Introduction to Financial Markets

PAOLO BRANDIMARTE is Full Professor at the Department of Mathematical Sciences of Politecnico di Torino in Italy, where he teaches Business Analytics and Financial Engineering. He is the author of several publications, including more than ten books on the application of optimization and simulation to diverse areas such as production and supply chain management, telecommunications, and finance.

• Preface xvAbout the Companion Website xixPart I Overview1 Financial Markets: Functions, Institutions, and Traded Assets 11.1 What is the purpose of finance? 21.2 Traded assets 121.2.1 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 291.3 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 511.4 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 581.5 Market indexes 60Problems 63Further reading 65Bibliography 652 Basic Problems in Quantitative Finance 672.1 Portfolio optimization 682.1.1 Static portfolio optimization: Mean–variance efficiency 702.1.2 Dynamic decision-making under uncertainty: A stylized consumption–saving model 752.2 Risk measurement and management 802.2.1 Sensitivity of asset prices to underlying risk factors 812.2.2 Risk measures in a non-normal world: Value-atrisk 842.2.3 Risk management: Introductory hedging examples 932.2.4 Financial vs. nonfinancial risk factors 1002.3 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 1162.4 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 125S2.1 Multiobjective optimization 129S2.2 Summary of LP duality 133Problems 137Further reading 139Bibliography 139Part II Fixed income assets3 Elementary Theory of Interest Rates 1433.1 The time value of money: Shifting money forward in time 1463.1.1 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 1533.2.1 Discount factors and pricing a zero-coupon bond 1543.2.2 Discount factors vs. interest rates 1583.3 Nominal vs. real interest rates 1613.4 The term structure of interest rates 1633.5 Elementary bond pricing 1653.5.1 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 1883.6 A digression: Elementary investment analysis 1903.6.1 Net present value 1913.6.2 Internal rate of return 1923.6.3 Real options 1933.7 Spot vs. forward interest rates 1933.7.1 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) 203 Problems 203Further reading 205Bibliography 2054 Forward Rate Agreements, Interest Rate Futures, and Vanilla Swaps 2074.1 LIBOR and EURIBOR rates 2084.2 Forward rate agreements 2094.2.1 A hedging view of forward rates 2104.2.2 FRAs as bond trades 2144.2.3 A numerical example 2154.3 Eurodollar futures 2164.4 Vanilla interest rate swaps 2204.4.1 Swap valuation: Approach 1 2214.4.2 Swap valuation: Approach 2 2234.4.3 The swap curve and the term structure 225Problems 226Further reading 226Bibliography 2265 Fixed-Income Markets 2295.1 Day count conventions 2305.2 Bond markets 2315.2.1 Bond credit ratings 2335.2.2 Quoting bond prices 2335.2.3 Bonds with embedded options 2355.3 Interest rate derivatives 2375.3.1 Swap markets 2375.3.2 Bond futures and options 2385.4 The repo market and other money market instruments 2395.5 Securitization 240Problems 244Further reading 244Bibliography 2446 Interest Rate Risk Management 2476.1 Duration as a first-order sensitivity measure 2486.1.1 Duration of fixed-coupon bonds 2506.1.2 Duration of a floater 2546.1.3 Dollar duration and interest rate swaps 2556.2 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 2606.3 Classical duration-based immunization 2616.3.1 Cash flow matching 2626.3.2 Duration matching 2636.4 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 273Part III Equity portfolios7 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 2847.3 A conceptual tool: The utility function 2887.3.1 A few standard utility functions 2937.3.2 Limitations of utility functions 2977.4 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 3097.5 Stochastic dominance 310S7.1 Theorem proofs 314S7.1.1 Proof of Theorem 7.2 314S7.1.2 Proof of Theorem 7.4 315Problems 315Further reading 317Bibliography 3178 Mean–Variance Efficient Portfolios 3198.1 Risk aversion and capital allocation to risky assets 3208.1.1 The role of risk aversion 3248.2 The mean–variance efficient frontier with risky assets 3258.2.1 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 3298.3 Mean–variance efficiency with a risk-free asset: The separation property 3328.4 Maximizing the Sharpe ratio 3378.4.1 Technical issues in Sharpe ratio maximization 3408.5 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 3499 Factor Models 3519.1 Statistical issues in mean–variance portfolio optimization 3529.2 The single-index model 3539.2.1 Estimating a factor model 3549.2.2 Portfolio optimization within the single-index model 3569.3 The Treynor–Black model 3589.3.1 A top-down/bottom-up optimization procedure 3629.4 Multifactor models 3659.5 Factor models in practice 367S9.1 Proof of Equation (9.17) 368Problems 369Further reading 371Bibliography 37110 Equilibrium Models: CAPM and APT 37310.1 What is an equilibrium model? 37410.2 The capital asset pricing model 37510.2.1 Proof of the CAPM formula 37710.2.2 Interpreting CAPM 37810.2.3 CAPM as a pricing formula and its practical relevance 38010.3 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 38610.4 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 39310.4.5 Interpreting and using APT 39410.5 The behavioral critique 39810.5.1 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 401S10.1Bayesian statistics 404S10.1.1 Bayesian estimation 405S10.1.2 Bayesian learning in coin flipping 407S10.1.3 The expected value of a normal distribution 408Problems 411Further reading 413Bibliography 413Part IV Derivatives11 Modeling Dynamic Uncertainty 41711.1 Stochastic processes 42011.1.1 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 43611.2 Stochastic processes in continuous time 43811.2.1 A fundamental building block: Standard Wiener process 43811.2.2 A generalization: Lévy processes 44011.3 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 44511.4 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 45311.5 Stochastic processes in financial modeling 45711.5.1 Geometric Brownian motion 45711.5.2 Generalizations 46011.6 Sample path generation 46211.6.1 Monte Carlo sampling 46311.6.2 Scenario trees 465S11.1Probability spaces, measurability, and information 468Problems 476Further reading 478Bibliography 47812 Forward and Futures Contracts 48112.1 Pricing forward contracts on equity and foreign currencies 48212.1.1 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 49312.3.1 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 499Problems 501Further reading 502Bibliography 50213 Option Pricing: Complete Markets 50513.1 Option terminology 50613.1.1 Vanilla options 50713.1.2 Exotic options 50813.2 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 51713.3 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 52713.4 A continuous-time model: The Black–Scholes–Merton pricing formula 53013.4.1 The delta-hedging view 53213.4.2 The risk-neutral view: Feynman–Ka¡c representation theorem 53913.4.3 Interpreting the factors in the BSM formula 54313.5 Option price sensitivities: The Greeks 54513.5.1 Delta and gamma 54613.5.2 Theta 55013.5.3 Relationship between delta, gamma, and theta 55113.5.4 Vega 55213.6 The role of volatility 55313.6.1 The implied volatility surface 55313.6.2 The impact of volatility on barrier options 55513.7 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 56213.8 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 56613.9 Option pricing by numerical methods 569Problems 570Further reading 575Bibliography 57614 Option Pricing: Incomplete Markets 57914.1 A PDE approach to incomplete markets 58114.1.1 Pricing a zero-coupon bond in a driftless world 58414.2 Pricing by short-rate models 58814.2.1 The Vasicek short-rate model 58914.2.2 The Cox–Ingersoll–Ross short-rate model 59414.3 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 60314.4 Issues in model calibration 60314.4.1 Bias–variance tradeoff and regularized least-squares 60414.4.2 Financial model calibration 609Further reading 612Bibliography 612Part V Advanced optimization models15 Optimization Model Building 61715.1 Classification of optimization models 61815.2 Linear programming 62515.2.1 Cash flow matching 62715.3 Quadratic programming 62815.3.1 Maximizing the Sharpe ratio 62915.3.2 Quadratically constrained quadratic programming 63115.4 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 64215.5.1 Convex cones 64415.5.2 Second-order cone programming 65015.5.3 Semidefinite programming 65315.6 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 67515.7.1 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 68615.9.1 Uncertain LPs: Polyhedral uncertainty 68915.9.2 Uncertain LPs: Ellipsoidal uncertainty 69015.10Nonlinear programming models in finance 69115.10.1 Fixed-mix asset allocation 692Problems 693Further reading 695Bibliography 69616 Optimization Model Solving 69916.1 Local methods for nonlinear programming 70016.1.1 Unconstrained nonlinear programming 70016.1.2 Penalty function methods 70316.1.3 Lagrange multipliers and constraint qualification conditions 70716.1.4 Duality theory 71316.2 Global methods for nonlinear programming 71516.2.1 Genetic algorithms 71616.2.2 Particle swarm optimization 71716.3 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 72816.4.1 Conic duality 72816.4.2 Interior-point methods for SOCP and SDP 73116.5 Branch-and-bound methods for integer programming 73216.5.1 A matheuristic approach: Fix-and-relax 73516.6 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 739Problems 739Further reading 740Bibliography 741Index 743