Interest rate modeling theory and practice, second edition

336 12 Market Model for Inflation Derivatives Modeling 343 12.1 CPI Index and Inflation Derivatives Market.. 391 14 Dual-Curve SABR-LMM Market Model for Post-Crisis Interest Rate Derivat

Interest Rate Modeling

Theory and Practice

Second Edition

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Interest Rate Modeling

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Lixin Wu

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Interest Rate Modeling

Theory and Practice

Second Edition

Lixin Wu

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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.

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Subjects: LCSH: Interest rates Mathematical models | Interest rate

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To Molly, Dorothy and Derek

Preface to the First Edition xv

1.1 Brownian Motion 1

1.1.1 Simple Random Walks 2

1.1.2 Brownian Motion 3

1.1.3 Adaptive and Non-Adaptive Functions 6

1.2 Stochastic Integrals 7

1.2.1 Evaluation of Stochastic Integrals 10

1.3 Stochastic Differentials and Ito’s Lemma 11

1.4 Multi-Factor Extensions 16

1.4.1 Multi-Factor Ito’s Process 16

1.4.2 Ito’s Lemma 17

1.4.3 Correlated Brownian Motions 17

1.4.4 The Multi-Factor Lognormal Model 18

1.5 Martingales 19

2 The Martingale Representation Theorem 23 2.1 Changing Measures with Binomial Models 23

2.1.1 A Motivating Example 23

2.1.2 Binomial Trees and Path Probabilities 26

2.2 Change of Measures under Brownian Filtration 29

2.2.1 The Radon–Nikodym Derivative of a Brownian Path 29 2.2.2 The CMG Theorem 31

2.3 The Martingale Representation Theorem 32

2.4 A Complete Market with Two Securities 33

2.5 Replicating and Pricing of Contingent Claims 34

2.6 Multi-Factor Extensions 36

vii

2.7 A Complete Market with Multiple Securities 37

2.7.1 Existence of a Martingale Measure 38

2.7.2 Pricing Contingent Claims 40

2.8 The Black–Scholes Formula 41

2.9 Notes 43

3 Interest Rates and Bonds 51 3.1 Interest Rates and Fixed-Income Instruments 51

3.1.1 Short Rate and Money Market Accounts 51

3.1.2 Term Rates and Certificates of Deposit 52

3.1.3 Bonds and Bond Markets 53

3.1.4 Quotation and Interest Accrual 55

3.2 Yields 57

3.2.1 Yield to Maturity 57

3.2.2 Par Bonds, Par Yields, and the Par Yield Curve 59

3.2.3 Yield Curves for U.S Treasuries 60

3.3 Zero-Coupon Bonds and Zero-Coupon Yields 61

3.3.1 Zero-Coupon Bonds 61

3.3.2 Bootstrapping the Zero-Coupon Yields 62

3.3.2.1 Future Value and Present Value 63

3.4 Forward Rates and Forward-Rate Agreements 64

3.5 Yield-Based Bond Risk Management 65

3.5.1 Duration and Convexity 65

3.5.2 Portfolio Risk Management 67

4 The Heath–Jarrow–Morton Model 71 4.1 Lognormal Model: The Starting Point 72

4.2 The HJM Model 75

4.3 Special Cases of the HJM Model 78

4.3.1 The Ho–Lee Model 78

4.3.2 The Hull–White (or Extended Vasicek) Model 79

4.4 Estimating the HJM Model from Yield Data 82

4.4.1 From a Yield Curve to a Forward-Rate Curve 82

4.4.2 Principal Component Analysis 87

4.5 A Case Study with a Two-Factor Model 92

4.6 Monte Carlo Implementations 93

4.7 Forward Prices 96

4.8 Forward Measure 99

4.9 Black’s Formula for Call and Put Options 102

4.9.1 Equity Options under the Hull–White Model 103

4.9.2 Options on Coupon Bonds 106

4.10 Numeraires and Changes of Measure 109

4.11 Linear Gaussian Models 110

4.12 Notes 111

5 Short-Rate Models and Lattice Implementation 119 5.1 From Short-Rate Models to Forward-Rate Models 120

5.2 General Markovian Models 122

5.2.1 One-Factor Models 128

5.2.2 Monte Carlo Simulations for Options Pricing 130

5.3 Binomial Trees of Interest Rates 131

5.3.1 A Binomial Tree for the Ho–Lee Model 132

5.3.2 Arrow–Debreu Prices 133

5.3.3 A Calibrated Tree for the Ho–Lee Model 135

5.4 A General Tree-Building Procedure 138

5.4.1 A Truncated Tree for the Hull–White Model 139

5.4.2 Trinomial Trees with Adaptive Time Steps 144

5.4.3 The Black–Karasinski Model 145

6 The LIBOR Market Model 149 6.1 LIBOR Market Instruments 149

6.1.1 LIBOR Rates 150

6.1.2 Forward-Rate Agreements 150

6.1.3 Repurchasing Agreement 152

6.1.4 Eurodollar Futures 152

6.1.5 Floating-Rate Notes 154

6.1.6 Swaps 155

6.1.7 Caps 157

6.1.8 Swaptions 158

6.1.9 Bermudan Swaptions 159

6.1.10 LIBOR Exotics 160

6.2 The LIBOR Market Model 162

6.3 Pricing of Caps and Floors 167

6.4 Pricing of Swaptions 168

6.5 Specifications of the LIBOR Market Model 175

6.6 Monte Carlo Simulation Method 178

6.6.1 The Log–Euler Scheme 178

6.6.2 Calculation of the Greeks 179

6.6.3 Early Exercise 180

6.7 Notes 185

7 Calibration of LIBOR Market Model 189 7.1 Implied Cap and Caplet Volatilities 190

7.2 Calibrating the LIBOR Market Model to Caps 192

7.3 Calibration to Caps, Swaptions, and Input

Correlations 195

7.4 Calibration Methodologies 200

7.4.1 Rank-Reduction Algorithm 200

7.4.2 The Eigenvalue Problem for Calibrating to Input Prices 211

7.5 Sensitivity with Respect to the Input Prices 223

8 Volatility and Correlation Adjustments 225 8.1 Adjustment due to Correlations 226

8.1.1 Futures Price versus Forward Price 226

8.1.2 Convexity Adjustment for LIBOR Rates 230

8.1.3 Convexity Adjustment under the Ho–Lee Model 232

8.1.4 An Example of Arbitrage 232

8.2 Adjustment due to Convexity 234

8.2.1 Payment in Arrears versus Payment in Advance 235

8.2.2 Geometric Explanation for Convexity Adjustment 236

8.2.3 General Theory of Convexity Adjustment 237

8.2.4 Convexity Adjustment for CMS and CMT Swaps 241

8.3 Timing Adjustment 243

8.4 Quanto Derivatives 244

8.5 Notes 249

9 Affine Term Structure Models 253 9.1 An Exposition with One-Factor Models 254

9.2 Analytical Solution of Riccarti Equations 261

9.3 Pricing Options on Coupon Bonds 265

9.4 Distributional Properties of Square-Root Processes 266

9.5 Multi-Factor Models 266

9.5.1 Admissible ATSMs 268

9.5.2 Three-Factor ATSMs 269

9.6 Swaption Pricing under ATSMs 272

9.7 Notes 278

10 Market Models with Stochastic Volatilities 281 10.1 SABR Model 282

10.2 The Wu and Zhang (2001) Model 289

10.3 Pricing of Caplets 293

10.4 Pricing of Swaptions 297

10.5 Model Calibration 301

10.6 Notes 308

11 L´evy Market Model 315

11.1 Introduction to L´evy Processes 315

11.1.1 Infinite Divisibility 315

11.1.2 Basic Examples of the L´evy Processes 317

11.1.2.1 Poisson Processes 317

11.1.2.2 Compound Poisson Processes 317

11.1.2.3 Linear Brownian Motion 318

11.1.3 Introduction of the Jump Measure 319

11.1.4 Characteristic Exponents for General L´evy Processes 319 11.2 The L´evy HJM Model 323

11.3 Market Model under L´evy Processes 328

11.4 Caplet Pricing 330

11.5 Swaption Pricing 332

11.6 Approximate Swaption Pricing via the Merton Formula 334

11.7 Notes 336

12 Market Model for Inflation Derivatives Modeling 343 12.1 CPI Index and Inflation Derivatives Market 345

12.1.1 TIPS 347

12.1.2 ZCIIS 347

12.1.3 YYIIS 348

12.1.4 Inflation Caps and Floors 349

12.1.5 Inflation Swaptions 349

12.2 Rebuilt Market Model and the New Paradigm 349

12.2.1 Inflation Discount Bonds and Inflation Forward Rates 349

12.2.2 The Compatibility Condition 351

12.2.3 Rebuilding the Market Model 353

12.2.4 The New Paradigm 354

12.2.5 Unifying the Jarrow-Yildirim Model 355

12.3 Pricing Inflation Derivatives 356

12.3.1 YYIIS 356

12.3.2 Caps 357

12.3.3 Swaptions 357

12.4 Model Calibration 360

12.5 Smile Modeling 361

12.6 Notes 362

13 Market Model for Credit Derivatives 363 13.1 Pricing of Risky Bonds: A New Perspective 365

13.2 Forward Spreads 367

13.3 Two Kinds of Default Protection Swaps 369

13.4 Par CDS Rates 371

13.5 Implied Survival Curve and Recovery-Rate Curve 373

13.6 Credit Default Swaptions and an Extended Market Model 378

13.7 Pricing of CDO Tranches under the Market Model 384

13.8 Notes 391

14 Dual-Curve SABR-LMM Market Model for Post-Crisis Interest Rate Derivatives Markets 393 14.1 LIBOR Market Model under Default Risks 395

14.2 Swaps and Basis Swaps 401

14.3 Option Pricing Using Heat Kernel Expansion 403

14.3.1 Derivation of the Heat Kernel 405

14.3.1.1 General Heat Kernel Expansion Formulae 405

14.3.1.2 Heat Kernel Expansion for the Dual-Curve SABR-LMM Model 407

14.3.2 Calculating the Volatility for Local Volatility Model 411 14.3.2.1 Calculation of the Local Volatility Function 411 14.3.2.2 Calculation of the Saddle Point 417

14.3.3 Calculation of the Implied Black’s Volatility 419

14.3.4 Numerical Results for 3M Caplets 420

14.4 Pricing 3M Swaptions 421

14.4.1 Dynamics of the State Variables 421

14.4.1.1 Swap Rate Dynamics under the Forward Swap Measure 425

14.4.2 Geometric Inputs 427

14.4.2.1 Inputs Parameter for the Heat Kernel Expansion 427

14.4.3 Local Volatility Function of Swap Rates 429

14.4.4 Calculation of the Saddle Point 430

14.4.4.1 Interpolation in High Dimensional Cases 430

14.4.5 Implied Black’s Volatility 432

14.4.6 Numerical Results for 3M Swaptions 432

14.5 Pricing Caps and Swaptions of Other Tenors 436

14.5.1 Linkage between 3M Rates and Rates of Other Tenors 436

14.5.1.1 The 6M Risk-Free OIS Rates 436

14.5.1.2 The 6M Expected Loss Rates 436

14.5.2 Dynamics of the 6M Risky LIBOR Rates 439

14.5.3 Dynamics of the 6M Swap Rates 440

14.5.4 Numerical Results of 6M Caplets and Swaptions 442

14.5.5 Model Calibration 442

14.6 Notes 442

15 xVA: Definition, Evaluation, and Risk Management 449

15.1 Pricing through Bilateral Replications 453

15.1.1 Margin Accounts, Collaterals, and Capitals 453

15.1.2 Pricing in the Absence of Funding Cost 454

15.2 The Rise of Other xVA 459

15.3 Examples 466

15.4 Notes 468

re-Interest-rate modeling has long been at the core of financial derivativestheory There are already quite a number of monographs and textbooks oninterest-rate models It is a good idea to write another book on the subjectonly if it will contribute significant added value to the literature This is why Ithought about this book This book portrays the theory of interest-rate mod-eling as a three-dimensional object of finance, mathematics, and computation.

In this book, all models are introduced with financial and economical fications; options are modeled along the so-called martingale approach; andoption 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 competentquantitative analyst in today’s market

justi-The book takes the top-down approach to introducing interest-rate models.The framework for no-arbitrage models is first established, and then the storyevolves around three representative types of models, namely, the Hull–Whitemodel, the market model, and affine models Relating individual models to thearbitrage framework helps to achieve better appreciation of the motivationsbehind each model, as well as better understanding of the interconnectionsamong different models Note that these three types of models coexist in themarket The adoption of any of these models or their variants may often be

xv

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 down approach, readers will get a clear picture of the status of this importantsubject, without being overwhelmed by too many specific models

top-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 mathematicalstyle 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 Theseexercises often require computer implementation Students not only learn themartingale approach for interest-rate modeling, but they also learn how to im-plement various models on computers The adoption of materials is influenced

by my experiences as a consultant and a lecturer for industrial courses Myearly 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 tains many results that are either new or exist only in recent research articles,including, as only a few examples, adaptive Hull–White lattice trees, mar-ket model calibration by quadratic programming, correlation adjustment, andswaption pricing under affine term structure models Many of the numericalmethods or schemes are very efficient or even optimized, owing to my orig-inal background as a numerical analyst In addition, notes are given at theend of most chapters to comment on cutting-edge research, which includesvolatility-smile modeling, convexity adjustment, and so on

con-Study Guide

When used as a textbook, this book can be covered in two 14-weeksemesters For a one-semester course, I recommend the coverage of Chap-

book for self-learning, readers should study short-rate models, market els, and affine models inChapters 5,6, and9, respectively For applications,

as it is particularly prepared for those readers who are interested in marketmodel calibration

modeling, where we introduce Ito’s calculus and the martingale representationtheorem The presentation of the theories is largely self-contained, except for

some omissions in the proof of the martingale representation theorem, which

I think is technically too demanding for this type of book

underlying securities or quantities of the interest-rate derivatives markets Ialso discuss the composition of bond markets and how they function Forcompleteness, I include the classical theory of risk management that is based

on parallel yield changes

Morton (HJM) model, the framework for no-arbitrage pricing models Withmarket data, we demonstrate the estimation of the HJM model Forwardmeasures, which are important devices for interest-rate options pricing, areintroduced Change of measures, a very useful technique for option pricing, isdiscussed in general

The theoretical part focuses on the issue of when the HJM model plies a Markovian short-rate model, and the numerical part is about theconstruction and calibration of short-rate lattice models A very efficientmethodology to construct and calibrate a truncated and adaptive lattice

im-is presented with the Hull–White model, which, after slight modifications,

is applicable to general Markovian models with the feature of mean sion

market and the LIBOR market model After the derivation of the marketmodel, I draw the connection between the model and the no-arbitrage frame-work of HJM This chapter contains perhaps the simplest yet most robustformula for swaption pricing in the literature Moreover, with the pricing

of Bermudan swaptions, I give an enlightening introduction to the popularLongstaff–Schwartz method for pricing American options in the context ofMonte Carlo simulations

markets—model calibration Model calibration is a procedure to fix theparameters of a model based on observed information from the deriva-tives market This issue is rarely dealt with in academic or theoreticalliterature, but in the real world it cannot be ignored With the LIBORmarket model, I show how a problem of calibration can be set up andsolved

and correlation adjustments Mathematically, these issues are about ing the expectation of a financial quantity under a non-martingale measure.With unprecedented generality and clarity, I offer analytical formulae for theseevaluation problems The adjustment formulae have widespread applications

comput-in priccomput-ing futures, non-vanilla swaps, and swaptions

Finally, inChapter 9, I introduce the class of affine term structure modelsfor interest rates Rooted in general equilibrium theory for asset pricing, the

affine term structure models are favored by many people, particularly those

in academic finance These models are parsimonious in parameterization, andthey have a high degree of analytical tractability The construction of themodels is demonstrated, followed by their applications to pricing options onbonds and interest rates

It has been almost ten years since the publication of the first edition of thisbook In responses to the 2008 financial crisis, major changes have takenplace over the past ten years in financial markets, from changes in regulations

to the practice of derivatives pricing and risk management Regulators havebeen pushing OTC trades to go through central counterparty clearing houses,which are subject to initial margin (IM) and variable margin (VM) For theremaining OTC trades, collaterals have become market standard, in addition

to risk capital requirements The funding costs for IM, VM, collaterals, andrisk capital have become a burden to many firms How to take into accountthe funding costs in trade prices has been a central issue to the practitioners,regulators, and researchers The current solution is to make various valuationadjustments, 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 tives Pre-crisis term structure models, which were based on a single forwardrate curve – so-called single curve modeling – were replaced by multiple curvemodels, which simultaneously model multiple forward rate curves Yet, mostmulti-curve models are at odds with the basis swap curves, which suggest thatthe forward rate curves cannot evolve separately in any usual ways There is

deriva-an affine solution of multi-curve modeling which is compatible with the basiscurves, 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 toits capacity to manage volatility smile risks

In the second edition, we will offer our solutions to xVA and post-crisisinterest rate modeling Specifically, we want to achieve three objectives First,

we will introduce the theories of major smile models for interest rate tives, and then adapt the most important one, the SABR-LMM model, to thepost-crisis markets Second, we will introduce models for inflation rate deriva-tives and credit derivatives Third, we will introduce our solution to the issue

deriva-of xVA Altogether, six new chapters will be added With exception deriva-of thelast chapter on xVA, all new chapters will be developed around the centraltheme: the LIBOR market model (this is a distinguished feature of the secondedition)

LMM model that feature the role of stochastic volatility in the formation

of volatility smiles Through these two models, we try to demonstrate the

xix

methodologies and techniques of smile models that are based on stochasticvolatilities.

models that captures volatility smiles based on the dynamics of jumps anddiffusions Although this topic has theoretically been complex, we will offer asimple exposition of model construction and pricing

pric-ing, 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 inflationmarket model to justify the current market practice in pricing and hedginginflation derivatives

inten-tion of pricing credit instruments, bonds, credit default swaps (CDS), CDSoptions (or credit swaption), and even collateralized debt obligation (CDO)using an LMM type model We will redefine risky zero-coupon bonds usingtradable securities, and then risky forward rates, and eventually the creditmarket model This model allows us to price all instruments except CDOs,for which we need additional tools like copulas to model correlated defaults

zero-coupon bonds, for the post-crisis interest rate derivative markets Based onthe new foundation, we redefine LIBOR in the presence of credit risk of LIBORpanel 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, thedual-curve SABR-LMM A large portion of the chapter is then devoted tothe pricing of caplets and swaptions under the dual-curve SABR-LMM, alongthe approach of the heat kernel expansion method of Henry-Labord`ere for theSABR-LMM model

Finally, in Chapter 15, we present an xVA theory, which is applicable togeneral derivatives pricing, including interest rate derivatives We will provethat the bilateral credit valuation adjustment is part of the fair price, anddemonstrate how funding costs enter the P&L of trades We show that onlythe market funding liquidity risk premium can enter into pricing, otherwiseprice asymmetry will occur

Second Edition

First I want to thank Mr Sarfraz Khan, the editor at Taylor & Francis whotook the initiative to contract with me for the second edition as otherwise thesecond 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 oncredit derivatives, Frederic Zhang for the joint work on xVA, and ShidongCui 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 ofcaplet 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 writingprocess of the second edition

Finally, I want to thank my daughter, Dorothy, who helped me to proofreadall six new chapters Her corrections and suggestions have definitely made thisbook better

Lixin WuHong Kong

xxi

Lixin Wu earned his PhD in applied ematics from UCLA in 1991 Originally aspecialist in numerical analysis, he switchedhis area of focus to financial mathematics in

math-1996 Since then, he has made notable butions to the area He co-developed the PDEmodel for soft barrier options and the finite-state Markov chain model for credit conta-gion He is, perhaps, best known in the fi-nancial engineering community for a series ofworks on market models, including an opti-mal calibration methodology for the standardmarket model, a market model with square-root volatility, a market model for credit derivatives, a market model forinflation derivatives, and a dual-curve SABR market model for post-crisisderivatives markets He also has made valuable contributions to the topic ofxVA Over the years, Dr Wu has been a consultant for financial institutionsand a lecturer for Risk Euromoney and Marco Evans, two professional educa-tion agencies He is currently a full professor at the Hong Kong University ofScience and Technology

contri-xxiii

Chapter 1

The Basics of Stochastic Calculus

The seemingly random fluctuation in stock prices is the most distinguishingfeature of financial markets and it creates both risks and opportunities forinvestors To model random fluctuations in stock prices as well as in other fi-nancial time series data (indexes, interest rates, exchange rates, etc.) has longbeen a central issue in the discipline of quantitative finance In mathematicalterms, a financial time series is a stochastic process In quantitative modeling,

a financial time series is treated as a function of other standardized stochasticprocesses, and these stochastic processes serve as the engines for the randomevolution of financial time series With these models, we can assess risk, valuerisky securities, design hedging strategies, and make decisions on asset alloca-tions in a scientific way Among the many standardized stochastic time seriesthat are available, the so-called Brownian motion is no doubt the most basicyet the most important one This chapter is devoted to describing Brownianmotion and its calculus

Financial time series data actually exist in discrete form, for example, tickdata, daily data, weekly data, and so on It may be intuitive to describe thesedata with discrete time series models Alternatively, we can also describe themwith continuous time series models, and then take discrete steps in time Itturns out that working with continuous-time financial time series is a lot moreefficient than working with discrete time series models, due to the existence of

an arsenal of stochastic analysis tools in continuous time The theory of nian motion is the single most important building block of continuous-timefinancial time series We proceed by introducing Brownian motion through alimiting process, starting with simple random walks

Brow-1

1.1.1 Simple Random Walks

Simple random walks are discrete time series, {Xi}, defined as

where ∆t > 0 stands for the interval of time for stepping forward One canverify that {Xi} have the following properties:

1 The increment of Xn+1− Xn is independent of {Xi} , ∀i ≤ n

2 E [Xn| Xm] = Xm, m ≤ n

An interesting feature of the simple random walk is the linearity of Xi’s ance in time: given X0, the variance of Xi is equal to i∆t, the time it takesthe time series to evolve from X0 to Xi

vari-Out of the simple Brownian random walk, we can construct a time process through linear interpolation:

Theorem 1.1.1 (The Lundeberg–Levi Central Limit Theorem) For the tinuous process, ¯X(t), there is

x

√n∆t)

= P

(1/nPn i=1(Xi− Xi−1) − 0

x

√t)

According to the central limit theorem, we have

lim

(1/nPn i=1(Xi− Xi−1) − 0

x

√t)

= P



ε ≤√xt



=

Z x/ √ t

−∞

1

√2πexp

let u =√tv= √1

A continuous stochastic process is a collection of real-valued randomvariables, {X(t, ω), 0 ≤ t ≤ T } or {Xt(ω), 0 ≤ t ≤ T }, that are defined on aprobability space (Ω, F, P) Here Ω is the collection of all ωs, which are so-called sample points, F the smallest σ-algebra that contains Ωs, and P aprobability measure on Ω Each random outcome, ω ∈ Ω, corresponds to anentire time series

which is called a path of Xt In view of Equation 1.7, we can regard Xt(ω)

as a function of two variables, ω and t For notational simplicity, however,

we often suppress the ω variable when its explicit appearance is not sary

neces-In the context of financial modeling, we are particularly interested inthe Brownian motion introduced earlier Its formal definition is given be-low

Definition 1.1.1 A Brownian motion or a Wiener process is a real-valuestochastic process, Wt or W (t), 0 ≤ t ≤ ∞, that has the following properties:

1 W (0) = 0

0 –0.2

0 0.2 0.4 0.6

3 For t ≥ 0 and s > 0, the increment W (t + s) − W (t) ∼ N (0, s)

4 W (t) is continuous almost surely (a.s.)

Here N (0, s) stands for a normal distribution with mean zero and variance

s Note that in some literature, property 4 is not part of the definition, as itcan be proved to be implied by the first three properties (Varadhan, 1980a orIkeda and Watanabe, 1989) A sample path of W (t) is shown in Figure 1.1,which is generated with a step size of ∆t = 2−10

Brownian motion plays a major role in continuous-time stochasticmodeling in physics, engineering and finance In finance, it has been used

to model the random behavior of asset returns Several major properties ofBrownian motion are listed below

Lemma 1.1.1 A Brownian motion, W (t), has the following properties:

1 Self-similarity: ∀λ > 0 W (t) 7→ (1/√λ)W (λt) = ˜W(t) is also a ian motion

Brown-2 Unbounded variation: for any T ≥ 0, lim

∆t→0

P

j,tj≤T |∆Wj| = ∞, where

∆Wj= W (tj+1) − W (tj)

3 Non-differentiability: W (t) is not differentiable at any t ≥ 0

The self-similarity property implies that W (t) is a fractal object Thiscan be proved straightforwardly and the proof is left to readers We will seethat unbounded variation implies non-differentiability To prove unboundedvariation, we will need the following lemma

Lemma 1.1.2 Let 0 = t0 < t1 < · · · < tn = T represent an arbitrarypartition of time interval [0, T ] , ∆t = max

j (tj+1− tj) and ∆Wj= W (tj+1) −

W(tj) Then lim

∆t→0

Pn−1 j=0(∆Wj)2= T almost surely

Proof: Let ∆tj= tj+1− tj, ∀j Then we can write

∆Wj = εjp∆tj, εj ∼ N (0, 1) iid (1.8)Here, iid stands for “independent with identical distribution.” By Kol-mogorov’s large number theorem,

[Proof of unbounded variation and non-differentiability]

We will do the proof by the method of contradiction Suppose that W (t)has bounded variation over a finite interval [0, T ] such that

which is a contradiction, so the variation must be unbounded

The property of non-differentiability follows from the unbounded ation property In fact, if W0(t) existed and was finite over any interval,say [0, T ], then there would be

|W0(t)| dt < ∞, (1.13)

where “=” means “is defined as.” Equation 1.13 contradicts to the property∆

1.1.3 Adaptive and Non-Adaptive Functions

We now define the class of functions of stochastic processes such that theirvalues at time t can be determined based on available information up to time

t Formally, we introduce the notion of filtration

Definition 1.1.2 Let Ft denote the smallest σ-algebra containing all sets ofthe form

{ω; Wt 1(ω) ∈ B1, , Wtk(ω) ∈ Bk} ⊂ Ω, (1.14)where k = 1, 2, , tj≤ t and Bj ⊂ R are Borel sets, where R stands for theset of real numbers Denote the σ-algebra as Ft= σ (W (s), 0 ≤ s ≤ t); we callthe collection of (Ft)t≥0 a Brownian filtration

For applications in mathematical finance, it suffices to think of Ft as

“information up to time t” or “history of Ws up to time t.” ing to the definition, Fs ⊂ Ft for s ≤ t, meaning that a filtration is

Accord-an increasing stream of information Readers cAccord-an find thorough sions of Brownian filtration in many previous works, for example, Øksendal(1992)

discus-Definition 1.1.3 A function, f (t), is said to be Ft-adaptive if

f(t) = ˜f({W (s), 0 ≤ s ≤ t} , t), ∀t, (1.15)that is, the value of the function at time t depends only on the path history up

to time t

Adaptive functions1 are natural candidates to work with in finance pose that the value of a function represents a decision in investment Then,such a decision has to be made based on the available information up to themoment of making the decision The next example gives a good idea of whatkind of function is or is not an Ft-adaptive function

is not Ft-adaptive, because f (t) cannot be determined at any time t < 1

1.2 Stochastic Integrals

Stochastic calculus considers the integration and differentiation of general

Ft-adaptive functions The purpose of developing such a stochastic calculus

is to model financial time series (with random dynamics) with either integral

or differential equations According to Lemma 1.1, a Brownian motion, W (t),

is nowhere differentiable in the usual sense of differentiation for deterministicfunctions To define differentials of stochastic processes in a proper sense, wemust first study the notion of stochastic integrals

Stochastic integrals can be defined for functions in the square-integrablespace, H2[0, T ] = L2(Ω × [0, T ], dP × dt), which is defined to be the collection

of functions satisfying

E

"

Z T 0

as a limit of integrals of elementary functions Finally, we define the integral of

a general square-integrable function as a limit of integrals of bounded uous functions The key in this three-step procedure is of course to ensure theconvergence of the limits in L2(Ω, F, P), the Hilbert space of random variablessatisfying

contin-EX2(ω) < ∞

This definition approach is taken by Øksendal (1992) Alternative treatments

of course also exist; see, for example, Mikosch (1998)

An elementary function has the form

ϕ(t, ω) =X

j

cj(ω)χ(tj,tj+1](t), (1.19)

where χA(t) is the indicator function such that χA(t) = 1 if t ∈ A, or otherwise

χA(t) = 0, and cj(ω) is adapted to Ft j For the elementary function, theintegral is defined in a rather natural way:

Z T 0

Lemma 1.2.1 (Ito isometry) If ϕ(t, ω) is a bounded elementary function,then

E





Z T 0

ϕ(t, ω) dW (t, ω)

!2

=

Z T 0

E[ϕ2(t, ω)] dt

For bounded continuous functions, we can define the stochastic integralintuitively through a limiting process:

Z T 0

E





Z T 0

f(t) dW (t, ω)

!2

=

Z T 0

which implies that ϕn(t) is a Cauchy sequence in L2

(Ω, F, P) By making useagain of Ito’s isometry for ϕn− ϕm, we can see that

Z T 0

ϕn(t, ω) dW (t, ω)

!2

=

Z T 0

Eϕ2

n(t, ω) dt,

we will arrive at Equation 1.24 Hence, Ito isometry holds for continuous

For a general function in L2

(Ω, F, P), definition (Equation 1.22) forstochastic integrals is no longer valid Nonetheless, we can approximate a gen-eral function of L2(Ω, F, P) by a sequence of bounded continuous functions inthe sense that

Z T 0

E( f(t) − fn(t))2 dt → 0 as n → ∞, (1.27)

and thus we define the stochastic integral or Ito’s integral for f (t) as the limit

Z T 0

f(t) dW (t)= lim∆

n→∞

Z T 0

Properties of Stochastic Integrals:

For an Ft-adaptive function, f ∈ L2

contin-1.2.1 Evaluation of Stochastic Integrals

We now consider the evaluation of stochastic integrals Suppose that weknow the anti-derivative of a function, f (t), such that

dF (t)

Could there be

Z t 0

determin-As a showcase of integral evaluation, we try to work out the integral of

f(t) = W (t) according to its definition Let tj = jt/n and denote Wj for

W(tj), j = 0, , n Start from the partial sum as follows:

We then have, by Kolmogorov’s large number theorem,

Similar to deterministic calculus, an integral equation implies a sponding differential equation To see that, we differentiate Equation 1.36with respect to t, obtaining

corre-W(t) dW (t) =1

or

Equation 1.38 is the first stochastic differential equation (SDE) to appear

in this book; it relates the differential of f (t) = W2(t) to the differential

of W (t) Knowing Equation 1.38, we can calculate the stochastic integral

Rt

0W(s) dW (s) easily, without going through the procedure from Equations1.34 through 1.36 In the next section, we study the dynamics of generalfunctions in a broader context

In this section, we study the differentials of functions of other stochasticprocesses In stochastic calculus, the so-called Ito’s process is most often used

as the basic stochastic process

Definition 1.3.1 Ito’s process is a continuous stochastic process of the form:

X(t) = X0+

Z t 0

σ(s) dW (s) +

Z t 0

where σ(s) and µ(s) are adaptive functions satisfying

E

Z t 0

We call σ(t) and µ(t) the volatility and drift of the SDE, respectively

We now consider a function of X(t), Y (t) = F (X(t), t) The next lemmadescribes the SDE satisfied by Y (t)

Lemma 1.3.1 (Ito’s Lemma) Let X(t) be Ito’s process with drift µ(t) andvolatility σ(t), and let F (x, t) be a smooth function with bounded second-orderderivatives Then Y (t) = F (X(t), t) is also Ito’s process with drift

Now, we sum up the increment for i = 0, 1, , n − 1, obtaining

Based on Equations 1.48 and 1.50, we conclude that the fourth term of tion 1.48 vanishes in L2

Equa-(Ω, F, P) Hence, as ∆t = maxj∆tj → 0, Equation1.47 becomes

Y(t) − Y (0) =

Z t 0

Next, we study the application of Ito’s lemma with two examples.Example 1.2 Consider again the differential of the function f (t) = W2

which reproduces Equation 1.38

Example 1.3 (A lognormal process) A lognormal model is perhaps themost authoritative model for asset prices in financial studies This model isbased on the assumption that the return on an asset over a fixed horizon obeys

a normal distribution Let S(t) denote the price of an asset at time t Then,the return over a horizon, (t, t + ∆t), is defined as

Note that the random term is proportional to √∆t, whereas the drift term

is proportional to ∆t Hence, for a small ∆t, the random term dominates thedeterministic term This is consistent to the widely held belief that, in anefficient market, asset price movements over a short horizon are pretty muchrandom Taking the limit ∆t → dt, we arrive at the model with a differentialequation:

d(ln St) = µtdt + σtdWt (1.56)

By integrating the above equation over the interval (0, t), we then obtain

ln St= ln S0+

Z t 0

or

S = S eR0tµsds+σsdWs (1.58)

To derive the differential equation for St, we let Xt = ln St/S0 and write

St= S0eXt Obviously, there are

a(s) (dWs)2=

Z t 0

for any Ft-adaptive square-integrable function, a(s) Based on Equation 1.62,

we define the following operational rules:

and

for the Brownian motion Note that Equation 1.62 is obviously at odds with

dWt= ε√∆t, ε ∼ N (0, 1), so it has to be interpreted according to Equation1.61

The above operation rules offer a great deal of convenience to stochasticdifferentiations Taking the derivation of Ito’s lemma for example, we maynow proceed as