# An Introduction to Financial Markets: A Quantitative Approach

# An Introduction to Financial Markets: A Quantitative Approach

ISBN: 978-1-118-59477-3 October 2017 784 Pages

## Description

**COVERS THE FUNDAMENTAL TOPICS IN MATHEMATICS, STATISTICS, AND FINANCIAL MANAGEMENT THAT ARE REQUIRED FOR A THOROUGH STUDY OF FINANCIAL MARKETS**

This comprehensive yet accessible book introduces students to financial markets and delves into more advanced material at a steady pace while providing motivating examples, poignant remarks, counterexamples, ideological clashes, and intuitive traps throughout. Tempered by real-life cases and actual market structures, *An Introduction to Financial Markets: A Quantitative Approach *accentuates theory through quantitative modeling whenever and wherever necessary. It focuses on the lessons learned from timely subject matter such as the impact of the recent subprime mortgage storm, the collapse of LTCM, and the harsh criticism on risk management and innovative finance. The book also provides the necessary foundations in stochastic calculus and optimization, alongside financial modeling concepts that are illustrated with relevant and hands-on examples.

*An Introduction to Financial Markets: A Quantitative Approach *starts with a complete overview of the subject matter. It then moves on to sections covering fixed income assets, equity portfolios, derivatives, and advanced optimization models. This book’s balanced and broad view of the state-of-the-art in financial decision-making helps provide readers with all the background and modeling tools needed to make “honest money” and, in the process, to become a sound professional.

- Stresses that gut feelings are not always sufficient and that “critical thinking” and real world applications are appropriate when dealing with complex social systems involving multiple players with conflicting incentives
- Features a related website that contains a solution manual for end-of-chapter problems
- Written in a modular style for tailored classroom use
- Bridges a gap for business and engineering students who are familiar with the problems involved, but are less familiar with the methodologies needed to make smart decisions

*An Introduction to Financial Markets: A Quantitative Approach *offers a balance between the need to illustrate mathematics in action and the need to understand the real life context. It is an ideal text for a first course in financial markets or investments for business, economic, statistics, engineering, decision science, and management science students.

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## Table of contents

Preface xv

About the Companion Website xix

**Part I Overview**

**1 Financial Markets: Functions, Institutions, and Traded Assets 1**

1.1 What is the purpose of finance? 2

1.2 Traded assets 12

1.2.1 The balance sheet 15

1.2.2 Assets vs. securities 20

1.2.3 Equity 22

1.2.4 Fixed income 24

1.2.5 FOREX markets 27

1.2.6 Derivatives 29

1.3 Market participants and their roles 46

1.3.1 Commercial vs. investment banks 48

1.3.2 Investment funds and insurance companies 49

1.3.3 Dealers and brokers 51

1.3.4 Hedgers, speculators, and arbitrageurs 51

1.4 Market structure and trading strategies 53

1.4.1 Primary and secondary markets 53

1.4.2 Over-the-counter vs. exchange-traded derivatives 53

1.4.3 Auction mechanisms and the limit order book 53

1.4.4 Buying on margin and leverage 55

1.4.5 Short-selling 58

1.5 Market indexes 60

Problems 63

Further reading 65

Bibliography 65

**2 Basic Problems in Quantitative Finance 67**

2.1 Portfolio optimization 68

2.1.1 Static portfolio optimization: Mean–variance efficiency 70

2.1.2 Dynamic decision-making under uncertainty: A stylized consumption–saving model 75

2.2 Risk measurement and management 80

2.2.1 Sensitivity of asset prices to underlying risk factors 81

2.2.2 Risk measures in a non-normal world: Value-atrisk 84

2.2.3 Risk management: Introductory hedging examples 93

2.2.4 Financial vs. nonfinancial risk factors 100

2.3 The no-arbitrage principle in asset pricing 102

2.3.1 Why do we need asset pricing models? 103

2.3.2 Arbitrage strategies 104

2.3.3 Pricing by no-arbitrage 108

2.3.4 Option pricing in a binomial model 112

2.3.5 The limitations of the no-arbitrage principle 116

2.4 The mathematics of arbitrage 117

2.4.1 Linearity of the pricing functional and law of one price 119

2.4.2 Dominant strategies 120

2.4.3 No-arbitrage principle and risk-neutral measures 125

S2.1 Multiobjective optimization 129

S2.2 Summary of LP duality 133

Problems 137

Further reading 139

Bibliography 139

**Part II Fixed income assets**

**3 Elementary Theory of Interest Rates 143**

3.1 The time value of money: Shifting money forward in time 146

3.1.1 Simple vs. compounded rates 147

3.1.2 Quoted vs. effective rates: Compounding frequencies 150

3.2 The time value of money: Shifting money backward in time 153

3.2.1 Discount factors and pricing a zero-coupon bond 154

3.2.2 Discount factors vs. interest rates 158

3.3 Nominal vs. real interest rates 161

3.4 The term structure of interest rates 163

3.5 Elementary bond pricing 165

3.5.1 Pricing coupon-bearing bonds 165

3.5.2 From bond prices to term structures, and vice versa 168

3.5.3 What is a risk-free rate, anyway? 171

3.5.4 Yield-to-maturity 174

3.5.5 Interest rate risk 180

3.5.6 Pricing floating rate bonds 188

3.6 A digression: Elementary investment analysis 190

3.6.1 Net present value 191

3.6.2 Internal rate of return 192

3.6.3 Real options 193

3.7 Spot vs. forward interest rates 193

3.7.1 The forward and the spot rate curves 197

3.7.2 Discretely compounded forward rates 197

3.7.3 Forward discount factors 198

3.7.4 The expectation hypothesis 199

3.7.5 A word of caution: Model risk and hidden assumptions 202

S3.1 Proof of Equation (3.42) 203 Problems 203

Further reading 205

Bibliography 205

**4 Forward Rate Agreements, Interest Rate Futures, and Vanilla Swaps 207**

4.1 LIBOR and EURIBOR rates 208

4.2 Forward rate agreements 209

4.2.1 A hedging view of forward rates 210

4.2.2 FRAs as bond trades 214

4.2.3 A numerical example 215

4.3 Eurodollar futures 216

4.4 Vanilla interest rate swaps 220

4.4.1 Swap valuation: Approach 1 221

4.4.2 Swap valuation: Approach 2 223

4.4.3 The swap curve and the term structure 225

Problems 226

Further reading 226

Bibliography 226

**5 Fixed-Income Markets 229**

5.1 Day count conventions 230

5.2 Bond markets 231

5.2.1 Bond credit ratings 233

5.2.2 Quoting bond prices 233

5.2.3 Bonds with embedded options 235

5.3 Interest rate derivatives 237

5.3.1 Swap markets 237

5.3.2 Bond futures and options 238

5.4 The repo market and other money market instruments 239

5.5 Securitization 240

Problems 244

Further reading 244

Bibliography 244

**6 Interest Rate Risk Management 247**

6.1 Duration as a first-order sensitivity measure 248

6.1.1 Duration of fixed-coupon bonds 250

6.1.2 Duration of a floater 254

6.1.3 Dollar duration and interest rate swaps 255

6.2 Further interpretations of duration 257

6.2.1 Duration and investment horizons 258

6.2.2 Duration and yield volatility 260

6.2.3 Duration and quantile-based risk measures 260

6.3 Classical duration-based immunization 261

6.3.1 Cash flow matching 262

6.3.2 Duration matching 263

6.4 Immunization by interest rate derivatives 265

6.4.1 Using interest rate swaps in asset–liability management 266

6.5 A second-order refinement: Convexity 266

6.6 Multifactor models in interest rate risk management 269

Problems 271

Further reading 272

Bibliography 273

**Part III Equity portfolios**

**7 Decision-Making under Uncertainty: The Static Case 277**

7.1 Introductory examples 278

7.2 Should we just consider expected values of returns and monetary outcomes? 282

7.2.1 Formalizing static decision-making under uncertainty 283

7.2.2 The flaw of averages 284

7.3 A conceptual tool: The utility function 288

7.3.1 A few standard utility functions 293

7.3.2 Limitations of utility functions 297

7.4 Mean–risk models 299

7.4.1 Coherent risk measures 300

7.4.2 Standard deviation and variance as risk measures 302

7.4.3 Quantile-based risk measures: V@R and CV@R 303

7.4.4 Formulation of mean–risk models 309

7.5 Stochastic dominance 310

S7.1 Theorem proofs 314

S7.1.1 Proof of Theorem 7.2 314

S7.1.2 Proof of Theorem 7.4 315

Problems 315

Further reading 317

Bibliography 317

**8 Mean–Variance Efficient Portfolios 319**

8.1 Risk aversion and capital allocation to risky assets 320

8.1.1 The role of risk aversion 324

8.2 The mean–variance efficient frontier with risky assets 325

8.2.1 Diversification and portfolio risk 325

8.2.2 The efficient frontier in the case of two risky assets 326

8.2.3 The efficient frontier in the case of n risky assets 329

8.3 Mean–variance efficiency with a risk-free asset: The separation property 332

8.4 Maximizing the Sharpe ratio 337

8.4.1 Technical issues in Sharpe ratio maximization 340

8.5 Mean–variance efficiency vs. expected utility 341

8.6 Instability in mean–variance portfolio optimization 343

S8.1 The attainable set for two risky assets is a hyperbola 345

S8.2 Explicit solution of mean–variance optimization in matrix form 346

Problems 348

Further reading 349

Bibliography 349

**9 Factor Models 351**

9.1 Statistical issues in mean–variance portfolio optimization 352

9.2 The single-index model 353

9.2.1 Estimating a factor model 354

9.2.2 Portfolio optimization within the single-index model 356

9.3 The Treynor–Black model 358

9.3.1 A top-down/bottom-up optimization procedure 362

9.4 Multifactor models 365

9.5 Factor models in practice 367

S9.1 Proof of Equation (9.17) 368

Problems 369

Further reading 371

Bibliography 371

**10 Equilibrium Models: CAPM and APT 373**

10.1 What is an equilibrium model? 374

10.2 The capital asset pricing model 375

10.2.1 Proof of the CAPM formula 377

10.2.2 Interpreting CAPM 378

10.2.3 CAPM as a pricing formula and its practical relevance 380

10.3 The Black–Litterman portfolio optimization model 381

10.3.1 Black–Litterman model: The role of CAPM and Bayesian Statistics 382

10.3.2 Black-Litterman model: A numerical example 386

10.4 Arbitrage pricing theory 388

10.4.1 The intuition 389

10.4.2 A not-so-rigorous proof of APT 391

10.4.3 APT for Well-Diversified Portfolios 392

10.4.4 APT for Individual Assets 393

10.4.5 Interpreting and using APT 394

10.5 The behavioral critique 398

10.5.1 The efficient market hypothesis 400

10.5.2 The psychology of choice by agents with limited rationality 400

10.5.3 Prospect theory: The aversion to sure loss 401

S10.1Bayesian statistics 404

S10.1.1 Bayesian estimation 405

S10.1.2 Bayesian learning in coin flipping 407

S10.1.3 The expected value of a normal distribution 408

Problems 411

Further reading 413

Bibliography 413

**Part IV Derivatives**

**11 Modeling Dynamic Uncertainty 417**

11.1 Stochastic processes 420

11.1.1 Introductory examples 422

11.1.2 Marginals do not tell the whole story 428

11.1.3 Modeling information: Filtration generated by a stochastic process 430

11.1.4 Markov processes 433

11.1.5 Martingales 436

11.2 Stochastic processes in continuous time 438

11.2.1 A fundamental building block: Standard Wiener process 438

11.2.2 A generalization: Lévy processes 440

11.3 Stochastic differential equations 441

11.3.1 A deterministic differential equation: The bank account process 442

11.3.2 The generalized Wiener process 443

11.3.3 Geometric Brownian motion and Itô processes 445

11.4 Stochastic integration and Itô’s lemma 447

11.4.1 A digression: Riemann and Riemann–Stieltjes integrals 447

11.4.2 Stochastic integral in the sense of Itô 448

11.4.3 Itô’s lemma 453

11.5 Stochastic processes in financial modeling 457

11.5.1 Geometric Brownian motion 457

11.5.2 Generalizations 460

11.6 Sample path generation 462

11.6.1 Monte Carlo sampling 463

11.6.2 Scenario trees 465

S11.1Probability spaces, measurability, and information 468

Problems 476

Further reading 478

Bibliography 478

**12 Forward and Futures Contracts 481**

12.1 Pricing forward contracts on equity and foreign currencies 482

12.1.1 The spot–forward parity theorem 482

12.1.2 The spot–forward parity theorem with dividend income 485

12.1.3 Forward contracts on currencies 487

12.1.4 Forward contracts on commodities or energy: Contango and backwardation 489

12.2 Forward vs. futures contracts 490

12.3 Hedging with linear contracts 493

12.3.1 Quantity-based hedging 493

12.3.2 Basis risk and minimum variance hedging 494

12.3.3 Hedging with index futures 496

12.3.4 Tailing the hedge 499

Problems 501

Further reading 502

Bibliography 502

**13 Option Pricing: Complete Markets 505**

13.1 Option terminology 506

13.1.1 Vanilla options 507

13.1.2 Exotic options 508

13.2 Model-free price restrictions 510

13.2.1 Bounds on call option prices 511

13.2.2 Bounds on put option prices: Early exercise and continuation regions 514

13.2.3 Parity relationships 517

13.3 Binomial option pricing 519

13.3.1 A hedging argument 520

13.3.2 Lattice calibration 523

13.3.3 Generalization to multiple steps 524

13.3.4 Binomial pricing of American-style options 527

13.4 A continuous-time model: The Black–Scholes–Merton pricing formula 530

13.4.1 The delta-hedging view 532

13.4.2 The risk-neutral view: Feynman–Ka¡c representation theorem 539

13.4.3 Interpreting the factors in the BSM formula 543

13.5 Option price sensitivities: The Greeks 545

13.5.1 Delta and gamma 546

13.5.2 Theta 550

13.5.3 Relationship between delta, gamma, and theta 551

13.5.4 Vega 552

13.6 The role of volatility 553

13.6.1 The implied volatility surface 553

13.6.2 The impact of volatility on barrier options 555

13.7 Options on assets providing income 556

13.7.1 Index options 557

13.7.2 Currency options 558

13.7.3 Futures options 559

13.7.4 The mechanics of futures options 559

13.7.5 A binomial view of futures options 560

13.7.6 A risk-neutral view of futures options 562

13.8 Portfolio strategies based on options 562

13.8.1 Portfolio insurance and the Black Monday of 1987 563

13.8.2 Volatility trading 564

13.8.3 Dynamic vs. Static hedging 566

13.9 Option pricing by numerical methods 569

Problems 570

Further reading 575

Bibliography 576

**14 Option Pricing: Incomplete Markets 579**

14.1 A PDE approach to incomplete markets 581

14.1.1 Pricing a zero-coupon bond in a driftless world 584

14.2 Pricing by short-rate models 588

14.2.1 The Vasicek short-rate model 589

14.2.2 The Cox–Ingersoll–Ross short-rate model 594

14.3 A martingale approach to incomplete markets 595

14.3.1 An informal approach to martingale equivalent measures 598

14.3.2 Choice of numeraire: The bank account 600

14.3.3 Choice of numeraire: The zero-coupon bond 601

14.3.4 Pricing options with stochastic interest rates: Black’s model 602

14.3.5 Extensions 603

14.4 Issues in model calibration 603

14.4.1 Bias–variance tradeoff and regularized least-squares 604

14.4.2 Financial model calibration 609

Further reading 612

Bibliography 612

**Part V Advanced optimization models**

**15 Optimization Model Building 617**

15.1 Classification of optimization models 618

15.2 Linear programming 625

15.2.1 Cash flow matching 627

15.3 Quadratic programming 628

15.3.1 Maximizing the Sharpe ratio 629

15.3.2 Quadratically constrained quadratic programming 631

15.4 Integer programming 632

15.4.1 A MIQP model to minimize TEV under a cardinality constraint 634

15.4.2 Good MILP model building: The role of tight model formulations 636

15.5 Conic optimization 642

15.5.1 Convex cones 644

15.5.2 Second-order cone programming 650

15.5.3 Semidefinite programming 653

15.6 Stochastic optimization 655

15.6.1 Chance-constrained LP models 656

15.6.2 Two-stage stochastic linear programming with recourse 657

15.6.3 Multistage stochastic linear programming with recourse 663

15.6.4 Scenario generation and stability in stochastic programming 670

15.7 Stochastic dynamic programming 675

15.7.1 The dynamic programming principle 676

15.7.2 Solving Bellman’s equation: The three curses of dimensionality 679

15.7.3 Application to pricing options with early exercise features 680

15.8 Decision rules for multistage SLPs 682

15.9 Worst-case robust models 686

15.9.1 Uncertain LPs: Polyhedral uncertainty 689

15.9.2 Uncertain LPs: Ellipsoidal uncertainty 690

15.10Nonlinear programming models in finance 691

15.10.1 Fixed-mix asset allocation 692

Problems 693

Further reading 695

Bibliography 696

**16 Optimization Model Solving 699**

16.1 Local methods for nonlinear programming 700

16.1.1 Unconstrained nonlinear programming 700

16.1.2 Penalty function methods 703

16.1.3 Lagrange multipliers and constraint qualification conditions 707

16.1.4 Duality theory 713

16.2 Global methods for nonlinear programming 715

16.2.1 Genetic algorithms 716

16.2.2 Particle swarm optimization 717

16.3 Linear programming 719

16.3.1 The simplex method 720

16.3.2 Duality in linear programming 723

16.3.3 Interior-point methods: Primal-dual barrier method for LP 726

16.4 Conic duality and interior-point methods 728

16.4.1 Conic duality 728

16.4.2 Interior-point methods for SOCP and SDP 731

16.5 Branch-and-bound methods for integer programming 732

16.5.1 A matheuristic approach: Fix-and-relax 735

16.6 Optimization software 736

16.6.1 Solvers 737

16.6.2 Interfacing through imperative programming languages 738

16.6.3 Interfacing through non-imperative algebraic languages 738

16.6.4 Additional interfaces 739

Problems 739

Further reading 740

Bibliography 741

Index 743