98
Default Intensity and Survival Probability	p. 100
Recovery Assumptions	p. 101
Risk-Neutral Valuation under the RMV Assumption	p. 102
Risk-Neutral Valuation under the RFV Assumption	p. 103
Valuing Credit Default Swaps Using the RFV Assumption	p. 104
A New Taxonomy of Term Structure Models	p. 106
Fundamental and Preference-Free Single-Factor Gaussian Models	p. 113
The Arbitrage-Free Pricing Framework of Vasicek	p. 115
The Term Structure Equation	p. 116
Risk-Neutral Valuation	p. 118
The Fundamental Vasicek Model	p. 120
Bond Price Solution	p. 124
Preference-Free Vasicek+, Vasicek++, and Vasicek+++ Models	p. 128
The Vasicek+ Model	p. 128
The Vasicek++, or the Extended Vasicek Model	p. 136
The Vasicek+++, or the Fully Extended Vasicek Model	p. 140
Valuing Futures	p. 144
Valuing Futures under the Vasicek, Vasicek+ and Vasicek++ Models	p. 145
Valuing Futures under the Vasicek+++ Model	p. 150
Valuing Options	p. 153
Options on Zero-Coupon Bonds	p. 153
Options on Coupon Bonds	p. 157
Valuing Interest Rate Contingent Claims Using Trees	p. 161
Binomial Trees	p. 163
Trinomial Trees	p. 165
Trinomial Tree under the Vasicek++ Model: An Example	p. 171
Trinomial Tree under the Vasicek+++ Model: An Example	p. 178
Bond Price Solution Using the Risk-Neutral Valuation Approach under the Fundamental Vasicek Model and the Preference-Free Vasicek+ Model	p. 181
Hull's Approximation to Convexity Bias under the Ho and Lee Model	p. 184
Fundamental and Preference-Free Jump-Extended Gaussian Models	p. 187
Fundamental Vasicek-GJ Model	p. 188
Bond Price Solution	p. 191
Jump-Diffusion Tree	p. 194
Preference-Free Vasicek-GJ+ and Vasicek-GJ++ Models	p. 201
The Vasicek-GJ+ Model	p. 202
The Vasicek-GJ++ Model	p. 203
Jump-Diffusion Tree	p. 205
Fundamental Vasicek-EJ Model	p. 206
Bond Price Solution	p. 207
Jump-Diffusion Tree	p. 209
Preference-Free Vasicek-EJ++ Model	p. 216
Jump-Diffusion Tree	p. 218
Valuing Futures and Options	p. 218
Valuing Futures	p. 219
Valuing Options: The Fourier Inversion Method	p. 222
Probability Transformations with a Damping Constant	p. 233
The Fundamental Cox, Ingersoll, and Ross Model with Exponential and Lognormal Jumps	p. 237
The Fundamental Cox, Ingersoll, and Ross Model	p. 239
Solution to Riccati Equation with Constant Coefficients	p. 242
CIR Bond Price Solution	p. 243
General Specifications of Market Prices of Risk	p. 244
Valuing Futures	p. 245
Valuing Options	p. 248
Interest Rate Trees for the Cox, Ingersoll, and Ross Model	p. 250
Binomial Tree for the CIR Model	p. 250
Trinomial Tree for the CIR Model	p. 263
Pricing Bond Options and Interest Rate Options with Trinomial Trees	p. 273
The CIR Model Extended with Jumps	p. 279
Valuing Futures	p. 283
Futures on a Time Deposit	p. 284
Valuing Options	p. 285
Jump-Diffusion Trees for the CIR Model Extended with Jumps	p. 287
Exponential Jumps	p. 287
Lognormal Jumps	p. 295
Preference-Free CIR and CEV Models with Jumps	p. 305
Mean-Calibrated CIR Model	p. 307
Preference-Free CIR+ and CIR++ Models	p. 309
A Common Notational Framework	p. 312
Probability Density and the Unconditional Moments	p. 313
Bond Price Solution	p. 315
Expected Bond Returns	p. 317
Constant Infinite-Maturity Forward Rate under Explosive CIR+ and CIR++ Models	p. 318
A Comparison with Other Markovian Preference-Free Models	p. 321
Calibration to the Market Prices of Bonds and Interest Rate Derivatives	p. 322
Valuing Futures	p. 323
Valuing Options	p. 325
Interest Rate Trees	p. 327
The CIR+ and CIR++ Models Extended with Jumps	p. 328
Preference-Free CIR-EJ+ and CIR-EJ++ Models	p. 329
Jump-Diffusion Trees	p. 331
Fundamental and Preference-Free Constant-Elasticity-of-Variance Models	p. 331
Forward Rate and Bond Return Volatilities under the CEV++ Models	p. 333
Valuing Interest Rate Derivatives Using Trinomial Trees	p. 336
Fundamental and Preference-Free Constant-Elasticity-of-Variance Models with Lognormal Jumps	p. 341
Fundamental and Preference-Free Two-Factor Affine Models	p. 345
Two-Factor Gaussian Models	p. 348
The Canonical, or the Ac, Form: The Dai and Singleton [2002] Approach	p. 349
The Ar Form: The Hull and White [1996] Approach	p. 353
The Ay Form: The Brigo and Mercuric [2001, 2006] Approach	p. 356
Relationship between the A[subscript 0c](2)++ Model and the A[subscript 0y](2)++ Model	p. 358
Relationship between the A[subscript 0r](2)++ Model and the A[subscript 0y](2)++ Model	p. 360
Bond Price Process and Forward Rate Process	p. 361
Probability Density of the Short Rate	p. 362
Valuing Options	p. 363
Two-Factor Gaussian Trees	p. 364
Two-Factor Hybrid Models	p. 373
Bond Price Process and Forward Rate Process	p. 377
Valuing Futures	p. 377
Valuing Options	p. 380
Two-Factor Stochastic Volatility Trees	p. 382
Two-Factor Square-Root Models	p. 393
The Ay Form	p. 393
The Ar Form	p. 399
Relationship between the Canonical Form and the Ar Form	p. 402
Two-Factor "Square-Root" Trees	p. 403
Hull and White Solution of [eta](t, T)	p. 410
Fundamental and Preference-Free Multifactor Affine Models	p. 413
Three-Factor Affine Term Structure Models	p. 416
The A[subscript 1r](3), A[subscript 1r](3)+, and A[subscript 1r](3)++ Models	p. 416
The A[subscript 2r](3), A[subscript 2r](3)+, and A[subscript 2r](3)++ Models	p. 421
Simple Multifactor Affine Models with Analytical Solutions	p. 425
The Simple A[subscript M](N) Models	p. 425
The Simple A[subscript M](N)+ and A[subscript M](N)++ Models	p. 427
The Nested ATSMs	p. 429
Valuing Futures	p. 429
Valuing Options on Zero-Coupon Bonds or Caplets: The Fourier Inversion Method	p. 433
Valuing Options on Coupon Bonds or Swaptions: The Cumulant Expansion Approximation	p. 435
Calibration to Interest Rate Caps Data	p. 448
Unspanned Stochastic Volatility	p. 455
Multifactor ATSMs for Pricing Credit Derivatives	p. 457
Simple Reduced-Form ATSMs under the RMV Assumption	p. 458
Simple Reduced-Form ATSMs under the RFV Assumption	p. 468
The Solution of [eta](t, T, [phiv]) for CDS Pricing Using Simple A[subscript M](N) Models under the RFV Assumption	p. 476
Stochastic Volatility Jump-Based Mixed-Sign A[subscript N](N)-EJ++ Model and A[subscript 1](3)-EJ++ Model	p. 477
The Mixed-Sign A[subscript N](N)-EJ++ Model	p. 478
The A[subscript 1](3)-EJ++Model	p. 479
Fundamental and Preference-Free Quadratic Models	p. 483
Single-Factor Quadratic Term Structure Model	p. 484
Duration and Convexity	p. 488
Preference-Free Single-Factor Quadratic Model	p. 492
Forward Rate Volatility	p. 495
Model Implementation Using Trees	p. 497
Extension to Jumps	p. 498
Fundamental Multifactor QTSMs	p. 501
Bond Price Formulas under Q[subscript 3](N) and Q[subscript 4](N) Models	p. 505
Parameter Estimates	p. 506
Preference-Free Multifactor QTSMs	p. 508
Forward Rate Volatility and Correlation Matrix	p. 515
Valuing Futures	p. 518
Valuing Options on Zero-Coupon Bonds or Caplets: The Fourier Inversion Method	p. 524
Valuing Options on Coupon Bonds or Swaptions: The Cumulant Expansion Approximation	p. 527
Calibration to Interest Rate Caps Data	p. 531
Multifactor QTSMs for Valuing Credit Derivatives	p. 537
Reduced-Form Q[subscript 3](N), Q[subscript 3](N)+, and Q[subscript 3](N)++ Models under the RMV Assumption	p. 537
Reduced-Form Q[subscript 3](N) and Q[subscript 3](N)+ Models under the RFV Assumption	p. 543
The Solution of [eta](t, T, [phiv]) for CDS Pricing Using the Q[subscript 3](N) Model under the RFV Assumption	p. 547
The HJM Forward Rate Model	p. 551
The HJM Forward Rate Model	p. 552
Numerical Implementation Using Nonrecombining Trees	p. 556
A One-Factor Nonrecombining Binomial Tree	p. 557
A Two-Factor Nonrecombining Trinomial Tree	p. 565
Recursive Programming	p. 569
A Recombining Tree for the Proportional Volatility HJM Model	p. 572
Forward Price Dynamics under the Forward Measure	p. 573
A Markovian Forward Price Process under the Proportional Volatility Model	p. 575
A Recombining Tree for the Proportional Volatility Model Using the Nelson and Ramaswamy Transform	p. 576
The LIBOR Market Model	p. 583
The Lognormal Forward LIBOR Model (LFM)	p. 585
Multifactor LFM under a Single Numeraire	p. 588
The Lognormal Forward Swap Model (LSM)	p. 591
A Joint Framework for Using Black's Formulas for Pricing Caps and Swaptions	p. 595
The Relationship between the Forward Swap Rate and Discrete Forward Rates	p. 596
Approximating the Black Implied Volatility of a Swaption under the LFM	p. 597
Specifying Volatilities and Correlations	p. 600
Forward Rate Volatilities: Some General Results	p. 600
Forward Rate Volatilities: Specific Functional Forms	p. 604
Instantaneous Correlations and Terminal Correlations	p. 608
Full-Rank Instantaneous Correlations	p. 612
Reduced-Rank Correlation Structures	p. 619
Terminal Correlations	p. 623
Explaining the Smile: The First Approach	p. 623
The CEV Extension of the LFM	p. 624
Displaced-Diffusion Extension of the LFM	p. 626
Unspanned Stochastic Volatility Jump Models	p. 629
Joshi and Rebonato [2003] Model	p. 630
Jarrow, Li, and Zhao [2007] Model	p. 631
An Extension of the JLZ Model	p. 636
Empirical Performance of the JLZ [2007] Model	p. 637
Reference	p. 647
About the CD-ROM	p. 658
Index	p. 661
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