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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: Levy 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 Ito processes 445
11.4 Stochastic integration and Ito’s lemma 447
11.4.1 A digression: Riemann and Riemann-Stieltjes integrals 447
11.4.2 Stochastic integral in the sense of Ito 448
11.4.3 Ito’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-Kai¿½c 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