Robert R. Reitano

INTRODUCTION TO

QUANTITATIVE

FINANCE

A MATH TOOL KIT

Introduction to Quantitative Finance

Introduction to Quantitative Finance

A Math Tool Kit

Robert R. Reitano

The MIT Press

Cambridge, Massachusetts

London, England

6 2010 Massachusetts Institute of Technology

All rights reserved. No part of this book may be reproduced in any form by any electronic or mechanical

means (including photocopying, recording, or information storage and retrieval) without permission in

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This book was set in Times New Roman on 3B2 by Asco Typesetters, Hong Kong and was printed and

bound in the United States of America.

Library of Congress Cataloging-in-Publication Data

Reitano, Robert R., 1950–

Introduction to quantitative ﬁnance : a math tool kit / Robert R. Reitano.

p. cm.

Includes index.

ISBN 978-0-262-01369-7 (hardcover : alk. paper) 1. Finance—Mathematical models. I. Title.

HG106.R45 2010

2009022214

332.01 0 5195—dc22

10 9 8 7

6 5 4 3

2 1

to Lisa

Contents

1

1.1

1.2

1.3

1.4

1.5

List of Figures and Tables

Introduction

xix

xxi

Mathematical Logic

Introduction

Axiomatic Theory

Inferences

Paradoxes

Propositional Logic

1.5.1 Truth Tables

1.5.2 Framework of a Proof

1.5.3 Methods of Proof

1

1

4

6

7

10

10

15

17

19

19

21

23

24

27

The Direct Proof

Proof by Contradiction

Proof by Induction

*1.6

1.7

2

2.1

2.2

2.3

Mathematical Logic

Applications to Finance

Exercises

Number Systems and Functions

Numbers: Properties and Structures

2.1.1 Introduction

2.1.2 Natural Numbers

2.1.3 Integers

2.1.4 Rational Numbers

2.1.5 Real Numbers

*2.1.6 Complex Numbers

Functions

Applications to Finance

2.3.1 Number Systems

2.3.2 Functions

Present Value Functions

Accumulated Value Functions

Nominal Interest Rate Conversion Functions

Bond-Pricing Functions

31

31

31

32

37

38

41

44

49

51

51

54

54

55

56

57

viii

Contents

Mortgage- and Loan-Pricing Functions

Preferred Stock-Pricing Functions

Common Stock-Pricing Functions

Portfolio Return Functions

Forward-Pricing Functions

Exercises

3

3.1

3.2

3.3

Euclidean and Other Spaces

Euclidean Space

3.1.1 Structure and Arithmetic

3.1.2 Standard Norm and Inner Product for Rn

*3.1.3 Standard Norm and Inner Product for C n

3.1.4 Norm and Inner Product Inequalities for Rn

*3.1.5 Other Norms and Norm Inequalities for Rn

Metric Spaces

3.2.1 Basic Notions

3.2.2 Metrics and Norms Compared

*3.2.3 Equivalence of Metrics

Applications to Finance

3.3.1 Euclidean Space

Asset Allocation Vectors

Interest Rate Term Structures

Bond Yield Vector Risk Analysis

Cash Flow Vectors and ALM

3.3.2

Metrics and Norms

Sample Statistics

Constrained Optimization

Tractability of the lp -Norms: An Optimization Example

General Optimization Framework

Exercises

4

4.1

4.2

Set Theory and Topology

Set Theory

4.1.1 Historical Background

*4.1.2 Overview of Axiomatic Set Theory

4.1.3 Basic Set Operations

Open, Closed, and Other Sets

59

59

60

61

62

64

71

71

71

73

74

75

77

82

82

84

88

93

93

94

95

99

100

101

101

103

105

110

112

117

117

117

118

121

122

Contents

4.3

5

5.1

*5.2

*5.3

5.4

5.5

6

6.1

6.2

ix

4.2.1 Open and Closed Subsets of R

4.2.2 Open and Closed Subsets of Rn

*4.2.3 Open and Closed Subsets in Metric Spaces

*4.2.4 Open and Closed Subsets in General Spaces

4.2.5 Other Properties of Subsets of a Metric Space

Applications to Finance

4.3.1 Set Theory

4.3.2 Constrained Optimization and Compactness

4.3.3 Yield of a Security

Exercises

122

127

128

129

130

134

134

135

137

139

Sequences and Their Convergence

Numerical Sequences

5.1.1 Deﬁnition and Examples

5.1.2 Convergence of Sequences

5.1.3 Properties of Limits

Limits Superior and Inferior

General Metric Space Sequences

Cauchy Sequences

5.4.1 Deﬁnition and Properties

*5.4.2 Complete Metric Spaces

Applications to Finance

5.5.1 Bond Yield to Maturity

5.5.2 Interval Bisection Assumptions Analysis

Exercises

145

145

145

146

149

152

157

162

162

165

167

167

170

172

Series and Their Convergence

Numerical Series

6.1.1 Deﬁnitions

6.1.2 Properties of Convergent Series

6.1.3 Examples of Series

*6.1.4 Rearrangements of Series

6.1.5 Tests of Convergence

The lp -Spaces

6.2.1 Deﬁnition and Basic Properties

*6.2.2 Banach Space

*6.2.3 Hilbert Space

177

177

177

178

180

184

190

196

196

199

202

x

6.3

6.4

7

7.1

7.2

Contents

Power Series

*6.3.1 Product of Power Series

*6.3.2 Quotient of Power Series

Applications to Finance

6.4.1 Perpetual Security Pricing: Preferred Stock

6.4.2 Perpetual Security Pricing: Common Stock

6.4.3 Price of an Increasing Perpetuity

6.4.4 Price of an Increasing Payment Security

6.4.5 Price Function Approximation: Asset Allocation

6.4.6 lp -Spaces: Banach and Hilbert

Exercises

Discrete Probability Theory

The Notion of Randomness

Sample Spaces

7.2.1 Undeﬁned Notions

7.2.2 Events

7.2.3 Probability Measures

7.2.4 Conditional Probabilities

Law of Total Probability

7.3

7.2.5 Independent Events

7.2.6 Independent Trials: One Sample Space

*7.2.7 Independent Trials: Multiple Sample Spaces

Combinatorics

7.3.1 Simple Ordered Samples

With Replacement

Without Replacement

7.3.2

General Orderings

Two Subset Types

Binomial Coe‰cients

The Binomial Theorem

r Subset Types

Multinomial Theorem

7.4

Random Variables

7.4.1 Quantifying Randomness

7.4.2 Random Variables and Probability Functions

206

209

212

215

215

217

218

220

222

223

224

231

231

233

233

234

235

238

239

240

241

245

247

247

247

247

248

248

249

250

251

252

252

252

254

Contents

7.5

xi

7.4.3 Random Vectors and Joint Probability Functions

7.4.4 Marginal and Conditional Probability Functions

7.4.5 Independent Random Variables

Expectations of Discrete Distributions

7.5.1 Theoretical Moments

Expected Values

Conditional and Joint Expectations

Mean

Variance

Covariance and Correlation

General Moments

General Central Moments

Absolute Moments

Moment-Generating Function

Characteristic Function

*7.5.2

Moments of Sample Data

Sample Mean

Sample Variance

Other Sample Moments

7.6

7.7

7.8

Discrete Probability Density Functions

7.6.1 Discrete Rectangular Distribution

7.6.2 Binomial Distribution

7.6.3 Geometric Distribution

7.6.4 Multinomial Distribution

7.6.5 Negative Binomial Distribution

7.6.6 Poisson Distribution

Generating Random Samples

Applications to Finance

7.8.1 Loan Portfolio Defaults and Losses

Individual Loss Model

Aggregate Loss Model

7.8.2

7.8.3

Insurance Loss Models

Insurance Net Premium Calculations

Generalized Geometric and Related Distributions

Life Insurance Single Net Premium

256

258

261

264

264

264

266

268

268

271

274

274

274

275

277

278

280

282

286

287

288

290

292

293

296

299

301

307

307

307

310

313

314

314

317

xii

Contents

Pension Beneﬁt Single Net Premium

Life Insurance Periodic Net Premiums

7.8.4

7.8.5

Asset Allocation Framework

Equity Price Models in Discrete Time

Stock Price Data Analysis

Binomial Lattice Model

Binomial Scenario Model

7.8.6

Discrete Time European Option Pricing: Lattice-Based

One-Period Pricing

Multi-period Pricing

7.8.7 Discrete Time European Option Pricing: Scenario Based

Exercises

8

8.1

8.2

8.3

8.4

8.5

8.6

*8.7

8.8

Fundamental Probability Theorems

Uniqueness of the m.g.f. and c.f.

Chebyshev’s Inequality

Weak Law of Large Numbers

Strong Law of Large Numbers

8.4.1 Model 1: Independent fX^n g

8.4.2 Model 2: Dependent fX^n g

8.4.3 The Strong Law Approach

*8.4.4 Kolmogorov’s Inequality

*8.4.5 Strong Law of Large Numbers

De Moivre–Laplace Theorem

8.5.1 Stirling’s Formula

8.5.2 De Moivre–Laplace Theorem

8.5.3 Approximating Binomial Probabilities I

The Normal Distribution

8.6.1 Deﬁnition and Properties

8.6.2 Approximating Binomial Probabilities II

The Central Limit Theorem

Applications to Finance

8.8.1 Insurance Claim and Loan Loss Tail Events

Risk-Free Asset Portfolio

Risky Assets

8.8.2

Binomial Lattice Equity Price Models as Dt ! 0

318

319

319

325

325

326

328

329

329

333

336

337

347

347

349

352

357

359

360

362

363

365

368

371

374

376

377

377

379

381

386

386

387

391

392

Contents

xiii

Parameter Dependence on Dt

Distributional Dependence on Dt

Real World Binomial Distribution as Dt ! 0

8.8.3

Lattice-Based European Option Prices as Dt ! 0

The Model

European Call Option Illustration

Black–Scholes–Merton Option-Pricing Formulas I

8.8.4

Scenario-Based European Option Prices as N ! y

The Model

Option Price Estimates as N ! y

Scenario-Based Prices and Replication

Exercises

9

9.1

9.2

Calculus I: Di¤erentiation

Approximating Smooth Functions

Functions and Continuity

9.2.1 Functions

9.2.2 The Notion of Continuity

The Meaning of ‘‘Discontinuous’’

*The Metric Notion of Continuity

Sequential Continuity

9.2.3

9.2.4

9.2.5

9.2.6

Basic Properties of Continuous Functions

Uniform Continuity

Other Properties of Continuous Functions

Ho¨lder and Lipschitz Continuity

‘‘Big O’’ and ‘‘Little o’’ Convergence

9.2.7

Convergence of a Sequence of Continuous Functions

*Series of Functions

*Interchanging Limits

9.3

*9.2.8 Continuity and Topology

Derivatives and Taylor Series

9.3.1 Improving an Approximation I

9.3.2 The First Derivative

9.3.3 Calculating Derivatives

A Discussion of e

9.3.4

Properties of Derivatives

394

395

396

400

400

402

404

406

406

407

409

411

417

417

418

418

420

425

428

429

430

433

437

439

440

442

445

445

448

450

450

452

454

461

462

xiv

Contents

9.3.5

9.3.6

9.3.7

Improving an Approximation II

Higher Order Derivatives

Improving an Approximation III: Taylor Series

Approximations

Analytic Functions

9.4

9.3.8 Taylor Series Remainder

Convergence of a Sequence of Derivatives

9.4.1 Series of Functions

9.4.2 Di¤erentiability of Power Series

Product of Taylor Series

*Division of Taylor Series

9.5

9.6

9.7

9.8

Critical Point Analysis

9.5.1 Second-Derivative Test

*9.5.2 Critical Points of Transformed Functions

Concave and Convex Functions

9.6.1 Deﬁnitions

9.6.2 Jensen’s Inequality

Approximating Derivatives

9.7.1 Approximating f 0 ðxÞ

9.7.2 Approximating f 00 ðxÞ

9.7.3 Approximating f ðnÞ ðxÞ, n > 2

Applications to Finance

9.8.1

Continuity of Price Functions

9.8.2

Constrained Optimization

9.8.3

Interval Bisection

9.8.4

Minimal Risk Asset Allocation

9.8.5

Duration and Convexity Approximations

Dollar-Based Measures

Embedded Options

Rate Sensitivity of Duration

9.8.6

Asset–Liability Management

Surplus Immunization, Time t ¼ 0

Surplus Immunization, Time t > 0

Surplus Ratio Immunization

9.8.7

The ‘‘Greeks’’

465

466

467

470

473

478

481

481

486

487

488

488

490

494

494

500

504

504

504

505

505

505

507

507

508

509

511

512

513

514

518

519

520

521

Contents

9.8.8

xv

Utility Theory

Investment Choices

Insurance Choices

Gambling Choices

Utility and Risk Aversion

Examples of Utility Functions

9.8.9

9.8.10

Optimal Risky Asset Allocation

Risk-Neutral Binomial Distribution as Dt ! 0

Analysis of the Risk-Neutral Probability: qðDtÞ

Risk-Neutral Binomial Distribution as Dt ! 0

*9.8.11

Special Risk-Averter Binomial Distribution as Dt ! 0

Analysis of the Special Risk-Averter Probability: qðDtÞ

Special Risk-Averter Binomial Distribution as Dt ! 0

Details of the Limiting Result

9.8.12 Black–Scholes–Merton Option-Pricing Formulas II

Exercises

10

10.1

10.2

Calculus II: Integration

Summing Smooth Functions

Riemann Integration of Functions

10.2.1 Riemann Integral of a Continuous Function

10.2.2 Riemann Integral without Continuity

Finitely Many Discontinuities

*Inﬁnitely Many Discontinuities

10.3

10.4

10.5

10.6

10.7

Examples of the Riemann Integral

Mean Value Theorem for Integrals

Integrals and Derivatives

10.5.1 The Integral of a Derivative

10.5.2 The Derivative of an Integral

Improper Integrals

10.6.1 Deﬁnitions

10.6.2 Integral Test for Series Convergence

Formulaic Integration Tricks

10.7.1 Method of Substitution

10.7.2 Integration by Parts

*10.7.3 Wallis’ Product Formula

522

523

523

524

524

527

528

532

533

538

543

543

545

546

547

549

559

559

560

560

566

566

569

574

579

581

581

585

587

587

588

592

592

594

596

xvi

Contents

10.8

10.9

Taylor Series with Integral Remainder

Convergence of a Sequence of Integrals

10.9.1 Review of Earlier Convergence Results

10.9.2 Sequence of Continuous Functions

10.9.3 Sequence of Integrable Functions

10.9.4 Series of Functions

10.9.5 Integrability of Power Series

10.10 Numerical Integration

10.10.1 Trapezoidal Rule

10.10.2 Simpson’s Rule

10.11 Continuous Probability Theory

10.11.1 Probability Space and Random Variables

10.11.2 Expectations of Continuous Distributions

*10.11.3 Discretization of a Continuous Distribution

10.11.4 Common Expectation Formulas

nth Moment

Mean

nth Central Moment

Variance

Standard Deviation

Moment-Generating Function

Characteristic Function

10.11.5

Continuous Probability Density Functions

Continuous Uniform Distribution

Beta Distribution

Exponential Distribution

Gamma Distribution

Cauchy Distribution

Normal Distribution

Lognormal Distribution

10.12

10.11.6 Generating Random Samples

Applications to Finance

10.12.1 Continuous Discounting

10.12.2 Continuous Term Structures

Bond Yields

598

602

602

603

605

606

607

609

609

612

613

613

618

620

624

624

624

624

624

625

625

625

626

627

628

630

630

632

634

637

640

641

641

644

644

Contents

xvii

Exercises

645

646

648

649

651

654

655

656

657

658

660

664

666

668

671

675

References

Index

685

689

Forward Rates

Fixed Income Investment Fund

Spot Rates

10.12.3

10.12.4

10.12.5

Continuous Stock Dividends and Reinvestment

Duration and Convexity Approximations

Approximating the Integral of the Normal Density

Power Series Method

Upper and Lower Riemann Sums

Trapezoidal Rule

Simpson’s Rule

*10.12.6

Generalized Black–Scholes–Merton Formula

The Piecewise ‘‘Continuitization’’ of the Binomial Distribution

The ‘‘Continuitization’’ of the Binomial Distribution

The Limiting Distribution of the ‘‘Continuitization’’

The Generalized Black–Scholes–Merton Formula

List of Figures and Tables

Figures

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

a2 þ b2

2.1

Pythagorean theorem: c ¼

2.2

a ¼ r cos t, b ¼ r sin t

47

3.1

lp -Balls: p ¼ 1; 1:25; 2; 5; y

86

3.2

lp -Ball: p ¼ 0:5

88

3.3

Equivalence of l1 - and l2 -metrics

93

3.4

f ðaÞ ¼ j5 À aj þ jÀ15 À aj

3.5

jx À 5j þ j y þ 15j ¼ 20

3.6

3

f ðaÞ ¼ j5 À aj þ jÀ15 À aj

3

46

108

109

3

3

110

3.7

jx À 5j þ j y þ 15j ¼ 2000

111

6.1

Positive integer lattice

190

7.1

F ðxÞ for Hs in three ﬂips

256

7.2

Binomial c.d.f.

304

7.3

Binomial stock price lattice

328

7.4

Binomial stock price path

2

f ðxÞ ¼ p1ﬃﬃﬃﬃ

eÀx =2

2p

&

sin x1 ; x 0 0

f ðxÞ ¼

0;

x¼0

&1

1

sin x ; x 0 0

gðxÞ ¼ x

0;

x¼0

329

f ðxÞ ¼ x 2 ðx 2 À 2Þ

Â

Ã

TðiÞ A Tði0 Þ 1 þ 12 C T ði0 Þði À i0 Þ 2

& 2

0ax < 1

x ;

f ðxÞ ¼

2

x þ 5; 1 a x a 2

450

Piecewise continuous sðxÞ

8

<1; x ¼ 0

f ðxÞ ¼ 1n ; x ¼ mn in lowest terms

:

0; x irrational

575

8.1

9.1

9.2

9.3

9.4

10.1

10.2

10.3

378

423

424

518

567

577

10.4

FðxÞ and Fd ðxÞ compared for d ¼ 0:5, midpoint tags

622

10.5

fU ðxÞ ¼ 14 , 1 a x a 5

x vÀ1 ð1ÀxÞ wÀ1

fb ðxÞ ¼ Bðv; wÞ

628

10.6

629

xx

10.7

10.8

10.9

10.10

10.11

10.12

List of Figures and Tables

fG ðxÞ ¼ 1b

ÀxÁcÀ1 eÀx=b

b

GðcÞ

1

1ﬃﬃﬃﬃ

x2

p

fC ðxÞ ¼ p1 1þx

,

fðxÞ

¼

exp

À

2

2s 2

s

2p

À ÁcÀ1 eÀx=b

ðln xÞ 2

1ﬃﬃﬃﬃ

p

fL ðxÞ ¼ x 2p exp À 2 , fG ðxÞ ¼ 1b xb

GðcÞ

2

ðÀx 2 =2Þ

jð2Þ ðxÞ ¼ p1ﬃﬃﬃﬃ

ðx

À

1Þe

2p

2

jð4Þ ðxÞ ¼ p1ﬃﬃﬃﬃ

ðx 4 À 6x 2 þ 3ÞeðÀx =2Þ

2p

631

634

638

658

660

Piecewise continuitization and continuitization of the binomial f ðxÞ

665

Interval bisection for bond yield

169

Table

5.1

Introduction

This book provides an accessible yet rigorous introduction to the ﬁelds of mathematics that are needed for success in investment and quantitative ﬁnance. The book’s

goal is to develop mathematics topics used in portfolio management and investment

banking, including basic derivatives pricing and risk management applications, that

are essential to quantitative investment ﬁnance, or more simply, investment ﬁnance.

A future book, Advanced Quantitative Finance: A Math Tool Kit, will cover more

advanced mathematical topics in these areas as used for investment modeling, derivatives pricing, and risk management. Collectively, these latter areas are called quantitative ﬁnance or mathematical ﬁnance.

The mathematics presented in this book would typically be learned by an undergraduate mathematics major. Each chapter of the book corresponds roughly to the

mathematical materials that are acquired in a one semester course. Naturally each

chapter presents only a subset of the materials from these traditional math courses,

since the goal is to emphasize the most important and relevant materials for the ﬁnance applications presented. However, more advanced topics are introduced earlier

than is customary so that the reader can become familiar with these materials in an

accessible setting.

My motivation for writing this text was to ﬁll two current gaps in the ﬁnancial and

mathematical literature as they apply to students, and practitioners, interested in

sharpening their mathematical skills and deepening their understanding of investment and quantitative ﬁnance applications. The gap in the mathematics literature is

that most texts are focused on a single ﬁeld of mathematics such as calculus. Anyone

interested in meeting the ﬁeld requirements in ﬁnance is left with the choice to either

pursue one or more degrees in mathematics or expend a signiﬁcant self-study e¤ort

on associated mathematics textbooks. Neither approach is e‰cient for business

school and ﬁnance graduate students nor for professionals working in investment

and quantitative ﬁnance and aiming to advance their mathematical skills. As the diligent reader quickly discovers, each such book presents more math than is needed for

ﬁnance, and it is nearly impossible to identify what math is essential for ﬁnance

applications. An additional complication is that math books rarely if ever provide

applications in ﬁnance, which further complicates the identiﬁcation of the relevant

theory.

The second gap is in the ﬁnance literature. Finance texts have e¤ectively become

bifurcated in terms of mathematical sophistication. One group of texts takes the

recipe-book approach to math ﬁnance often presenting mathematical formulas with

only simpliﬁed or heuristic derivations. These books typically neglect discussion of

the mathematical framework that derivations require, as well as e¤ects of assumptions by which the conclusions are drawn. While such treatment may allow more

xxii

Introduction

discussion of the ﬁnancial applications, it does not adequately prepare the student

who will inevitably be investigating quantitative problems for which the answers are

unknown.

The other group of ﬁnance textbooks are mathematically rigorous but inaccessible

to students who are not in a mathematics degree program. Also, while rigorous, such

books depend on sophisticated results developed elsewhere, and hence the discussions

are incomplete and inadequate even for a motivated student without additional classroom instruction. Here, again, the unprepared student must take on faith referenced

results without adequate understanding, which is essentially another form of recipe

book.

With this book I attempt to ﬁll some of these gaps by way of a reasonably economic, yet rigorous and accessible, review of many of the areas of mathematics

needed in quantitative investment ﬁnance. My objective is to help the reader acquire

a deep understanding of relevant mathematical theory and the tools that can be effectively put in practice. In each chapter I provide a concluding section on ﬁnance

applications of the presented materials to help the reader connect the chapter’s mathematical theory to ﬁnance applications and work in the ﬁnance industry.

What Does It Take to Be a ‘‘Quant’’?

In some sense, the emphasis of this book is on the development of the math tools one

needs to succeed in mathematical modeling applications in ﬁnance. The imagery

implied by ‘‘math tool kit’’ is deliberate, and it reﬂects my belief that the study of

mathematics is an intellectually rewarding endeavor, and it provides an enormously

ﬂexible collection of tools that allow users to answer a wide variety of important and

practical questions.

By tools, however, I do not mean a collection of formulas that should be memorized for later application. Of course, some memorization is mandatory in mathematics, as in any language, to understand what the words mean and to facilitate

accurate communication. But most formulas are outside this mandatorily memorized

collection. Indeed, although mathematics texts are full of formulas, the memorization of formulas should be relatively low on the list of priorities of any student or

user of these books. The student should instead endeavor to learn the mathematical

frameworks and the application of these frameworks to real world problems.

In other words, the student should focus on the thought process and mathematics

used to develop each result. These are the ‘‘tools,’’ that is, the mathematical methods

of each discipline of explicitly identifying assumptions, formally developing the

needed insights and formulas, and understanding the relationships between formulas

Introduction

xxiii

and the underlying assumptions. The tools so deﬁned and studied in this book will

equip the student with fairly robust frameworks for their applications in investment

and quantitative ﬁnance.

Despite its large size, this book has the relatively modest ambition of teaching a

very speciﬁc application of mathematics, that being to ﬁnance, and so the selection

of materials in every subdiscipline has been made parsimoniously. This selection of

materials was the most di‰cult aspect of developing this book. In general, the selection criterion I used was that a topic had to be either directly applicable to ﬁnance, or

needed for the understanding of a later topic that was directly applicable to ﬁnance.

Because my objective was to make this book more than a collection of mathematical

formulas, or just another ﬁnance recipe book, I devote considerable space to discussion on how the results are derived, and how they relate to their mathematical

assumptions. Ideally the students of this book should never again accept a formulaic

result as an immutable truth separate from any assumptions made by its originator.

The motivation for this approach is that in investment and quantitative ﬁnance,

there are few good careers that depend on the application of standard formulas in

standard situations. All such applications tend to be automated and run in companies’ computer systems with little or no human intervention. Think ‘‘program trading’’ as an example of this statement. While there is an interesting and deep theory

related to identifying so-called arbitrage opportunities, these can be formulaically

listed and programmed, and their implementation automated with little further analyst intervention.

Equally, if not more important, with new ﬁnancial products developed regularly,

there are increased demands on quants and all ﬁnance practitioners to apply the previous methodologies and adapt them appropriately to ﬁnancial analyses, pricing, risk

modeling, and risk management. Today, in practice, standard results may or may

not apply, and the most critical job of the ﬁnance quant is to determine if the traditional approach applies, and if not, to develop an appropriate modiﬁcation or even

an entirely new approach. In other words, for today’s ﬁnance quants, it has become

critical to be able to think in mathematics, and not simply to do mathematics by

rote.

The many ﬁnance applications developed in the chapters present enough detail to

be understood by someone new to the given application but in less detail than would

be appropriate for mastering the application. Ideally the reader will be familiar with

some applications and will be introduced to other applications that can, as needed,

be enhanced by further study. On my selection of mathematical topics and ﬁnance

applications, I hope to beneﬁt from the valuable comments of ﬁnance readers, whether

student or practitioner. All such feedback will be welcomed and acknowledged in future editions.

xxiv

Introduction

Plan of the Book

The ten chapters of this book are arranged so that each topic is developed based on

materials previously discussed. In a few places, however, a formula or result is introduced that could not be fully developed until a much later chapter. In fewer places, I

decided to not prove a deep result that would have brought the book too far aﬁeld

from its intended purpose. Overall, the book is intended to be self-contained, complete with respect to the materials discussed, and mathematically rigorous. The only

mathematical background required of the reader is competent skill in algebraic

manipulations and some knowledge of pre-calculus topics of graphing, exponentials

and logarithms. Thus the topics developed in this book are interrelated and applied

with the understanding that the student will be motivated to work through, with pen

or pencil and paper or by computer simulation, any derivation or example that may

be unclear and that the student has the algebraic skills and self-discipline to do so.

Of course, even when a proof or example appears clear, the student will beneﬁt in

using pencil and paper and computer simulation to clarify any missing details in derivations. Such informal exercises provide essential practice in the application of the

tools discussed, and analytical skills can be progressively sharpened by way of the

book’s formal exercises and ultimately in real world situations. While not every derivation in the book o¤ers the same amount of enlightenment on the mathematical

tools studied, or should be studied in detail before proceeding, developing the habit

of ﬁlling in details can deepen mathematical knowledge and the understanding of

how this knowledge can be applied.

I have identiﬁed the more advanced sections by an asterisk (*). The beginning

student may ﬁnd it useful to scan these sections on ﬁrst reading. These sections can

then be returned to if needed for a later application of interest. The more advanced

student may ﬁnd these sections to provide some insights on the materials they are

already familiar with. For beginning practitioners and professors of students new to

the materials, it may be useful to only scan the reasoning in the longer proofs on a

ﬁrst review before turning to the applications.

There are a number of productive approaches to the chapter sequencing of this

book for both self-study and formal classroom presentation. Professors and practitioners with good prior exposure might pick and choose chapters out of order to e‰ciently address pressing educational needs. For ﬁnance applications, again the best

approach is the one that suits the needs of the student or practitioner. Those familiar

with ﬁnance applications and aware of the math skills that need to be developed will

focus on the appropriate math sections, then proceed to the ﬁnance applications to

better understand the connections between the math and the ﬁnance. Those less fa-

INTRODUCTION TO

QUANTITATIVE

FINANCE

A MATH TOOL KIT

Introduction to Quantitative Finance

Introduction to Quantitative Finance

A Math Tool Kit

Robert R. Reitano

The MIT Press

Cambridge, Massachusetts

London, England

6 2010 Massachusetts Institute of Technology

All rights reserved. No part of this book may be reproduced in any form by any electronic or mechanical

means (including photocopying, recording, or information storage and retrieval) without permission in

writing from the publisher.

MIT Press books may be purchased at special quantity discounts for business or sales promotional use.

For information, please email special_sales@mitpress.mit.edu or write to Special Sales Department, The

MIT Press, 55 Hayward Street, Cambridge, MA 02142.

This book was set in Times New Roman on 3B2 by Asco Typesetters, Hong Kong and was printed and

bound in the United States of America.

Library of Congress Cataloging-in-Publication Data

Reitano, Robert R., 1950–

Introduction to quantitative ﬁnance : a math tool kit / Robert R. Reitano.

p. cm.

Includes index.

ISBN 978-0-262-01369-7 (hardcover : alk. paper) 1. Finance—Mathematical models. I. Title.

HG106.R45 2010

2009022214

332.01 0 5195—dc22

10 9 8 7

6 5 4 3

2 1

to Lisa

Contents

1

1.1

1.2

1.3

1.4

1.5

List of Figures and Tables

Introduction

xix

xxi

Mathematical Logic

Introduction

Axiomatic Theory

Inferences

Paradoxes

Propositional Logic

1.5.1 Truth Tables

1.5.2 Framework of a Proof

1.5.3 Methods of Proof

1

1

4

6

7

10

10

15

17

19

19

21

23

24

27

The Direct Proof

Proof by Contradiction

Proof by Induction

*1.6

1.7

2

2.1

2.2

2.3

Mathematical Logic

Applications to Finance

Exercises

Number Systems and Functions

Numbers: Properties and Structures

2.1.1 Introduction

2.1.2 Natural Numbers

2.1.3 Integers

2.1.4 Rational Numbers

2.1.5 Real Numbers

*2.1.6 Complex Numbers

Functions

Applications to Finance

2.3.1 Number Systems

2.3.2 Functions

Present Value Functions

Accumulated Value Functions

Nominal Interest Rate Conversion Functions

Bond-Pricing Functions

31

31

31

32

37

38

41

44

49

51

51

54

54

55

56

57

viii

Contents

Mortgage- and Loan-Pricing Functions

Preferred Stock-Pricing Functions

Common Stock-Pricing Functions

Portfolio Return Functions

Forward-Pricing Functions

Exercises

3

3.1

3.2

3.3

Euclidean and Other Spaces

Euclidean Space

3.1.1 Structure and Arithmetic

3.1.2 Standard Norm and Inner Product for Rn

*3.1.3 Standard Norm and Inner Product for C n

3.1.4 Norm and Inner Product Inequalities for Rn

*3.1.5 Other Norms and Norm Inequalities for Rn

Metric Spaces

3.2.1 Basic Notions

3.2.2 Metrics and Norms Compared

*3.2.3 Equivalence of Metrics

Applications to Finance

3.3.1 Euclidean Space

Asset Allocation Vectors

Interest Rate Term Structures

Bond Yield Vector Risk Analysis

Cash Flow Vectors and ALM

3.3.2

Metrics and Norms

Sample Statistics

Constrained Optimization

Tractability of the lp -Norms: An Optimization Example

General Optimization Framework

Exercises

4

4.1

4.2

Set Theory and Topology

Set Theory

4.1.1 Historical Background

*4.1.2 Overview of Axiomatic Set Theory

4.1.3 Basic Set Operations

Open, Closed, and Other Sets

59

59

60

61

62

64

71

71

71

73

74

75

77

82

82

84

88

93

93

94

95

99

100

101

101

103

105

110

112

117

117

117

118

121

122

Contents

4.3

5

5.1

*5.2

*5.3

5.4

5.5

6

6.1

6.2

ix

4.2.1 Open and Closed Subsets of R

4.2.2 Open and Closed Subsets of Rn

*4.2.3 Open and Closed Subsets in Metric Spaces

*4.2.4 Open and Closed Subsets in General Spaces

4.2.5 Other Properties of Subsets of a Metric Space

Applications to Finance

4.3.1 Set Theory

4.3.2 Constrained Optimization and Compactness

4.3.3 Yield of a Security

Exercises

122

127

128

129

130

134

134

135

137

139

Sequences and Their Convergence

Numerical Sequences

5.1.1 Deﬁnition and Examples

5.1.2 Convergence of Sequences

5.1.3 Properties of Limits

Limits Superior and Inferior

General Metric Space Sequences

Cauchy Sequences

5.4.1 Deﬁnition and Properties

*5.4.2 Complete Metric Spaces

Applications to Finance

5.5.1 Bond Yield to Maturity

5.5.2 Interval Bisection Assumptions Analysis

Exercises

145

145

145

146

149

152

157

162

162

165

167

167

170

172

Series and Their Convergence

Numerical Series

6.1.1 Deﬁnitions

6.1.2 Properties of Convergent Series

6.1.3 Examples of Series

*6.1.4 Rearrangements of Series

6.1.5 Tests of Convergence

The lp -Spaces

6.2.1 Deﬁnition and Basic Properties

*6.2.2 Banach Space

*6.2.3 Hilbert Space

177

177

177

178

180

184

190

196

196

199

202

x

6.3

6.4

7

7.1

7.2

Contents

Power Series

*6.3.1 Product of Power Series

*6.3.2 Quotient of Power Series

Applications to Finance

6.4.1 Perpetual Security Pricing: Preferred Stock

6.4.2 Perpetual Security Pricing: Common Stock

6.4.3 Price of an Increasing Perpetuity

6.4.4 Price of an Increasing Payment Security

6.4.5 Price Function Approximation: Asset Allocation

6.4.6 lp -Spaces: Banach and Hilbert

Exercises

Discrete Probability Theory

The Notion of Randomness

Sample Spaces

7.2.1 Undeﬁned Notions

7.2.2 Events

7.2.3 Probability Measures

7.2.4 Conditional Probabilities

Law of Total Probability

7.3

7.2.5 Independent Events

7.2.6 Independent Trials: One Sample Space

*7.2.7 Independent Trials: Multiple Sample Spaces

Combinatorics

7.3.1 Simple Ordered Samples

With Replacement

Without Replacement

7.3.2

General Orderings

Two Subset Types

Binomial Coe‰cients

The Binomial Theorem

r Subset Types

Multinomial Theorem

7.4

Random Variables

7.4.1 Quantifying Randomness

7.4.2 Random Variables and Probability Functions

206

209

212

215

215

217

218

220

222

223

224

231

231

233

233

234

235

238

239

240

241

245

247

247

247

247

248

248

249

250

251

252

252

252

254

Contents

7.5

xi

7.4.3 Random Vectors and Joint Probability Functions

7.4.4 Marginal and Conditional Probability Functions

7.4.5 Independent Random Variables

Expectations of Discrete Distributions

7.5.1 Theoretical Moments

Expected Values

Conditional and Joint Expectations

Mean

Variance

Covariance and Correlation

General Moments

General Central Moments

Absolute Moments

Moment-Generating Function

Characteristic Function

*7.5.2

Moments of Sample Data

Sample Mean

Sample Variance

Other Sample Moments

7.6

7.7

7.8

Discrete Probability Density Functions

7.6.1 Discrete Rectangular Distribution

7.6.2 Binomial Distribution

7.6.3 Geometric Distribution

7.6.4 Multinomial Distribution

7.6.5 Negative Binomial Distribution

7.6.6 Poisson Distribution

Generating Random Samples

Applications to Finance

7.8.1 Loan Portfolio Defaults and Losses

Individual Loss Model

Aggregate Loss Model

7.8.2

7.8.3

Insurance Loss Models

Insurance Net Premium Calculations

Generalized Geometric and Related Distributions

Life Insurance Single Net Premium

256

258

261

264

264

264

266

268

268

271

274

274

274

275

277

278

280

282

286

287

288

290

292

293

296

299

301

307

307

307

310

313

314

314

317

xii

Contents

Pension Beneﬁt Single Net Premium

Life Insurance Periodic Net Premiums

7.8.4

7.8.5

Asset Allocation Framework

Equity Price Models in Discrete Time

Stock Price Data Analysis

Binomial Lattice Model

Binomial Scenario Model

7.8.6

Discrete Time European Option Pricing: Lattice-Based

One-Period Pricing

Multi-period Pricing

7.8.7 Discrete Time European Option Pricing: Scenario Based

Exercises

8

8.1

8.2

8.3

8.4

8.5

8.6

*8.7

8.8

Fundamental Probability Theorems

Uniqueness of the m.g.f. and c.f.

Chebyshev’s Inequality

Weak Law of Large Numbers

Strong Law of Large Numbers

8.4.1 Model 1: Independent fX^n g

8.4.2 Model 2: Dependent fX^n g

8.4.3 The Strong Law Approach

*8.4.4 Kolmogorov’s Inequality

*8.4.5 Strong Law of Large Numbers

De Moivre–Laplace Theorem

8.5.1 Stirling’s Formula

8.5.2 De Moivre–Laplace Theorem

8.5.3 Approximating Binomial Probabilities I

The Normal Distribution

8.6.1 Deﬁnition and Properties

8.6.2 Approximating Binomial Probabilities II

The Central Limit Theorem

Applications to Finance

8.8.1 Insurance Claim and Loan Loss Tail Events

Risk-Free Asset Portfolio

Risky Assets

8.8.2

Binomial Lattice Equity Price Models as Dt ! 0

318

319

319

325

325

326

328

329

329

333

336

337

347

347

349

352

357

359

360

362

363

365

368

371

374

376

377

377

379

381

386

386

387

391

392

Contents

xiii

Parameter Dependence on Dt

Distributional Dependence on Dt

Real World Binomial Distribution as Dt ! 0

8.8.3

Lattice-Based European Option Prices as Dt ! 0

The Model

European Call Option Illustration

Black–Scholes–Merton Option-Pricing Formulas I

8.8.4

Scenario-Based European Option Prices as N ! y

The Model

Option Price Estimates as N ! y

Scenario-Based Prices and Replication

Exercises

9

9.1

9.2

Calculus I: Di¤erentiation

Approximating Smooth Functions

Functions and Continuity

9.2.1 Functions

9.2.2 The Notion of Continuity

The Meaning of ‘‘Discontinuous’’

*The Metric Notion of Continuity

Sequential Continuity

9.2.3

9.2.4

9.2.5

9.2.6

Basic Properties of Continuous Functions

Uniform Continuity

Other Properties of Continuous Functions

Ho¨lder and Lipschitz Continuity

‘‘Big O’’ and ‘‘Little o’’ Convergence

9.2.7

Convergence of a Sequence of Continuous Functions

*Series of Functions

*Interchanging Limits

9.3

*9.2.8 Continuity and Topology

Derivatives and Taylor Series

9.3.1 Improving an Approximation I

9.3.2 The First Derivative

9.3.3 Calculating Derivatives

A Discussion of e

9.3.4

Properties of Derivatives

394

395

396

400

400

402

404

406

406

407

409

411

417

417

418

418

420

425

428

429

430

433

437

439

440

442

445

445

448

450

450

452

454

461

462

xiv

Contents

9.3.5

9.3.6

9.3.7

Improving an Approximation II

Higher Order Derivatives

Improving an Approximation III: Taylor Series

Approximations

Analytic Functions

9.4

9.3.8 Taylor Series Remainder

Convergence of a Sequence of Derivatives

9.4.1 Series of Functions

9.4.2 Di¤erentiability of Power Series

Product of Taylor Series

*Division of Taylor Series

9.5

9.6

9.7

9.8

Critical Point Analysis

9.5.1 Second-Derivative Test

*9.5.2 Critical Points of Transformed Functions

Concave and Convex Functions

9.6.1 Deﬁnitions

9.6.2 Jensen’s Inequality

Approximating Derivatives

9.7.1 Approximating f 0 ðxÞ

9.7.2 Approximating f 00 ðxÞ

9.7.3 Approximating f ðnÞ ðxÞ, n > 2

Applications to Finance

9.8.1

Continuity of Price Functions

9.8.2

Constrained Optimization

9.8.3

Interval Bisection

9.8.4

Minimal Risk Asset Allocation

9.8.5

Duration and Convexity Approximations

Dollar-Based Measures

Embedded Options

Rate Sensitivity of Duration

9.8.6

Asset–Liability Management

Surplus Immunization, Time t ¼ 0

Surplus Immunization, Time t > 0

Surplus Ratio Immunization

9.8.7

The ‘‘Greeks’’

465

466

467

470

473

478

481

481

486

487

488

488

490

494

494

500

504

504

504

505

505

505

507

507

508

509

511

512

513

514

518

519

520

521

Contents

9.8.8

xv

Utility Theory

Investment Choices

Insurance Choices

Gambling Choices

Utility and Risk Aversion

Examples of Utility Functions

9.8.9

9.8.10

Optimal Risky Asset Allocation

Risk-Neutral Binomial Distribution as Dt ! 0

Analysis of the Risk-Neutral Probability: qðDtÞ

Risk-Neutral Binomial Distribution as Dt ! 0

*9.8.11

Special Risk-Averter Binomial Distribution as Dt ! 0

Analysis of the Special Risk-Averter Probability: qðDtÞ

Special Risk-Averter Binomial Distribution as Dt ! 0

Details of the Limiting Result

9.8.12 Black–Scholes–Merton Option-Pricing Formulas II

Exercises

10

10.1

10.2

Calculus II: Integration

Summing Smooth Functions

Riemann Integration of Functions

10.2.1 Riemann Integral of a Continuous Function

10.2.2 Riemann Integral without Continuity

Finitely Many Discontinuities

*Inﬁnitely Many Discontinuities

10.3

10.4

10.5

10.6

10.7

Examples of the Riemann Integral

Mean Value Theorem for Integrals

Integrals and Derivatives

10.5.1 The Integral of a Derivative

10.5.2 The Derivative of an Integral

Improper Integrals

10.6.1 Deﬁnitions

10.6.2 Integral Test for Series Convergence

Formulaic Integration Tricks

10.7.1 Method of Substitution

10.7.2 Integration by Parts

*10.7.3 Wallis’ Product Formula

522

523

523

524

524

527

528

532

533

538

543

543

545

546

547

549

559

559

560

560

566

566

569

574

579

581

581

585

587

587

588

592

592

594

596

xvi

Contents

10.8

10.9

Taylor Series with Integral Remainder

Convergence of a Sequence of Integrals

10.9.1 Review of Earlier Convergence Results

10.9.2 Sequence of Continuous Functions

10.9.3 Sequence of Integrable Functions

10.9.4 Series of Functions

10.9.5 Integrability of Power Series

10.10 Numerical Integration

10.10.1 Trapezoidal Rule

10.10.2 Simpson’s Rule

10.11 Continuous Probability Theory

10.11.1 Probability Space and Random Variables

10.11.2 Expectations of Continuous Distributions

*10.11.3 Discretization of a Continuous Distribution

10.11.4 Common Expectation Formulas

nth Moment

Mean

nth Central Moment

Variance

Standard Deviation

Moment-Generating Function

Characteristic Function

10.11.5

Continuous Probability Density Functions

Continuous Uniform Distribution

Beta Distribution

Exponential Distribution

Gamma Distribution

Cauchy Distribution

Normal Distribution

Lognormal Distribution

10.12

10.11.6 Generating Random Samples

Applications to Finance

10.12.1 Continuous Discounting

10.12.2 Continuous Term Structures

Bond Yields

598

602

602

603

605

606

607

609

609

612

613

613

618

620

624

624

624

624

624

625

625

625

626

627

628

630

630

632

634

637

640

641

641

644

644

Contents

xvii

Exercises

645

646

648

649

651

654

655

656

657

658

660

664

666

668

671

675

References

Index

685

689

Forward Rates

Fixed Income Investment Fund

Spot Rates

10.12.3

10.12.4

10.12.5

Continuous Stock Dividends and Reinvestment

Duration and Convexity Approximations

Approximating the Integral of the Normal Density

Power Series Method

Upper and Lower Riemann Sums

Trapezoidal Rule

Simpson’s Rule

*10.12.6

Generalized Black–Scholes–Merton Formula

The Piecewise ‘‘Continuitization’’ of the Binomial Distribution

The ‘‘Continuitization’’ of the Binomial Distribution

The Limiting Distribution of the ‘‘Continuitization’’

The Generalized Black–Scholes–Merton Formula

List of Figures and Tables

Figures

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

a2 þ b2

2.1

Pythagorean theorem: c ¼

2.2

a ¼ r cos t, b ¼ r sin t

47

3.1

lp -Balls: p ¼ 1; 1:25; 2; 5; y

86

3.2

lp -Ball: p ¼ 0:5

88

3.3

Equivalence of l1 - and l2 -metrics

93

3.4

f ðaÞ ¼ j5 À aj þ jÀ15 À aj

3.5

jx À 5j þ j y þ 15j ¼ 20

3.6

3

f ðaÞ ¼ j5 À aj þ jÀ15 À aj

3

46

108

109

3

3

110

3.7

jx À 5j þ j y þ 15j ¼ 2000

111

6.1

Positive integer lattice

190

7.1

F ðxÞ for Hs in three ﬂips

256

7.2

Binomial c.d.f.

304

7.3

Binomial stock price lattice

328

7.4

Binomial stock price path

2

f ðxÞ ¼ p1ﬃﬃﬃﬃ

eÀx =2

2p

&

sin x1 ; x 0 0

f ðxÞ ¼

0;

x¼0

&1

1

sin x ; x 0 0

gðxÞ ¼ x

0;

x¼0

329

f ðxÞ ¼ x 2 ðx 2 À 2Þ

Â

Ã

TðiÞ A Tði0 Þ 1 þ 12 C T ði0 Þði À i0 Þ 2

& 2

0ax < 1

x ;

f ðxÞ ¼

2

x þ 5; 1 a x a 2

450

Piecewise continuous sðxÞ

8

<1; x ¼ 0

f ðxÞ ¼ 1n ; x ¼ mn in lowest terms

:

0; x irrational

575

8.1

9.1

9.2

9.3

9.4

10.1

10.2

10.3

378

423

424

518

567

577

10.4

FðxÞ and Fd ðxÞ compared for d ¼ 0:5, midpoint tags

622

10.5

fU ðxÞ ¼ 14 , 1 a x a 5

x vÀ1 ð1ÀxÞ wÀ1

fb ðxÞ ¼ Bðv; wÞ

628

10.6

629

xx

10.7

10.8

10.9

10.10

10.11

10.12

List of Figures and Tables

fG ðxÞ ¼ 1b

ÀxÁcÀ1 eÀx=b

b

GðcÞ

1

1ﬃﬃﬃﬃ

x2

p

fC ðxÞ ¼ p1 1þx

,

fðxÞ

¼

exp

À

2

2s 2

s

2p

À ÁcÀ1 eÀx=b

ðln xÞ 2

1ﬃﬃﬃﬃ

p

fL ðxÞ ¼ x 2p exp À 2 , fG ðxÞ ¼ 1b xb

GðcÞ

2

ðÀx 2 =2Þ

jð2Þ ðxÞ ¼ p1ﬃﬃﬃﬃ

ðx

À

1Þe

2p

2

jð4Þ ðxÞ ¼ p1ﬃﬃﬃﬃ

ðx 4 À 6x 2 þ 3ÞeðÀx =2Þ

2p

631

634

638

658

660

Piecewise continuitization and continuitization of the binomial f ðxÞ

665

Interval bisection for bond yield

169

Table

5.1

Introduction

This book provides an accessible yet rigorous introduction to the ﬁelds of mathematics that are needed for success in investment and quantitative ﬁnance. The book’s

goal is to develop mathematics topics used in portfolio management and investment

banking, including basic derivatives pricing and risk management applications, that

are essential to quantitative investment ﬁnance, or more simply, investment ﬁnance.

A future book, Advanced Quantitative Finance: A Math Tool Kit, will cover more

advanced mathematical topics in these areas as used for investment modeling, derivatives pricing, and risk management. Collectively, these latter areas are called quantitative ﬁnance or mathematical ﬁnance.

The mathematics presented in this book would typically be learned by an undergraduate mathematics major. Each chapter of the book corresponds roughly to the

mathematical materials that are acquired in a one semester course. Naturally each

chapter presents only a subset of the materials from these traditional math courses,

since the goal is to emphasize the most important and relevant materials for the ﬁnance applications presented. However, more advanced topics are introduced earlier

than is customary so that the reader can become familiar with these materials in an

accessible setting.

My motivation for writing this text was to ﬁll two current gaps in the ﬁnancial and

mathematical literature as they apply to students, and practitioners, interested in

sharpening their mathematical skills and deepening their understanding of investment and quantitative ﬁnance applications. The gap in the mathematics literature is

that most texts are focused on a single ﬁeld of mathematics such as calculus. Anyone

interested in meeting the ﬁeld requirements in ﬁnance is left with the choice to either

pursue one or more degrees in mathematics or expend a signiﬁcant self-study e¤ort

on associated mathematics textbooks. Neither approach is e‰cient for business

school and ﬁnance graduate students nor for professionals working in investment

and quantitative ﬁnance and aiming to advance their mathematical skills. As the diligent reader quickly discovers, each such book presents more math than is needed for

ﬁnance, and it is nearly impossible to identify what math is essential for ﬁnance

applications. An additional complication is that math books rarely if ever provide

applications in ﬁnance, which further complicates the identiﬁcation of the relevant

theory.

The second gap is in the ﬁnance literature. Finance texts have e¤ectively become

bifurcated in terms of mathematical sophistication. One group of texts takes the

recipe-book approach to math ﬁnance often presenting mathematical formulas with

only simpliﬁed or heuristic derivations. These books typically neglect discussion of

the mathematical framework that derivations require, as well as e¤ects of assumptions by which the conclusions are drawn. While such treatment may allow more

xxii

Introduction

discussion of the ﬁnancial applications, it does not adequately prepare the student

who will inevitably be investigating quantitative problems for which the answers are

unknown.

The other group of ﬁnance textbooks are mathematically rigorous but inaccessible

to students who are not in a mathematics degree program. Also, while rigorous, such

books depend on sophisticated results developed elsewhere, and hence the discussions

are incomplete and inadequate even for a motivated student without additional classroom instruction. Here, again, the unprepared student must take on faith referenced

results without adequate understanding, which is essentially another form of recipe

book.

With this book I attempt to ﬁll some of these gaps by way of a reasonably economic, yet rigorous and accessible, review of many of the areas of mathematics

needed in quantitative investment ﬁnance. My objective is to help the reader acquire

a deep understanding of relevant mathematical theory and the tools that can be effectively put in practice. In each chapter I provide a concluding section on ﬁnance

applications of the presented materials to help the reader connect the chapter’s mathematical theory to ﬁnance applications and work in the ﬁnance industry.

What Does It Take to Be a ‘‘Quant’’?

In some sense, the emphasis of this book is on the development of the math tools one

needs to succeed in mathematical modeling applications in ﬁnance. The imagery

implied by ‘‘math tool kit’’ is deliberate, and it reﬂects my belief that the study of

mathematics is an intellectually rewarding endeavor, and it provides an enormously

ﬂexible collection of tools that allow users to answer a wide variety of important and

practical questions.

By tools, however, I do not mean a collection of formulas that should be memorized for later application. Of course, some memorization is mandatory in mathematics, as in any language, to understand what the words mean and to facilitate

accurate communication. But most formulas are outside this mandatorily memorized

collection. Indeed, although mathematics texts are full of formulas, the memorization of formulas should be relatively low on the list of priorities of any student or

user of these books. The student should instead endeavor to learn the mathematical

frameworks and the application of these frameworks to real world problems.

In other words, the student should focus on the thought process and mathematics

used to develop each result. These are the ‘‘tools,’’ that is, the mathematical methods

of each discipline of explicitly identifying assumptions, formally developing the

needed insights and formulas, and understanding the relationships between formulas

Introduction

xxiii

and the underlying assumptions. The tools so deﬁned and studied in this book will

equip the student with fairly robust frameworks for their applications in investment

and quantitative ﬁnance.

Despite its large size, this book has the relatively modest ambition of teaching a

very speciﬁc application of mathematics, that being to ﬁnance, and so the selection

of materials in every subdiscipline has been made parsimoniously. This selection of

materials was the most di‰cult aspect of developing this book. In general, the selection criterion I used was that a topic had to be either directly applicable to ﬁnance, or

needed for the understanding of a later topic that was directly applicable to ﬁnance.

Because my objective was to make this book more than a collection of mathematical

formulas, or just another ﬁnance recipe book, I devote considerable space to discussion on how the results are derived, and how they relate to their mathematical

assumptions. Ideally the students of this book should never again accept a formulaic

result as an immutable truth separate from any assumptions made by its originator.

The motivation for this approach is that in investment and quantitative ﬁnance,

there are few good careers that depend on the application of standard formulas in

standard situations. All such applications tend to be automated and run in companies’ computer systems with little or no human intervention. Think ‘‘program trading’’ as an example of this statement. While there is an interesting and deep theory

related to identifying so-called arbitrage opportunities, these can be formulaically

listed and programmed, and their implementation automated with little further analyst intervention.

Equally, if not more important, with new ﬁnancial products developed regularly,

there are increased demands on quants and all ﬁnance practitioners to apply the previous methodologies and adapt them appropriately to ﬁnancial analyses, pricing, risk

modeling, and risk management. Today, in practice, standard results may or may

not apply, and the most critical job of the ﬁnance quant is to determine if the traditional approach applies, and if not, to develop an appropriate modiﬁcation or even

an entirely new approach. In other words, for today’s ﬁnance quants, it has become

critical to be able to think in mathematics, and not simply to do mathematics by

rote.

The many ﬁnance applications developed in the chapters present enough detail to

be understood by someone new to the given application but in less detail than would

be appropriate for mastering the application. Ideally the reader will be familiar with

some applications and will be introduced to other applications that can, as needed,

be enhanced by further study. On my selection of mathematical topics and ﬁnance

applications, I hope to beneﬁt from the valuable comments of ﬁnance readers, whether

student or practitioner. All such feedback will be welcomed and acknowledged in future editions.

xxiv

Introduction

Plan of the Book

The ten chapters of this book are arranged so that each topic is developed based on

materials previously discussed. In a few places, however, a formula or result is introduced that could not be fully developed until a much later chapter. In fewer places, I

decided to not prove a deep result that would have brought the book too far aﬁeld

from its intended purpose. Overall, the book is intended to be self-contained, complete with respect to the materials discussed, and mathematically rigorous. The only

mathematical background required of the reader is competent skill in algebraic

manipulations and some knowledge of pre-calculus topics of graphing, exponentials

and logarithms. Thus the topics developed in this book are interrelated and applied

with the understanding that the student will be motivated to work through, with pen

or pencil and paper or by computer simulation, any derivation or example that may

be unclear and that the student has the algebraic skills and self-discipline to do so.

Of course, even when a proof or example appears clear, the student will beneﬁt in

using pencil and paper and computer simulation to clarify any missing details in derivations. Such informal exercises provide essential practice in the application of the

tools discussed, and analytical skills can be progressively sharpened by way of the

book’s formal exercises and ultimately in real world situations. While not every derivation in the book o¤ers the same amount of enlightenment on the mathematical

tools studied, or should be studied in detail before proceeding, developing the habit

of ﬁlling in details can deepen mathematical knowledge and the understanding of

how this knowledge can be applied.

I have identiﬁed the more advanced sections by an asterisk (*). The beginning

student may ﬁnd it useful to scan these sections on ﬁrst reading. These sections can

then be returned to if needed for a later application of interest. The more advanced

student may ﬁnd these sections to provide some insights on the materials they are

already familiar with. For beginning practitioners and professors of students new to

the materials, it may be useful to only scan the reasoning in the longer proofs on a

ﬁrst review before turning to the applications.

There are a number of productive approaches to the chapter sequencing of this

book for both self-study and formal classroom presentation. Professors and practitioners with good prior exposure might pick and choose chapters out of order to e‰ciently address pressing educational needs. For ﬁnance applications, again the best

approach is the one that suits the needs of the student or practitioner. Those familiar

with ﬁnance applications and aware of the math skills that need to be developed will

focus on the appropriate math sections, then proceed to the ﬁnance applications to

better understand the connections between the math and the ﬁnance. Those less fa-

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