An Introduction to Financial Markets A Quantitative Approach

An Introduction to Financial MarketsA Quantitative Approach

Paolo Brandimarte

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Contents

Part IOverview

1 Financial Markets: Functions, Institutions, and Traded Assets 11.1 What is the purpose of finance? 21.2 Traded assets 12

1.2.1 The balance sheet 151.2.2 Assets vs. securities 201.2.3 Equity 221.2.4 Fixed income 241.2.5 FOREX markets 271.2.6 Derivatives 29

1.3 Market participants and their roles 461.3.1 Commercial vs. investment banks 481.3.2 Investment funds and insurance companies 491.3.3 Dealers and brokers 511.3.4 Hedgers, speculators, and arbitrageurs 51

1.4 Market structure and trading strategies 531.4.1 Primary and secondary markets 531.4.2 Over-the-counter vs. exchange-traded derivatives 531.4.3 Auction mechanisms and the limit order book 531.4.4 Buying on margin and leverage 551.4.5 Short-selling 58

1.5 Market indexes 60Problems 63Further reading 65Bibliography 65

2 Basic Problems in Quantitative Finance 672.1 Portfolio optimization 68

2.1.1 Static portfolio optimization: Mean–varianceefficiency 70

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

2.2 Risk measurement and management 80

v

Preface xvAbout the Companion Website xix

vi CONTENTS

2.2.1 Sensitivity of asset prices to underlying risk factors 812.2.2 Risk measures in a non-normal world: Value-at-

risk 842.2.3 Risk management: Introductory hedging examples 932.2.4 Financial vs. nonfinancial risk factors 100

2.3 The no-arbitrage principle in asset pricing 1022.3.1 Why do we need asset pricing models? 1032.3.2 Arbitrage strategies 1042.3.3 Pricing by no-arbitrage 1082.3.4 Option pricing in a binomial model 1122.3.5 The limitations of the no-arbitrage principle 116

2.4 The mathematics of arbitrage 1172.4.1 Linearity of the pricing functional and law of one

price 1192.4.2 Dominant strategies 1202.4.3 No-arbitrage principle and risk-neutral measures 125

S2.1 Multiobjective optimization 129S2.2 Summary of LP duality 133Problems 137Further reading 139Bibliography 139

Part IIFixed-income assets

3 Elementary Theory of Interest Rates 1433.1 The time value of money: Shifting money forward in time 146

3.1.1 Simple vs. compounded rates 1473.1.2 Quoted vs. effective rates: Compounding

frequencies 1503.2 The time value of money: Shifting money backward in time 153

3.2.1 Discount factors and pricing a zero-coupon bond 1543.2.2 Discount factors vs. interest rates 158

3.3 Nominal vs. real interest rates 1613.4 The term structure of interest rates 1633.5 Elementary bond pricing 165

3.5.1 Pricing coupon-bearing bonds 1653.5.2 From bond prices to term structures, and vice versa 1683.5.3 What is a risk-free rate, anyway? 1713.5.4 Yield-to-maturity 1743.5.5 Interest rate risk 1803.5.6 Pricing floating rate bonds 188

3.6 A digression: Elementary investment analysis 1903.6.1 Net present value 1913.6.2 Internal rate of return 192

CONTENTS vii

3.6.3 Real options 1933.7 Spot vs. forward interest rates 193

3.7.1 The forward and the spot rate curves 1973.7.2 Discretely compounded forward rates 1973.7.3 Forward discount factors 1983.7.4 The expectation hypothesis 1993.7.5 A word of caution: Model risk and hidden

assumptions 202S3.1 Proof of Equation (3.42) 203Problems 203Further reading 205Bibliography 205

4 Forward Rate Agreements, Interest Rate Futures, and Vanilla Swaps 2074.1 LIBOR and EURIBOR rates 2084.2 Forward rate agreements 209

4.2.1 A hedging view of forward rates 2104.2.2 FRAs as bond trades 2144.2.3 A numerical example 215

4.3 Eurodollar futures 2164.4 Vanilla interest rate swaps 220

4.4.1 Swap valuation: Approach 1 2214.4.2 Swap valuation: Approach 2 2234.4.3 The swap curve and the term structure 225

Problems 226Further reading 226Bibliography 226

5 Fixed-Income Markets 2295.1 Day count conventions 2305.2 Bond markets 231

5.2.1 Bond credit ratings 2335.2.2 Quoting bond prices 2335.2.3 Bonds with embedded options 235

5.3 Interest rate derivatives 2375.3.1 Swap markets 2375.3.2 Bond futures and options 238

5.4 The repo market and other money market instruments 2395.5 Securitization 240Problems 244Further reading 244Bibliography 244

6 Interest Rate Risk Management 2476.1 Duration as a first-order sensitivity measure 248

6.1.1 Duration of fixed-coupon bonds 250

viii CONTENTS

6.1.2 Duration of a floater 2546.1.3 Dollar duration and interest rate swaps 255

6.2 Further interpretations of duration 2576.2.1 Duration and investment horizons 2586.2.2 Duration and yield volatility 2606.2.3 Duration and quantile-based risk measures 260

6.3 Classical duration-based immunization 2616.3.1 Cash flow matching 2626.3.2 Duration matching 263

6.4 Immunization by interest rate derivatives 2656.4.1 Using interest rate swaps in asset–liability

management 2666.5 A second-order refinement: Convexity 2666.6 Multifactor models in interest rate risk management 269Problems 271Further reading 272Bibliography 273

Part IIIEquity portfolios

7 Decision-Making under Uncertainty: The Static Case 2777.1 Introductory examples 2787.2 Should we just consider expected values of returns and monetary

outcomes? 2827.2.1 Formalizing static decision-making under uncertainty 2837.2.2 The flaw of averages 284

7.3 A conceptual tool: The utility function 2887.3.1 A few standard utility functions 2937.3.2 Limitations of utility functions 297

7.4 Mean–risk models 2997.4.1 Coherent risk measures 3007.4.2 Standard deviation and variance as risk measures 3027.4.3 Quantile-based risk measures: V@R and CV@R 3037.4.4 Formulation of mean–risk models 309

7.5 Stochastic dominance 310S7.1 Theorem proofs 314

S7.1.1 Proof of Theorem 7.2 314S7.1.2 Proof of Theorem 7.4 315

Problems 315Further reading 317Bibliography 317

8 Mean–Variance Efficient Portfolios 3198.1 Risk aversion and capital allocation to risky assets 320

CONTENTS ix

8.1.1 The role of risk aversion 3248.2 The mean–variance efficient frontier with risky assets 325

8.2.1 Diversification and portfolio risk 3258.2.2 The efficient frontier in the case of two risky assets 3268.2.3 The efficient frontier in the case of n risky assets 329

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

8.4 Maximizing the Sharpe ratio 3378.4.1 Technical issues in Sharpe ratio maximization 340

8.5 Mean–variance efficiency vs. expected utility 3418.6 Instability in mean–variance portfolio optimization 343S8.1 The attainable set for two risky assets is a hyperbola 345S8.2 Explicit solution of mean–variance optimization in matrix form 346Problems 348Further reading 349Bibliography 349

9 Factor Models 3519.1 Statistical issues in mean–variance portfolio optimization 3529.2 The single-index model 353

9.2.1 Estimating a factor model 3549.2.2 Portfolio optimization within the single-index model 356

9.3 The Treynor–Black model 3589.3.1 A top-down/bottom-up optimization procedure 362

9.4 Multifactor models 3659.5 Factor models in practice 367S9.1 Proof of Equation (9.17) 368Problems 369Further reading 371Bibliography 371

10 Equilibrium Models: CAPM and APT 37310.1 What is an equilibrium model? 37410.2 The capital asset pricing model 375

10.2.1 Proof of the CAPM formula 37710.2.2 Interpreting CAPM 37810.2.3 CAPM as a pricing formula and its practical relevance 380

10.3 The Black–Litterman portfolio optimization model 38110.3.1 Black–Litterman model: The role of CAPM and

Bayesian Statistics 38210.3.2 Black-Litterman model: A numerical example 386

10.4 Arbitrage pricing theory 38810.4.1 The intuition 38910.4.2 A not-so-rigorous proof of APT 39110.4.3 APT for Well-Diversified Portfolios 39210.4.4 APT for Individual Assets 393

x CONTENTS

10.4.5 Interpreting and using APT 39410.5 The behavioral critique 398

10.5.1 The efficient market hypothesis 40010.5.2 The psychology of choice by agents with limited

rationality 40010.5.3 Prospect theory: The aversion to sure loss 401

S10.1Bayesian statistics 404S10.1.1 Bayesian estimation 405S10.1.2 Bayesian learning in coin flipping 407S10.1.3 The expected value of a normal distribution 408

Problems 411Further reading 413Bibliography 413

Part IVDerivatives

11 Modeling Dynamic Uncertainty 41711.1 Stochastic processes 420

11.1.1 Introductory examples 42211.1.2 Marginals do not tell the whole story 42811.1.3 Modeling information: Filtration generated by a

stochastic process 43011.1.4 Markov processes 43311.1.5 Martingales 436

11.2 Stochastic processes in continuous time 43811.2.1 A fundamental building block: Standard Wiener

process 43811.2.2 A generalization: Lévy processes 440

11.3 Stochastic differential equations 44111.3.1 A deterministic differential equation: The bank

account process 44211.3.2 The generalized Wiener process 44311.3.3 Geometric Brownian motion and Itô processes 445

11.4 Stochastic integration and Itô’s lemma 44711.4.1 A digression: Riemann and Riemann–Stieltjes integrals 44711.4.2 Stochastic integral in the sense of Itô 44811.4.3 Itô’s lemma 453

11.5 Stochastic processes in financial modeling 45711.5.1 Geometric Brownian motion 45711.5.2 Generalizations 460

11.6 Sample path generation 46211.6.1 Monte Carlo sampling 46311.6.2 Scenario trees 465

S11.1Probability spaces, measurability, and information 468

CONTENTS xi

Problems 476Further reading 478Bibliography 478

12 Forward and Futures Contracts 48112.1 Pricing forward contracts on equity and foreign currencies 482

12.1.1 The spot–forward parity theorem 48212.1.2 The spot–forward parity theorem with dividend

income 48512.1.3 Forward contracts on currencies 48712.1.4 Forward contracts on commodities or energy: Contango

and backwardation 48912.2 Forward vs. futures contracts 49012.3 Hedging with linear contracts 493

12.3.1 Quantity-based hedging 49312.3.2 Basis risk and minimum variance hedging 49412.3.3 Hedging with index futures 49612.3.4 Tailing the hedge 499

Problems 501Further reading 502Bibliography 502

13 Option Pricing: Complete Markets 50513.1 Option terminology 506

13.1.1 Vanilla options 50713.1.2 Exotic options 508

13.2 Model-free price restrictions 51013.2.1 Bounds on call option prices 51113.2.2 Bounds on put option prices: Early exercise and

continuation regions 51413.2.3 Parity relationships 517

13.3 Binomial option pricing 51913.3.1 A hedging argument 52013.3.2 Lattice calibration 52313.3.3 Generalization to multiple steps 52413.3.4 Binomial pricing of American-style options 527

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

13.4.1 The delta-hedging view 53213.4.2 The risk-neutral view: Feynman–Kač representation

theorem 53913.4.3 Interpreting the factors in the BSM formula 543

13.5 Option price sensitivities: The Greeks 54513.5.1 Delta and gamma 54613.5.2 Theta 55013.5.3 Relationship between delta, gamma, and theta 551

xii CONTENTS

13.5.4 Vega 55213.6 The role of volatility 553

13.6.1 The implied volatility surface 55313.6.2 The impact of volatility on barrier options 555

13.7 Options on assets providing income 55613.7.1 Index options 55713.7.2 Currency options 55813.7.3 Futures options 55913.7.4 The mechanics of futures options 55913.7.5 A binomial view of futures options 56013.7.6 A risk-neutral view of futures options 562

13.8 Portfolio strategies based on options 56213.8.1 Portfolio insurance and the Black Monday of 1987 56313.8.2 Volatility trading 56413.8.3 Dynamic vs. Static hedging 566

13.9 Option pricing by numerical methods 569Problems 570Further reading 575Bibliography 576

14 Option Pricing: Incomplete Markets 57914.1 A PDE approach to incomplete markets 581

14.1.1 Pricing a zero-coupon bond in a driftless world 58414.2 Pricing by short-rate models 588

14.2.1 The Vasicek short-rate model 58914.2.2 The Cox–Ingersoll–Ross short-rate model 594

14.3 A martingale approach to incomplete markets 59514.3.1 An informal approach to martingale equivalent

measures 59814.3.2 Choice of numeraire: The bank account 60014.3.3 Choice of numeraire: The zero-coupon bond 60114.3.4 Pricing options with stochastic interest rates: Black’s

model 60214.3.5 Extensions 603

14.4 Issues in model calibration 60314.4.1 Bias–variance tradeoff and regularized least-squares 60414.4.2 Financial model calibration 609

Further reading 612Bibliography 612

Part VAdvanced optimization models

15 Optimization Model Building 61715.1 Classification of optimization models 618

CONTENTS xiii

15.2 Linear programming 62515.2.1 Cash flow matching 627

15.3 Quadratic programming 62815.3.1 Maximizing the Sharpe ratio 62915.3.2 Quadratically constrained quadratic programming 631

15.4 Integer programming 63215.4.1 A MIQP model to minimize TEV under a cardinality

constraint 63415.4.2 Good MILP model building: The role of tight model

formulations 63615.5 Conic optimization 642

15.5.1 Convex cones 64415.5.2 Second-order cone programming 65015.5.3 Semidefinite programming 653

15.6 Stochastic optimization 65515.6.1 Chance-constrained LP models 65615.6.2 Two-stage stochastic linear programming with

recourse 65715.6.3 Multistage stochastic linear programming with

recourse 66315.6.4 Scenario generation and stability in stochastic

programming 67015.7 Stochastic dynamic programming 675

15.7.1 The dynamic programming principle 67615.7.2 Solving Bellman’s equation: The three curses of

dimensionality 67915.7.3 Application to pricing options with early exercise

features 68015.8 Decision rules for multistage SLPs 68215.9 Worst-case robust models 686

15.9.1 Uncertain LPs: Polyhedral uncertainty 68915.9.2 Uncertain LPs: Ellipsoidal uncertainty 690

15.10Nonlinear programming models in finance 69115.10.1 Fixed-mix asset allocation 692

Problems 693Further reading 695Bibliography 696

16 Optimization Model Solving 69916.1 Local methods for nonlinear programming 700

16.1.1 Unconstrained nonlinear programming 70016.1.2 Penalty function methods 70316.1.3 Lagrange multipliers and constraint qualification

conditions 70716.1.4 Duality theory 713

16.2 Global methods for nonlinear programming 715

xiv CONTENTS

16.2.1 Genetic algorithms 71616.2.2 Particle swarm optimization 717

16.3 Linear programming 71916.3.1 The simplex method 72016.3.2 Duality in linear programming 72316.3.3 Interior-point methods: Primal-dual barrier method

for LP 72616.4 Conic duality and interior-point methods 728

16.4.1 Conic duality 72816.4.2 Interior-point methods for SOCP and SDP 731

16.5 Branch-and-bound methods for integer programming 73216.5.1 A matheuristic approach: Fix-and-relax 735

16.6 Optimization software 73616.6.1 Solvers 73716.6.2 Interfacing through imperative programming

languages 73816.6.3 Interfacing through non-imperative algebraic

languages 73816.6.4 Additional interfaces 739

Problems 739Further reading 740Bibliography 741

Index 743

Preface

This book arises from slides and lecture notes that I have used over the yearsin my courses Financial Markets and Instruments and Financial Engineering,which were offered at Politecnico di Torino to graduate students in Mathemat-ical Engineering. Given the audience, the treatment is naturally geared towarda mathematically inclined reader. Nevertheless, the required prerequisites arerelatively modest, and any student in engineering, mathematics, and statisticsshould be well-equipped to tackle the contents of this introductory book.1 Thebook should also be of interest to students in economics, as well as junior prac-titioners with a suitable quantitative background.

We begin with quite elementary concepts, and material is introduced pro-gressively, always paying due attention to the practical side of things. Mathe-matical modeling is an art of selective simplification, which must be supportedby intuition building, as well as by a healthy dose of skepticism. This is theaim of remarks, counterexamples, and financial horror stories that the book isinterspersed with. Occasionally, we also touch upon current research topics.

Book structure

The book is organized into five parts.

1. Part One, Overview, consists of two chapters. Chapter 1 aims at get-ting unfamiliar readers acquainted with the role and structure of finan-cial markets, the main classes of traded assets (equity, fixed income, andderivatives), and the main types of market participants, both in terms ofinstitutions (e.g., investments banks and pension funds) and roles (e.g.,speculators, hedgers, and arbitrageurs). We try to give a practical flavorthat is essential to students of quantitative disciplines, setting the stagefor the application of quantitative models. Chapter 2 overviews the basicproblems in finance, like asset allocation, pricing, and risk management,which may be tackled by quantitative models. We also introduce the fun-damental concepts related to arbitrage theory, including market complete-ness and risk-neutral measures, in a simple static and discrete setting.

2. Part Two, Fixed-income assets, consists of four chapters and introducesthe simplest assets depending on interest rates, starting with plain bonds.The fundamental concepts of interest rate modeling, including the term

1In case of need, the mathematical prerequisites are covered in my other book: QuantitativeMethods: An Introduction for Business Management. Wiley, 2011.

xv

xvi PREFACE

structure and forward rates, as well as bond pricing, are covered in Chap-ter 3. The simplest interest rate derivatives (forward rate agreements andvanilla swaps) are covered in Chapter 4, whereas Chapter 5 aims at pro-viding the reader with a flavor of real-life markets, where details like daycount and quoting conventions are relevant. Chapter 6 concludes this partby showing how quantitative models may be used to manage interest raterisk. In this part, we do not consider interest rate options, which require astronger mathematical background and are discussed later.

3. Part Three, Equity portfolios, consists of four chapters, where we dis-cuss equity markets and portfolios of stock shares. Actually, this is notthe largest financial market, but it is arguably the kind of market thatthe layman is more familiar with. Chapter 7 is a bit more theoreticaland lays down the foundations of static decision-making under uncer-tainty. By static, we mean that we make one decision and then we waitfor its consequences, finger crossed. Multistage decision models are dis-cussed later. In this chapter, we also introduce the basics of risk aversionand risk measurement. Chapter 8 is quite classical and covers traditionalmean–variance portfolio optimization. The impact of statistical estima-tion issues on portfolio management motivates the introduction of factormodels, which are the subject of Chapter 9. Finally, in Chapter 10, wediscuss equilibrium models in their simplest forms, the capital asset pric-ing model (CAPM), which is related to a single-index factor model, andarbitrage pricing theory (APT), which is related to a multifactor model.We do not discuss further developments in equilibrium models, but wehint at some criticism based on behavioral finance.

4. Part Four, Derivatives, includes four chapters. We discuss dynamic un-certainty models in Chapter 11, which is more challenging than previouschapters, as we have to introduce the necessary foundations of optionpricing models, namely, stochastic differential equations and stochasticintegrals. Chapter 12 describes simple forward and futures contracts, ex-tending concepts that were introduced in Chapter 4, when dealing withforward and futures interest rates. Chapter 13 covers option pricing inthe case of complete markets, including the celebrated and controversialBlack–Scholes–Merton formula, whereas Chapter 14 extends the basicconcepts to the more realistic setting of incomplete markets.

5. Part Five, Advanced optimization models, is probably the less standardpart of this book, when compared to typical textbooks on financial mar-kets. We deal with optimization model building, in Chapter 15, and op-timization model solving, in Chapter 16. Actually, it is difficult to drawa sharp line between model building and model solving, but it is a factof life that advanced software is available for solving quite sophisticatedmodels, and the average user does not need a very deep knowledge of theinvolved algorithms, whereas she must be able to build a model. This isthe motivation for separating the two chapters.

PREFACE xvii

Needless to say, the choice of which topics should be included or omit-ted is debatable and based on authors’ personal bias, not to mention the needto keep a book size within a sensible limit. With respect to introductory text-books on financial markets, there is a deeper treatment of derivative models. Onthe other hand, more challenging financial engineering textbooks do not cover,e.g., equilibrium models and portfolio optimization. We aim at an intermediatetreatment, whose main limitations include the following:

We only hint at criticism put forward by behavioral finance and do notcover market microstructure and algorithmic trading strategies.From a mathematical viewpoint, we pursue an intuitive treatment of fi-nancial engineering models, as well as a simplified coverage of the re-lated tools of stochastic calculus. We do not rely on rigorous argumentsinvolving self-financing strategies, martingale representation theorems, orchange of probability measures.From a financial viewpoint, by far, the most significant omission concernscredit risk and credit derivatives. Counterparty and liquidity risk playa prominent role in post-Lehman Brothers financial markets and, as aconsequence of the credit crunch started in 2007, new concepts like CVA,DVA, and FVA have been introduced. This is still a field in flux, and thematter is arguably not quite assessed yet.Another major omission is econometric time series models.

Adequate references on these topics are provided for the benefit of the interestedreaders.

My choices are also influenced by the kind of students this book is mainlyaimed at. The coverage of optimization models and methods is deeper thanusual, and I try to open readers’ critical eye by carefully crafted examples andcounterexamples. I try to strike a satisfactory balance between the need to il-lustrate mathematics in action and the need to understand the real-life context,without which quantitative methods boil down to a solution in search of a prob-lem (or a hammer looking for nails, if you prefer). I also do not disdain just a bitof repetition and redundancy, when it may be convenient to readers who wishto jump from chapter to chapter. More advanced sections, which may be safelyskipped by readers, are referred to as supplements and their number is markedby an initial “S.”

In my Financial Engineering course, I also give some more information onnumerical methods. The interested reader might refer to my other books:

P. Brandimarte, Numerical Methods in Finance and Economics: A MAT-LAB-Based Introduction (2nd ed.), Wiley, 2006P. Brandimarte, Handbook in Monte Carlo Simulation: Applications inFinancial Engineering, Risk Management, and Economics, Wiley, 2014

xviii PREFACE

Acknowledgements

In the past years, I have adopted the following textbooks (or earlier editions) inmy courses. I have learned a lot from them, and they have definitely influencedthe writing of this book:

Z. Bodie, A. Kane, and A. Marcus, Investments (9th ed.), McGraw-Hill,2010J.C. Hull, Options, Futures, and Other Derivatives (8th ed.), PrenticeHall, 2011P. Veronesi, Fixed Income Securities: Valuation, Risk, and Risk Manage-ment, Wiley, 2010

Other specific acknowledgements are given in the text. I apologize in advancefor any unintentional omission.

Additional material

Some end-of-chapter problems are included and fully worked solutions will beposted on a web page. My current URL is

http://staff.polito.it/paolo.brandimarte/

A hopefully short list of errata will be posted there as well. One of the manycorollaries of Murphy’s law states that my URL is going to change shortly afterpublication of the book. An up-to-date link will be maintained on the Wileyweb page:

http://www.wiley.com/

For comments, suggestions, and criticisms, all of which are quite welcome,my e-mail address is

paolo.brandimarte@polito.it

PAOLO BRANDIMARTE

Turin, September 2017

http://staff.polito.it/paolo.brandimarte/http://www.wiley.com/mailto:paolo.brandimarte@polito.it

About the Companion ebsite

This book is accompanied by a companion website:

www.wiley.com/go/brandimarte/financialmarkets

The website includes:• Solutions manual for end-of-chapter problems

xix

http://www.wiley.com/go/brandimarte/financialmarkets

Part One

Overview

Chapter One

Financial Markets: Functions,Institutions, and TradedAssets

Providing a simple, yet exhaustive definition of finance is no quite easy task,but a possible attempt, at least from a conceptual viewpoint, is the following:1

Finance is the study of how people and organizations allocate scarceresources over time, subject to uncertainty.

This definition might sound somewhat generic, but it does involve the two es-sential ingredients that we shall deal with in practically every single page of thisbook: Time and uncertainty. Appreciating their role is essential in understand-ing why finance was born in the past and is so pervasive now. The time value ofmoney is reflected in the interest rates that define how much money we have topay over the time span of our mortgage, or the increase in wealth that we obtainby locking up our capital in a certificate of deposit issued by a bank. It is com-mon wisdom that the value of $1 now is larger than the value of $1 in one year.This is not only a consequence of the potential loss of value due to inflation.2

A dollar now, rather than in the future, paves the way to earlier investment op-portunities, and it may also serve as a precautionary cushion against unforeseenneeds. Uncertainty is related, e.g., to the impossibility of forecasting the returnthat we obtain from investing in stock shares, but also to the risk of adversemovements in currency exchange rates for an import/export firm, or longevityrisk for a worker approaching retirement. As we show in Chapter 2, we maymodel issues related to time and uncertainty within a mathematical framework,applying principles from financial economics and tools from probability, statis-tics, and optimization theory. Before doing that, we need a more concrete view

1This definition is taken from [2].2This holds under common economic conditions; the exception to the rule is deflation, which is(at the time of writing) a possibility in Euroland. In this book, we will assume that the standardeconomic conditions prevail.

1

2 CHAPTER 1 Financial Markets: Functions, Institutions, and Traded Assets

in order to understand how financial markets work, which kinds of assets areexchanged, and which actors play a role in them and what their incentives are.We pursue this “institutional” approach to get acquainted with finance in thischapter. Some of the more mathematically inclined students tend to considerthis side of the coin modestly exciting, but a firm understanding of it is neces-sary to put models in the right perspective and to appreciate their pitfalls andlimitations.

In Section 1.1, we discuss the role of time and uncertainty in a rather ab-stract way that, nevertheless, lays down some essential concepts. A more con-crete view is taken in Section 1.2, where we describe the fundamental classes ofassets that are traded on financial markets, namely, stock shares, bonds, curren-cies, and the basic classes of derivatives, like forward/futures contracts and op-tions. In order to provide a proper framework, we also hint at the essential shapeof a balance sheet, in terms of assets, liabilities, and equity, and we empha-size the difference between standardized assets traded on regulated exchangesand less liquid assets, possibly engineered to meet specific client requirements,which are traded over-the-counter. In Section 1.3, we describe the classes ofplayers involved in financial markets, such as investment/commercial banks,common/hedge/pension funds, insurance companies, brokers, and dealers. Weinsist on the separation between the institutional form and the role of those play-ers: A single player may be of one given kind, in institutional terms, but it mayplay different roles. For instance, an investment bank can, among many otherthings, play the role of a prime broker for a hedge fund. Furthermore, dependingon circumstances, players may act as hedgers, speculators, or arbitrageurs. Theexact organization of financial markets is far from trivial, especially in the lightof extensive use of information technology, and a full description is beyond thescope of this book. Nevertheless, some essential concepts are needed, such asthe difference between primary and secondary markets, which is explained inSection 1.4. There, we also introduce some trading strategies, like buying onmargin and short-selling, which are essential to interpret what happens on finan-cial markets in practice, as well as to understand some mathematical argumentsthat we will use over and over in this book. Finally, in Section 1.5 we con-sider market indexes and describe some basic features explaining, for instance,the difference between an index like the Dow Jones Industrial Average and theStandard & Poor 500.

1.1 What is the purpose of finance?

If you are reading this book, chances are that it is because you would like toland a rewarding job in finance. Even if this is not the case, one of the reasonswhy we aim at finding a good job is because we need to earn some income inorder to purchase goods and services, for ourselves and possibly other peoplewe care about. Every month (hopefully) we receive some income, and we mustplan its use. The old grasshopper and ant fable teaches that we should actually

1.1 What is the purpose of finance? 3

(a) Income equals consumption

(b) Shifting consumption forward by saving

(c) Shifting consumption backward by borrowing

t = 1 t = 2 t = 3

I1 I2 I3

C1 C2 C3

C1 C2 C3

C1 C2 C3

t = 1 t = 2 t = 3

I1

L2,3

B3,2

I2 I3

I1 I2 I3

t = 1 t = 2 t = 3

(1+R)

1/(1+R)

FIGURE 1.1 Shifting consumption forward and backward in time.

plan ahead with care. Part of that income should be saved to allow consumptionat some later time. Sometimes, we might need to use more income than weare earning at present, e.g., in order to finance the purchase of our home sweethome.

Now, imagine a world in which we cannot “store” money, and we have toconsume whatever our income is immediately, no more, no less, just like wewould do with perishable food, if no one had invented refrigerators and otherconservation techniques. This unpleasing situation is depicted in Fig. 1.1(a).There, time is discretized in T = 3 time periods, indexed by t = 1, . . . , T .3 Theincome during time period t is denoted by It, and it is equal to the consumptionCt during the same period:

It = Ct, t = 1, . . . , T.

3Sometimes, time discretization requires careful thinking about events. Do we earn incomeat the beginning or at the end of a time period? In other words, is income earned during timeperiod t immediately available for consumption during the same time period? We may argue thatincome during time period t is available for consumption only during time period t+1. We shalldiscuss more precise notation and concepts in Section 2.1.2. Here, for the sake of simplicity,we assume that every event during a time period is concentrated at some time instant. Wesometimes use the rather awkward term epoch to refer to a specific point in time. We also oftenuse the term time bucket to refer to a time period delimited by two time instants.

4 CHAPTER 1 Financial Markets: Functions, Institutions, and Traded Assets

This state of the matter is not quite satisfactory, if we have excess income insome period and would like to delay consumption to a later time period. InFig. 1.1(b), part of income I2, denoted by L2,3, is shifted forward from timeperiod 2 to time period 3. This results in an increase of C3 and a decrease ofC2. The amount of income saved can be regarded as money invested or lent tosomeone else. By a similar token, we might wish to anticipate consumption toan earlier time period. In Fig. 1.1(c), consumption C2 is increased by shiftingincome backward in time from time period t = 3, which means borrowing anamount of money B3,2, to be used in time period t = 2 and repaid in time pe-riod t = 3. Savers and borrowers may be individuals or institutions, and wemay play both of these roles at different stages of our working life. Clearly, allof this may happen if there is a way to match savers and borrowers, so that all ofthem may improve their consumption timing. This is one of the many roles offinancial markets; more specifically, we use the term money markets when thetime span of the loan is short. In other cases, the investment may stretch over aconsiderable time span, especially if savers/borrowers are not just households,but corporations, innovative startups, or public administrations that have to fi-nance the development of a new product, the building of a new hospital, or anessential infrastructure. In this case, we talk about capital markets.

Needless to say, if we accept to delay consumption, it is because we expectto be compensated in some way. Informally, we exchange an egg for a chicken;formally, we earn some interest rate R along the time period involved in theshift.4 We may interpret the shift as a flow of money over a network in timebut, unlike other network flows involved in transportation over space, we do nothave exact conservation of flows. With reference to Fig. 1.1(b), we have thefollowing flow balance equations at nodes 2 and 3:

C2 = I2 − L2,3,C3 = I3 + L2,3(1 +R),

stating that we give up an amount L2,3 of consumption at time 2 in exchangefor an increase (1 + R)L2,3 in later consumption. The factor 1 + R is a gainassociated with the flow of money along the arc connecting node t = 2 to nodet = 3. This is what the time value of money is all about. The exact value of theinterest rate R, as we shall see in Chapter 3, may be related to the possibilityof default (i.e., the borrower may not repay the full amount of his debt) and toinflation risk, among other things.

Clearly, there must be another side of the coin: The increase in later con-sumption must be paid by a counterparty in an exchange. We delay consump-tion while someone else anticipates it. With reference to Fig. 1.1(c), we have

4In financial practice, whenever an interest rate is quoted, it is always an annual rate. For now,let us associate the rate with an arbitrary time period.

1.1 What is the purpose of finance? 5

the following flow balance equations at nodes 2 and 3:

C2 = I2 +B3,2

1 +R,

C3 = I3 −B3,2.

Note that we are expressing the borrowed amount B3,2 in terms of the money attime t = 3, when the debt is repaid; in other words B3,2 is a flow out of node 3.This is not essential at all: If we use money at time t = 2, i.e., we consider theflow B∗3,2 into node 2, the flow balance would simply read

C2 = I2 +B∗3,2,

C3 = I3 −B∗3,2(1 +R).

The two sides of the coin must be somehow matched by a market mechanism.In practice, funds are channeled by financial intermediaries, which must becompensated for their job. In fact, there is a difference between lending andborrowing rates, called bid–ask (or bid–offer) spread, which applies to otherkinds of financial assets as well. Lending and borrowing money through a bankis what we are familiar with as individuals, whereas a large corporation and asovereign government have the alternative of raising funds by issuing securi-ties like bonds, typically promising the payment of periodic interest, as wellas the refund of the capital at some prespecified point in time, the maturity ofthe bond. Corporations may also raise funds by issuing stock shares. Buyinga stock share does not mean that we lend money to a firm; hence, we are notentitled to the payment of any interest. Rather, we own a share of the firm andmay receive a corresponding share of earnings that may be distributed in theform of dividends to stockholders. However, the amount that we will receive israndom and no promise is made about dividends, as they depend on how wellthe business is doing, as well as the decision of reinvesting part of the earningin new business ventures, rather than distributing the whole of it.

After being first issued, securities like bonds and stock shares may be ex-changed among market participants, at prices that may depend on several under-lying risk factors. Since the values of these factors are not known with certainty,the future prices of bonds and stock shares are random. In fact, time is inter-twined with another fundamental dimension in finance, namely, uncertainty.When we lend or borrow money at a given interest rate, the future cash flowsare known with certainty, if we do not consider the possibility of a default ondebt. However, when we buy a stock share at time t = 0 and plan to sell itat time t = T , randomness comes into play. Let us denote the initial price byS(0).5 The future price S(T ) is a random variable, which we may denoteas S(T, ω) to emphasize its dependence on the random outcome (scenario) ω.We recall that, in probability theory, a random variable is a function mappingunderlying random outcomes, corresponding to future scenarios or states of na-ture, to numeric values. Let ωi, i = 1, . . . ,m, denote the i-th outcome, which

5Depending on notational convenience, we shall write S(t) or St, as no ambiguity should arise.

6 CHAPTER 1 Financial Markets: Functions, Institutions, and Traded Assets

t = 0 t = T

11

2

3

4

5

2

3

4

5

FIGURE 1.2 Representing uncertain states of the world by a scenario fan.

occurs with probability πi. For the sake of simplicity, we are considering a dis-crete and finite set of possible outcomes, whereas later we will deal extensivelywith continuous random variables. A simple way to depict this kind of discreteuncertainty is by a scenario fan like the one depicted in Fig. 1.2. Therefore,S(T, ω) is a random variable, and we associate a future price S(T, ωi) with eachfuture state of the world. The corresponding holding period return is defined asfollows.

DEFINITION 1.1 (Holding period return) Let us consider a holding period[0, T ], where the initial asset price is S(0) and the terminal random asset priceis S(T, ω). We define the holding period return as

R(ω).=S(T, ω)− S(0)

S(0)(1.1)

and the holding period gain as

G(ω).=S(T, ω)

S(0)= 1 +R(ω). (1.2)

The gain and the holding period return (return for short) are clearly related. Areturn of 10% means that the stock price was multiplied by a gain factor of 1.10.

Remark. The term gain is not so common in finance textbooks. Usually, termslike total return or gross return are used, rather than gain. On the contrary, termslike rate of return and net return are used to refer to (holding period) return. Theproblem is that these terms may ring different bells, especially to practitioners.We may use the qualifier “total” when we want to emphasize a return includ-ing dividend income, besides the capital gain related to price changes. Termslike “gross” and “net” may be related with taxation issues, which we shall al-ways disregard. This is why we prefer using “gain,” even though this usageis less common. We shall not confuse gain, which is a multiplicative factor,