Interest rate modeling theory and practice, second edition

Interest Rate
Modeling
Theory and Practice
Second Edition


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Interest Rate Modeling
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Interest Rate
Modeling
Theory and Practice
Second Edition

Lixin Wu


CRC Press
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Library of Congress Cataloging-in-Publication Data
Names: Wu, Lixin, 1961- author.
Title: Interest rate modeling : theory and practice / Lixin Wu.
Description: 2nd edition. | Boca Raton, Florida : CRC Press, [2019] |
Includes bibliographical references and index.
Identifiers: LCCN 2018050904| ISBN 9780815378914 (hardback : alk.
paper)| ISBN 9781351227421 (ebook : alk. paper)
Subjects: LCSH: Interest rates--Mathematical models. | Interest rate
futures--Mathematical models.
Classification: LCC HG6024.5 .W82 2019 | DDC 332.801/5195--dc23
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To my parents,
To Molly,
Dorothy and Derek



Contents

Preface to the First Edition

xv

Preface to the Second Edition

xix

Acknowledgments to the Second Edition

xxi

Author

xxiii

1 The Basics of Stochastic Calculus
1.1

1.2
1.3
1.4

1.5

Brownian Motion . . . . . . . . . . . . . . .
1.1.1 Simple Random Walks . . . . . . . . .
1.1.2 Brownian Motion . . . . . . . . . . . .
1.1.3 Adaptive and Non-Adaptive Functions
Stochastic Integrals . . . . . . . . . . . . . .
1.2.1 Evaluation of Stochastic Integrals . . .
Stochastic Differentials and Ito’s Lemma . .
Multi-Factor Extensions . . . . . . . . . . . .
1.4.1 Multi-Factor Ito’s Process . . . . . . .
1.4.2 Ito’s Lemma . . . . . . . . . . . . . .
1.4.3 Correlated Brownian Motions . . . . .
1.4.4 The Multi-Factor Lognormal Model .
Martingales . . . . . . . . . . . . . . . . . . .

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2 The Martingale Representation Theorem
2.1

2.2

2.3

2.4
2.5
2.6

Changing Measures with Binomial Models . . . . . . . . .
2.1.1 A Motivating Example . . . . . . . . . . . . . . . . .
2.1.2 Binomial Trees and Path Probabilities . . . . . . . .
Change of Measures under Brownian Filtration . . . . . . .
2.2.1 The Radon–Nikodym Derivative of a Brownian Path
2.2.2 The CMG Theorem . . . . . . . . . . . . . . . . . .
The Martingale Representation Theorem . . . . . . . . . .
A Complete Market with Two Securities . . . . . . . . . .
Replicating and Pricing of Contingent Claims . . . . . . .
Multi-Factor Extensions . . . . . . . . . . . . . . . . . . . .

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viii

Contents
2.7

2.8

2.9

A Complete Market with Multiple Securities
2.7.1 Existence of a Martingale Measure . .
2.7.2 Pricing Contingent Claims . . . . . . .
The Black–Scholes Formula . . . . . . . . . .
Notes . . . . . . . . . . . . . . . . . . . . . .

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3 Interest Rates and Bonds
3.1

3.2

3.3

3.4
3.5

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Interest Rates and Fixed-Income Instruments . . . . . .
3.1.1 Short Rate and Money Market Accounts . . . . .
3.1.2 Term Rates and Certificates of Deposit . . . . .
3.1.3 Bonds and Bond Markets . . . . . . . . . . . . .
3.1.4 Quotation and Interest Accrual . . . . . . . . . .
Yields . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Yield to Maturity . . . . . . . . . . . . . . . . .
3.2.2 Par Bonds, Par Yields, and the Par Yield Curve
3.2.3 Yield Curves for U.S. Treasuries . . . . . . . . .
Zero-Coupon Bonds and Zero-Coupon Yields . . . . . .
3.3.1 Zero-Coupon Bonds . . . . . . . . . . . . . . . .
3.3.2 Bootstrapping the Zero-Coupon Yields . . . . . .
3.3.2.1 Future Value and Present Value . . . .
Forward Rates and Forward-Rate Agreements . . . . .
Yield-Based Bond Risk Management . . . . . . . . . .
3.5.1 Duration and Convexity . . . . . . . . . . . . . .

3.5.2 Portfolio Risk Management . . . . . . . . . . . .

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4 The Heath–Jarrow–Morton Model
4.1
4.2
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Lognormal Model: The Starting Point . . . . . . . . .
The HJM Model . . . . . . . . . . . . . . . . . . . . .
Special Cases of the HJM Model . . . . . . . . . . . .
4.3.1 The Ho–Lee Model . . . . . . . . . . . . . . . .
4.3.2 The Hull–White (or Extended Vasicek) Model
4.4 Estimating the HJM Model from Yield Data . . . . .
4.4.1 From a Yield Curve to a Forward-Rate Curve .
4.4.2 Principal Component Analysis . . . . . . . . .
4.5 A Case Study with a Two-Factor Model . . . . . . . .
4.6 Monte Carlo Implementations . . . . . . . . . . . . .
4.7 Forward Prices . . . . . . . . . . . . . . . . . . . . . .
4.8 Forward Measure . . . . . . . . . . . . . . . . . . . .
4.9 Black’s Formula for Call and Put Options . . . . . . .
4.9.1 Equity Options under the Hull–White Model .
4.9.2 Options on Coupon Bonds . . . . . . . . . . . .
4.10 Numeraires and Changes of Measure . . . . . . . . . .

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109


Contents

ix

4.11 Linear Gaussian Models . . . . . . . . . . . . . . . . . . . . .
4.12 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5 Short-Rate Models and Lattice Implementation
5.1
5.2

5.3

5.4

119

From Short-Rate Models to Forward-Rate Models .
General Markovian Models . . . . . . . . . . . . . .
5.2.1 One-Factor Models . . . . . . . . . . . . . . .
5.2.2 Monte Carlo Simulations for Options Pricing
Binomial Trees of Interest Rates . . . . . . . . . . .

5.3.1 A Binomial Tree for the Ho–Lee Model . . .
5.3.2 Arrow–Debreu Prices . . . . . . . . . . . . .
5.3.3 A Calibrated Tree for the Ho–Lee Model . . .
A General Tree-Building Procedure . . . . . . . . .
5.4.1 A Truncated Tree for the Hull–White Model
5.4.2 Trinomial Trees with Adaptive Time Steps .
5.4.3 The Black–Karasinski Model . . . . . . . . .

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6 The LIBOR Market Model
6.1

6.2
6.3
6.4
6.5
6.6

6.7

LIBOR Market Instruments . . . . . . . .
6.1.1 LIBOR Rates . . . . . . . . . . . . .
6.1.2 Forward-Rate Agreements . . . . . .
6.1.3 Repurchasing Agreement . . . . . .
6.1.4 Eurodollar Futures . . . . . . . . . .
6.1.5 Floating-Rate Notes . . . . . . . . .

6.1.6 Swaps . . . . . . . . . . . . . . . . .
6.1.7 Caps . . . . . . . . . . . . . . . . . .
6.1.8 Swaptions . . . . . . . . . . . . . . .
6.1.9 Bermudan Swaptions . . . . . . . . .
6.1.10 LIBOR Exotics . . . . . . . . . . . .
The LIBOR Market Model . . . . . . . . .
Pricing of Caps and Floors . . . . . . . . .
Pricing of Swaptions . . . . . . . . . . . . .
Specifications of the LIBOR Market Model
Monte Carlo Simulation Method . . . . . .
6.6.1 The Log–Euler Scheme . . . . . . .
6.6.2 Calculation of the Greeks . . . . . .
6.6.3 Early Exercise . . . . . . . . . . . .
Notes . . . . . . . . . . . . . . . . . . . . .

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7 Calibration of LIBOR Market Model

7.1
7.2

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Implied Cap and Caplet Volatilities . . . . . . . . . . . . . .
Calibrating the LIBOR Market Model to Caps . . . . . . . .

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192


x

Contents
7.3
7.4

7.5

Calibration to Caps, Swaptions, and Input
Correlations . . . . . . . . . . . . . . . . . . .
Calibration Methodologies . . . . . . . . . . .
7.4.1 Rank-Reduction Algorithm . . . . . . .
7.4.2 The Eigenvalue Problem for Calibrating
to Input Prices . . . . . . . . . . . . . .
Sensitivity with Respect to the Input Prices .

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8 Volatility and Correlation Adjustments
8.1

8.2

8.3
8.4
8.5

225

Adjustment due to Correlations . . . . . . . . . . . . . .
8.1.1 Futures Price versus Forward Price . . . . . . . . .
8.1.2 Convexity Adjustment for LIBOR Rates . . . . . .
8.1.3 Convexity Adjustment under the Ho–Lee Model .
8.1.4 An Example of Arbitrage . . . . . . . . . . . . . .
Adjustment due to Convexity . . . . . . . . . . . . . . .
8.2.1 Payment in Arrears versus Payment in Advance .
8.2.2 Geometric Explanation for Convexity Adjustment
8.2.3 General Theory of Convexity Adjustment . . . . .
8.2.4 Convexity Adjustment for CMS and CMT Swaps .
Timing Adjustment . . . . . . . . . . . . . . . . . . . . .
Quanto Derivatives . . . . . . . . . . . . . . . . . . . . .
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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9 Affine Term Structure Models
9.1
9.2

9.3
9.4
9.5

9.6
9.7

253

An Exposition with One-Factor Models . . . . . .
Analytical Solution of Riccarti Equations . . . . .
Pricing Options on Coupon Bonds . . . . . . . . .
Distributional Properties of Square-Root Processes
Multi-Factor Models . . . . . . . . . . . . . . . . .
9.5.1 Admissible ATSMs . . . . . . . . . . . . . .
9.5.2 Three-Factor ATSMs . . . . . . . . . . . . .
Swaption Pricing under ATSMs . . . . . . . . . .
Notes . . . . . . . . . . . . . . . . . . . . . . . . .

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10 Market Models with Stochastic Volatilities
10.1
10.2
10.3
10.4
10.5
10.6

SABR Model . . . . . . .
The Wu and Zhang (2001)
Pricing of Caplets . . . .

Pricing of Swaptions . . .
Model Calibration . . . .
Notes . . . . . . . . . . .

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Contents

xi

11 L´
evy Market Model

315

11.1 Introduction to L´evy Processes . . . . . . . . . . . . . . . .
11.1.1 Infinite Divisibility . . . . . . . . . . . . . . . . . . .
11.1.2 Basic Examples of the L´evy Processes . . . . . . . .
11.1.2.1 Poisson Processes . . . . . . . . . . . . . .
11.1.2.2 Compound Poisson Processes . . . . . . . .
11.1.2.3 Linear Brownian Motion . . . . . . . . . .
11.1.3 Introduction of the Jump Measure . . . . . . . . . .
11.1.4 Characteristic Exponents for General L´evy Processes
11.2 The L´evy HJM Model . . . . . . . . . . . . . . . . . . . . .
11.3 Market Model under L´evy Processes . . . . . . . . . . . . .
11.4 Caplet Pricing . . . . . . . . . . . . . . . . . . . . . . . . .
11.5 Swaption Pricing . . . . . . . . . . . . . . . . . . . . . . . .
11.6 Approximate Swaption Pricing via the Merton
Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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12 Market Model for Inflation Derivatives Modeling
12.1 CPI Index and Inflation Derivatives Market .
12.1.1 TIPS . . . . . . . . . . . . . . . . . . .
12.1.2 ZCIIS . . . . . . . . . . . . . . . . . .

12.1.3 YYIIS . . . . . . . . . . . . . . . . . .
12.1.4 Inflation Caps and Floors . . . . . . .
12.1.5 Inflation Swaptions . . . . . . . . . . .
12.2 Rebuilt Market Model and the New Paradigm
12.2.1 Inflation Discount Bonds and Inflation
Forward Rates . . . . . . . . . . . . .
12.2.2 The Compatibility Condition . . . . .
12.2.3 Rebuilding the Market Model . . . . .
12.2.4 The New Paradigm . . . . . . . . . . .
12.2.5 Unifying the Jarrow-Yildirim Model .
12.3 Pricing Inflation Derivatives . . . . . . . . .
12.3.1 YYIIS . . . . . . . . . . . . . . . . . .
12.3.2 Caps . . . . . . . . . . . . . . . . . . .
12.3.3 Swaptions . . . . . . . . . . . . . . . .
12.4 Model Calibration . . . . . . . . . . . . . . .
12.5 Smile Modeling . . . . . . . . . . . . . . . .
12.6 Notes . . . . . . . . . . . . . . . . . . . . . .

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349
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13 Market Model for Credit Derivatives
13.1 Pricing of Risky Bonds: A New Perspective . . . . . . . . . .
13.2 Forward Spreads . . . . . . . . . . . . . . . . . . . . . . . . .

363
365
367


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Contents
13.3
13.4
13.5
13.6

Two Kinds of Default Protection Swaps . . . . . .
Par CDS Rates . . . . . . . . . . . . . . . . . . . .
Implied Survival Curve and Recovery-Rate Curve
Credit Default Swaptions and an Extended Market
Model . . . . . . . . . . . . . . . . . . . . . . . . .
13.7 Pricing of CDO Tranches under the Market Model
13.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . .

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373

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378
384
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14 Dual-Curve SABR-LMM Market Model for Post-Crisis
Interest Rate Derivatives Markets
14.1 LIBOR Market Model under Default Risks . . . . . . . . . .
14.2 Swaps and Basis Swaps . . . . . . . . . . . . . . . . . . . . .
14.3 Option Pricing Using Heat Kernel Expansion . . . . . . . . .
14.3.1 Derivation of the Heat Kernel . . . . . . . . . . . . . .
14.3.1.1 General Heat Kernel Expansion Formulae . .
14.3.1.2 Heat Kernel Expansion for the Dual-Curve
SABR-LMM Model . . . . . . . . . . . . . .
14.3.2 Calculating the Volatility for Local Volatility Model .
14.3.2.1 Calculation of the Local Volatility Function .
14.3.2.2 Calculation of the Saddle Point . . . . . . . .
14.3.3 Calculation of the Implied Black’s Volatility . . . . . .
14.3.4 Numerical Results for 3M Caplets . . . . . . . . . . .
14.4 Pricing 3M Swaptions . . . . . . . . . . . . . . . . . . . . . .
14.4.1 Dynamics of the State Variables . . . . . . . . . . . .
14.4.1.1 Swap Rate Dynamics under the Forward
Swap Measure . . . . . . . . . . . . . . . . .
14.4.2 Geometric Inputs . . . . . . . . . . . . . . . . . . . . .
14.4.2.1 Inputs Parameter for the Heat Kernel
Expansion . . . . . . . . . . . . . . . . . . .
14.4.3 Local Volatility Function of Swap Rates . . . . . . . .
14.4.4 Calculation of the Saddle Point . . . . . . . . . . . . .
14.4.4.1 Interpolation in High Dimensional Cases . .
14.4.5 Implied Black’s Volatility . . . . . . . . . . . . . . . .
14.4.6 Numerical Results for 3M Swaptions . . . . . . . . . .
14.5 Pricing Caps and Swaptions of Other Tenors . . . . . . . . .
14.5.1 Linkage between 3M Rates and Rates of
Other Tenors . . . . . . . . . . . . . . . . . . . . . . .
14.5.1.1 The 6M Risk-Free OIS Rates . . . . . . . . .

14.5.1.2 The 6M Expected Loss Rates . . . . . . . . .
14.5.2 Dynamics of the 6M Risky LIBOR Rates . . . . . . .
14.5.3 Dynamics of the 6M Swap Rates . . . . . . . . . . . .
14.5.4 Numerical Results of 6M Caplets and Swaptions . . .
14.5.5 Model Calibration . . . . . . . . . . . . . . . . . . . .
14.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Contents

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15 xVA: Definition, Evaluation, and Risk Management
15.1 Pricing through Bilateral Replications . . . . . . .
15.1.1 Margin Accounts, Collaterals, and Capitals
15.1.2 Pricing in the Absence of Funding Cost . .
15.2 The Rise of Other xVA . . . . . . . . . . . . . . .
15.3 Examples . . . . . . . . . . . . . . . . . . . . . . .
15.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . .

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453
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454
459
466
468

References

471

Index

489



Preface to the First Edition

Motivations
This book was motivated by my teaching of the subject to graduate students at the Hong Kong University of Science and Technology (HKUST).
My interest-rate class usually consists of students working toward both research degrees and professional degrees; their interests and levels of mathematical sophistication vary quite a bit. To meet the needs of the students
with diverse backgrounds, I must choose materials that are interesting to
the majority of them, and strike a balance between theory and application when delivering the course materials. These considerations, together
with my own preferences, have shaped a coherent course curriculum that

seems to work well. Given this success, I decided to write a book based on
that curriculum.
Interest-rate modeling has long been at the core of financial derivatives
theory. There are already quite a number of monographs and textbooks on
interest-rate models. It is a good idea to write another book on the subject
only if it will contribute significant added value to the literature. This is why I
thought about this book. This book portrays the theory of interest-rate modeling as a three-dimensional object of finance, mathematics, and computation.
In this book, all models are introduced with financial and economical justifications; options are modeled along the so-called martingale approach; and
option evaluations are handled with fine numerical methods. With this book,
the reader may capture the interdisciplinary nature of the field of interest-rate
(or fixed-income) modeling, and understand what it takes to be a competent
quantitative analyst in today’s market.
The book takes the top-down approach to introducing interest-rate models.
The framework for no-arbitrage models is first established, and then the story
evolves around three representative types of models, namely, the Hull–White
model, the market model, and affine models. Relating individual models to the
arbitrage framework helps to achieve better appreciation of the motivations
behind each model, as well as better understanding of the interconnections
among different models. Note that these three types of models coexist in the
market. The adoption of any of these models or their variants may often be
xv


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Preface to the First Edition

determined by products or sectors rather than by the subjective will. Hence,
a quant must have flexibility in adopting models. The premise, of course, is
a thorough understanding of the models. It is hoped that, through the topdown approach, readers will get a clear picture of the status of this important

subject, without being overwhelmed by too many specific models.
This book can serve as a textbook. Inherited from my lecture notes for
a diverse pool of students, the book is not written in a strict mathematical
style. Were that the case, there would be a lot more lemmas and theorems
in the text. But efforts were indeed made to make the book self-contained
in mathematics, and rigorous justifications are given for almost all results.
There are quite a number of examples in the text; many of them are based
on real market data. Exercise sets are provided for all but one chapter. These
exercises often require computer implementation. Students not only learn the
martingale approach for interest-rate modeling, but they also learn how to implement various models on computers. The adoption of materials is influenced
by my experiences as a consultant and a lecturer for industrial courses. My
early students often noted that materials in the course were directly relevant
to their work in institutions. Those materials are included here.
To a large extent, this book can also serve as a research monograph. It contains many results that are either new or exist only in recent research articles,
including, as only a few examples, adaptive Hull–White lattice trees, market model calibration by quadratic programming, correlation adjustment, and
swaption pricing under affine term structure models. Many of the numerical
methods or schemes are very efficient or even optimized, owing to my original background as a numerical analyst. In addition, notes are given at the
end of most chapters to comment on cutting-edge research, which includes
volatility-smile modeling, convexity adjustment, and so on.

Study Guide
When used as a textbook, this book can be covered in two 14-week
semesters. For a one-semester course, I recommend the coverage of Chapters 1–4, half of Chapter 5 on lattice trees, and Chapter 6. As a reference
book for self-learning, readers should study short-rate models, market models, and affine models in Chapters 5, 6, and 9, respectively. For applications,
Chapter 8 is also very useful. Chapter 7, meanwhile, is special in this book,
as it is particularly prepared for those readers who are interested in market
model calibration.
Chapters 1 and 2 contain the mathematical foundations for interest-rate
modeling, where we introduce Ito’s calculus and the martingale representation
theorem. The presentation of the theories is largely self-contained, except for

Preface to the First Edition

xvii

some omissions in the proof of the martingale representation theorem, which
I think is technically too demanding for this type of book.
In Chapter 3, I introduce bonds and bond yields, which constitute the
underlying securities or quantities of the interest-rate derivatives markets. I
also discuss the composition of bond markets and how they function. For
completeness, I include the classical theory of risk management that is based
on parallel yield changes.
Chapter 4 is a cornerstone of the book, as it introduces the Heath–Jarrow–
Morton (HJM) model, the framework for no-arbitrage pricing models. With
market data, we demonstrate the estimation of the HJM model. Forward
measures, which are important devices for interest-rate options pricing, are
introduced. Change of measures, a very useful technique for option pricing, is
discussed in general.
Chapter 5 consists of two parts: a theoretical part and a numerical part.
The theoretical part focuses on the issue of when the HJM model implies a Markovian short-rate model, and the numerical part is about the
construction and calibration of short-rate lattice models. A very efficient
methodology to construct and calibrate a truncated and adaptive lattice
is presented with the Hull–White model, which, after slight modifications,
is applicable to general Markovian models with the feature of mean reversion.
Chapter 6 is another cornerstone of the book, where I introduce the LIBOR
market and the LIBOR market model. After the derivation of the market
model, I draw the connection between the model and the no-arbitrage framework of HJM. This chapter contains perhaps the simplest yet most robust
formula for swaption pricing in the literature. Moreover, with the pricing
of Bermudan swaptions, I give an enlightening introduction to the popular

Longstaff–Schwartz method for pricing American options in the context of
Monte Carlo simulations.
Chapter 7 discusses an important aspect of model applications in the
markets—model calibration. Model calibration is a procedure to fix the
parameters of a model based on observed information from the derivatives market. This issue is rarely dealt with in academic or theoretical
literature, but in the real world it cannot be ignored. With the LIBOR
market model, I show how a problem of calibration can be set up and
solved.
In Chapter 8, I address two intriguing industrial issues, namely, volatility
and correlation adjustments. Mathematically, these issues are about computing the expectation of a financial quantity under a non-martingale measure.
With unprecedented generality and clarity, I offer analytical formulae for these
evaluation problems. The adjustment formulae have widespread applications
in pricing futures, non-vanilla swaps, and swaptions.
Finally, in Chapter 9, I introduce the class of affine term structure models
for interest rates. Rooted in general equilibrium theory for asset pricing, the


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Preface to the First Edition

affine term structure models are favored by many people, particularly those
in academic finance. These models are parsimonious in parameterization, and
they have a high degree of analytical tractability. The construction of the
models is demonstrated, followed by their applications to pricing options on
bonds and interest rates.


Preface to the Second Edition

It has been almost ten years since the publication of the first edition of this
book. In responses to the 2008 financial crisis, major changes have taken
place over the past ten years in financial markets, from changes in regulations
to the practice of derivatives pricing and risk management. Regulators have
been pushing OTC trades to go through central counterparty clearing houses,
which are subject to initial margin (IM) and variable margin (VM). For the
remaining OTC trades, collaterals have become market standard, in addition
to risk capital requirements. The funding costs for IM, VM, collaterals, and
risk capital have become a burden to many firms. How to take into account
the funding costs in trade prices has been a central issue to the practitioners,
regulators, and researchers. The current solution is to make various valuation
adjustments, so-called xVA, to either the trade prices or accounting books,
which has been controversial and is still debated today.
Major changes also occurred to the modeling of the interest rate derivatives. Pre-crisis term structure models, which were based on a single forward
rate curve – so-called single curve modeling – were replaced by multiple curve
models, which simultaneously model multiple forward rate curves. Yet, most
multi-curve models are at odds with the basis swap curves, which suggest that
the forward rate curves cannot evolve separately in any usual ways. There is
an affine solution of multi-curve modeling which is compatible with the basis
curves, but such a model is very different from models quants are used to, such
as the SABR-LIBOR market model (SABR-LMM) that is popular owing to
its capacity to manage volatility smile risks.
In the second edition, we will offer our solutions to xVA and post-crisis
interest rate modeling. Specifically, we want to achieve three objectives. First,
we will introduce the theories of major smile models for interest rate derivatives, and then adapt the most important one, the SABR-LMM model, to the
post-crisis markets. Second, we will introduce models for inflation rate derivatives and credit derivatives. Third, we will introduce our solution to the issue
of xVA. Altogether, six new chapters will be added. With exception of the
last chapter on xVA, all new chapters will be developed around the central
theme: the LIBOR market model (this is a distinguished feature of the second
edition).

In Chapter 10, we will introduce the SABR model and the Heston’s type
LMM model that feature the role of stochastic volatility in the formation
of volatility smiles. Through these two models, we try to demonstrate the
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Preface to the Second Edition

methodologies and techniques of smile models that are based on stochastic
volatilities.
In Chapter 11, we will introduce the L´evy market model, a framework of
models that captures volatility smiles based on the dynamics of jumps and
diffusions. Although this topic has theoretically been complex, we will offer a
simple exposition of model construction and pricing.
In Chapter 12, we take readers to inflation derivatives modeling and pricing, a two-decade-old theoretical subject not yet fully understood. Our aim is
to first build a solid foundation, and then on top of that develop the inflation
market model to justify the current market practice in pricing and hedging
inflation derivatives.
In Chapter 13, we deal with single-name credit derivatives, with the intention of pricing credit instruments, bonds, credit default swaps (CDS), CDS
options (or credit swaption), and even collateralized debt obligation (CDO)
using an LMM type model. We will redefine risky zero-coupon bonds using
tradable securities, and then risky forward rates, and eventually the credit
market model. This model allows us to price all instruments except CDOs,
for which we need additional tools like copulas to model correlated defaults.
In Chapter 14, we will rebuild the foundation, namely, the risky zerocoupon bonds, for the post-crisis interest rate derivative markets. Based on
the new foundation, we redefine LIBOR in the presence of credit risk of LIBOR
panel banks, and demonstrate that such a risk is responsible for the emergence
of the basis curves. We then define a dual-curve LMM and, more notably, the

dual-curve SABR-LMM. A large portion of the chapter is then devoted to
the pricing of caplets and swaptions under the dual-curve SABR-LMM, along
the approach of the heat kernel expansion method of Henry-Labord`ere for the
SABR-LMM model.
Finally, in Chapter 15, we present an xVA theory, which is applicable to
general derivatives pricing, including interest rate derivatives. We will prove
that the bilateral credit valuation adjustment is part of the fair price, and
demonstrate how funding costs enter the P&L of trades. We show that only
the market funding liquidity risk premium can enter into pricing, otherwise
price asymmetry will occur.


Acknowledgments to the
Second Edition

First I want to thank Mr. Sarfraz Khan, the editor at Taylor & Francis who
took the initiative to contract with me for the second edition as otherwise the
second edition would not have reached readers in 2019.
Three chapters are based on the joint publications that I worked on with
my former PhD students. I want to thank Ho Siu Lam for the joint work on
credit derivatives, Frederic Zhang for the joint work on xVA, and Shidong
Cui for the joint work on the dual-curve SABR-LMM model. I would like
to especially mention Shidong for his work with the very complex results of
caplet and swaption pricing; his ability to manage the details is truly amazing.
I also want to thank my wife, Molly, for her support throughout the writing
process of the second edition.
Finally, I want to thank my daughter, Dorothy, who helped me to proofread
all six new chapters. Her corrections and suggestions have definitely made this
book better.
Lixin Wu

Hong Kong

xxi



Author

Lixin Wu earned his PhD in applied mathematics from UCLA in 1991. Originally a
specialist in numerical analysis, he switched
his area of focus to financial mathematics in
1996. Since then, he has made notable contributions to the area. He co-developed the PDE
model for soft barrier options and the finitestate Markov chain model for credit contagion. He is, perhaps, best known in the financial engineering community for a series of
works on market models, including an optimal calibration methodology for the standard
market model, a market model with squareroot volatility, a market model for credit derivatives, a market model for
inflation derivatives, and a dual-curve SABR market model for post-crisis
derivatives markets. He also has made valuable contributions to the topic of
xVA. Over the years, Dr. Wu has been a consultant for financial institutions
and a lecturer for Risk Euromoney and Marco Evans, two professional education agencies. He is currently a full professor at the Hong Kong University of
Science and Technology.

xxiii