Introduction to Quantitative Finance: A Math Tool Kit

A MATH TOOL KIT

INTRODUCTION TO

QUANTITATIVE

FINANCE

Robert R. Reitano

Reitano_JKT.indd 1 1/12/10 10:00 AM

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

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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

332.010

5195—dc22 2009022214

10 9 8 7 6 5 4 3 2 1

to Lisa

Contents

List of Figures and Tables xix

Introduction xxi

1 Mathematical Logic 1

1.1 Introduction 1

1.2 Axiomatic Theory 4

1.3 Inferences 6

1.4 Paradoxes 7

1.5 Propositional Logic 10

1.5.1 Truth Tables 10

1.5.2 Framework of a Proof 15

1.5.3 Methods of Proof 17

The Direct Proof 19

Proof by Contradiction 19

Proof by Induction 21

*1.6 Mathematical Logic 23

1.7 Applications to Finance 24

Exercises 27

2 Number Systems and Functions 31

2.1 Numbers: Properties and Structures 31

2.1.1 Introduction 31

2.1.2 Natural Numbers 32

2.1.3 Integers 37

2.1.4 Rational Numbers 38

2.1.5 Real Numbers 41

*2.1.6 Complex Numbers 44

2.2 Functions 49

2.3 Applications to Finance 51

2.3.1 Number Systems 51

2.3.2 Functions 54

Present Value Functions 54

Accumulated Value Functions 55

Nominal Interest Rate Conversion Functions 56

Bond-Pricing Functions 57

Mortgage- and Loan-Pricing Functions 59

Preferred Stock-Pricing Functions 59

Common Stock-Pricing Functions 60

Portfolio Return Functions 61

Forward-Pricing Functions 62

Exercises 64

3 Euclidean and Other Spaces 71

3.1 Euclidean Space 71

3.1.1 Structure and Arithmetic 71

3.1.2 Standard Norm and Inner Product for Rn 73

*3.1.3 Standard Norm and Inner Product for Cn 74

3.1.4 Norm and Inner Product Inequalities for Rn 75

*3.1.5 Other Norms and Norm Inequalities for Rn 77

3.2 Metric Spaces 82

3.2.1 Basic Notions 82

3.2.2 Metrics and Norms Compared 84

*3.2.3 Equivalence of Metrics 88

3.3 Applications to Finance 93

3.3.1 Euclidean Space 93

Asset Allocation Vectors 94

Interest Rate Term Structures 95

Bond Yield Vector Risk Analysis 99

Cash Flow Vectors and ALM 100

3.3.2 Metrics and Norms 101

Sample Statistics 101

Constrained Optimization 103

Tractability of the lp-Norms: An Optimization Example 105

General Optimization Framework 110

Exercises 112

4 Set Theory and Topology 117

4.1 Set Theory 117

4.1.1 Historical Background 117

*4.1.2 Overview of Axiomatic Set Theory 118

4.1.3 Basic Set Operations 121

4.2 Open, Closed, and Other Sets 122

viii Contents

4.2.1 Open and Closed Subsets of R 122

4.2.2 Open and Closed Subsets of Rn 127

*4.2.3 Open and Closed Subsets in Metric Spaces 128

*4.2.4 Open and Closed Subsets in General Spaces 129

4.2.5 Other Properties of Subsets of a Metric Space 130

4.3 Applications to Finance 134

4.3.1 Set Theory 134

4.3.2 Constrained Optimization and Compactness 135

4.3.3 Yield of a Security 137

Exercises 139

5 Sequences and Their Convergence 145

5.1 Numerical Sequences 145

5.1.1 Definition and Examples 145

5.1.2 Convergence of Sequences 146

5.1.3 Properties of Limits 149

*5.2 Limits Superior and Inferior 152

*5.3 General Metric Space Sequences 157

5.4 Cauchy Sequences 162

5.4.1 Definition and Properties 162

*5.4.2 Complete Metric Spaces 165

5.5 Applications to Finance 167

5.5.1 Bond Yield to Maturity 167

5.5.2 Interval Bisection Assumptions Analysis 170

Exercises 172

6 Series and Their Convergence 177

6.1 Numerical Series 177

6.1.1 Definitions 177

6.1.2 Properties of Convergent Series 178

6.1.3 Examples of Series 180

*6.1.4 Rearrangements of Series 184

6.1.5 Tests of Convergence 190

6.2 The lp-Spaces 196

6.2.1 Definition and Basic Properties 196

*6.2.2 Banach Space 199

*6.2.3 Hilbert Space 202

Contents ix

6.3 Power Series 206

*6.3.1 Product of Power Series 209

*6.3.2 Quotient of Power Series 212

6.4 Applications to Finance 215

6.4.1 Perpetual Security Pricing: Preferred Stock 215

6.4.2 Perpetual Security Pricing: Common Stock 217

6.4.3 Price of an Increasing Perpetuity 218

6.4.4 Price of an Increasing Payment Security 220

6.4.5 Price Function Approximation: Asset Allocation 222

6.4.6 lp-Spaces: Banach and Hilbert 223

Exercises 224

7 Discrete Probability Theory 231

7.1 The Notion of Randomness 231

7.2 Sample Spaces 233

7.2.1 Undefined Notions 233

7.2.2 Events 234

7.2.3 Probability Measures 235

7.2.4 Conditional Probabilities 238

Law of Total Probability 239

7.2.5 Independent Events 240

7.2.6 Independent Trials: One Sample Space 241

*7.2.7 Independent Trials: Multiple Sample Spaces 245

7.3 Combinatorics 247

7.3.1 Simple Ordered Samples 247

With Replacement 247

Without Replacement 247

7.3.2 General Orderings 248

Two Subset Types 248

Binomial Coe‰cients 249

The Binomial Theorem 250

r Subset Types 251

Multinomial Theorem 252

7.4 Random Variables 252

7.4.1 Quantifying Randomness 252

7.4.2 Random Variables and Probability Functions 254

x Contents

7.4.3 Random Vectors and Joint Probability Functions 256

7.4.4 Marginal and Conditional Probability Functions 258

7.4.5 Independent Random Variables 261

7.5 Expectations of Discrete Distributions 264

7.5.1 Theoretical Moments 264

Expected Values 264

Conditional and Joint Expectations 266

Mean 268

Variance 268

Covariance and Correlation 271

General Moments 274

General Central Moments 274

Absolute Moments 274

Moment-Generating Function 275

Characteristic Function 277

*7.5.2 Moments of Sample Data 278

Sample Mean 280

Sample Variance 282

Other Sample Moments 286

7.6 Discrete Probability Density Functions 287

7.6.1 Discrete Rectangular Distribution 288

7.6.2 Binomial Distribution 290

7.6.3 Geometric Distribution 292

7.6.4 Multinomial Distribution 293

7.6.5 Negative Binomial Distribution 296

7.6.6 Poisson Distribution 299

7.7 Generating Random Samples 301

7.8 Applications to Finance 307

7.8.1 Loan Portfolio Defaults and Losses 307

Individual Loss Model 307

Aggregate Loss Model 310

7.8.2 Insurance Loss Models 313

7.8.3 Insurance Net Premium Calculations 314

Generalized Geometric and Related Distributions 314

Life Insurance Single Net Premium 317

Contents xi

Pension Benefit Single Net Premium 318

Life Insurance Periodic Net Premiums 319

7.8.4 Asset Allocation Framework 319

7.8.5 Equity Price Models in Discrete Time 325

Stock Price Data Analysis 325

Binomial Lattice Model 326

Binomial Scenario Model 328

7.8.6 Discrete Time European Option Pricing: Lattice-Based 329

One-Period Pricing 329

Multi-period Pricing 333

7.8.7 Discrete Time European Option Pricing: Scenario Based 336

Exercises 337

8 Fundamental Probability Theorems 347

8.1 Uniqueness of the m.g.f. and c.f. 347

8.2 Chebyshev’s Inequality 349

8.3 Weak Law of Large Numbers 352

8.4 Strong Law of Large Numbers 357

8.4.1 Model 1: Independent fX^ng 359

8.4.2 Model 2: Dependent fX^ng 360

8.4.3 The Strong Law Approach 362

*8.4.4 Kolmogorov’s Inequality 363

*8.4.5 Strong Law of Large Numbers 365

8.5 De Moivre–Laplace Theorem 368

8.5.1 Stirling’s Formula 371

8.5.2 De Moivre–Laplace Theorem 374

8.5.3 Approximating Binomial Probabilities I 376

8.6 The Normal Distribution 377

8.6.1 Definition and Properties 377

8.6.2 Approximating Binomial Probabilities II 379

*8.7 The Central Limit Theorem 381

8.8 Applications to Finance 386

8.8.1 Insurance Claim and Loan Loss Tail Events 386

Risk-Free Asset Portfolio 387

Risky Assets 391

8.8.2 Binomial Lattice Equity Price Models as Dt ! 0 392

xii Contents

Parameter Dependence on Dt 394

Distributional Dependence on Dt 395

Real World Binomial Distribution as Dt ! 0 396

8.8.3 Lattice-Based European Option Prices as Dt ! 0 400

The Model 400

European Call Option Illustration 402

Black–Scholes–Merton Option-Pricing Formulas I 404

8.8.4 Scenario-Based European Option Prices as N ! y 406

The Model 406

Option Price Estimates as N ! y 407

Scenario-Based Prices and Replication 409

Exercises 411

9 Calculus I: Di¤erentiation 417

9.1 Approximating Smooth Functions 417

9.2 Functions and Continuity 418

9.2.1 Functions 418

9.2.2 The Notion of Continuity 420

The Meaning of ‘‘Discontinuous’’ 425

*The Metric Notion of Continuity 428

Sequential Continuity 429

9.2.3 Basic Properties of Continuous Functions 430

9.2.4 Uniform Continuity 433

9.2.5 Other Properties of Continuous Functions 437

9.2.6 Ho¨lder and Lipschitz Continuity 439

‘‘Big O’’ and ‘‘Little o’’ Convergence 440

9.2.7 Convergence of a Sequence of Continuous Functions 442

*Series of Functions 445

*Interchanging Limits 445

*9.2.8 Continuity and Topology 448

9.3 Derivatives and Taylor Series 450

9.3.1 Improving an Approximation I 450

9.3.2 The First Derivative 452

9.3.3 Calculating Derivatives 454

A Discussion of e 461

9.3.4 Properties of Derivatives 462

Contents xiii

9.3.5 Improving an Approximation II 465

9.3.6 Higher Order Derivatives 466

9.3.7 Improving an Approximation III: Taylor Series

Approximations 467

Analytic Functions 470

9.3.8 Taylor Series Remainder 473

9.4 Convergence of a Sequence of Derivatives 478

9.4.1 Series of Functions 481

9.4.2 Di¤erentiability of Power Series 481

Product of Taylor Series 486

*Division of Taylor Series 487

9.5 Critical Point Analysis 488

9.5.1 Second-Derivative Test 488

*9.5.2 Critical Points of Transformed Functions 490

9.6 Concave and Convex Functions 494

9.6.1 Definitions 494

9.6.2 Jensen’s Inequality 500

9.7 Approximating Derivatives 504

9.7.1 Approximating f 0

ðxÞ 504

9.7.2 Approximating f 00ðxÞ 504

9.7.3 Approximating f ðnÞ

ðxÞ, n > 2 505

9.8 Applications to Finance 505

9.8.1 Continuity of Price Functions 505

9.8.2 Constrained Optimization 507

9.8.3 Interval Bisection 507

9.8.4 Minimal Risk Asset Allocation 508

9.8.5 Duration and Convexity Approximations 509

Dollar-Based Measures 511

Embedded Options 512

Rate Sensitivity of Duration 513

9.8.6 Asset–Liability Management 514

Surplus Immunization, Time t ¼ 0 518

Surplus Immunization, Time t > 0 519

Surplus Ratio Immunization 520

9.8.7 The ‘‘Greeks’’ 521

xiv Contents