# Introduction to quantitative finance

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
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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.
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
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 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
Generalized Geometric and Related Distributions

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

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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