内容

目次

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