The Mathematics of Financial Modeling and Investment Management

Contents......Page 6
Preface......Page 17
Acknowledgments......Page 19
About the Authors......Page 21
Commonly Used Symbols......Page 22
Abbreviations and Acronyms......Page 23
1 From Art to Engineering in Finance......Page 26
Step 2: Establishing an Investment Policy......Page 27
Step 3: Selecting a Portfolio Strategy......Page 31
Step 4: Selecting the Specific Assets......Page 32
Step 5: Measuring and Evaluating Performance......Page 34
Financial Engineering in Historical Perspective......Page 35
The Role of Information Technology......Page 36
Industry’s Evaluation of Modeling Tools......Page 38
Integrating Qualitative and Quantitative Information......Page 40
Principles for Engineering a Suite of Models......Page 42
Summary......Page 43
Financial Assets......Page 46
Classification of Financial Markets......Page 50
Economic Functions of Financial Markets......Page 51
Secondary Markets......Page 52
Overview of Market Participants......Page 59
Role of Financial Intermediaries......Page 60
Institutional Investors......Page 62
Pension Funds......Page 66
Investment Companies......Page 67
Depository Institutions......Page 68
Trading Locations......Page 70
Stock Market Indicators......Page 71
Trading Arrangements......Page 73
Maturity......Page 76
Coupon Rate......Page 77
Provisions for Paying off Bonds......Page 80
Options Granted to Bondholders......Page 81
Futures and Forward Contracts......Page 82
Futures versus Forward Contracts......Page 83
Pricing of Futures Contracts......Page 84
The Role of Futures in Financial Markets......Page 88
Options......Page 89
The Option Price......Page 91
Swaps......Page 94
Caps and Floors......Page 95
Summary......Page 96
3 Milestones in Financial Modeling and Investment Management......Page 100
The Precursors: Pareto, Walras, and the Lausanne School......Page 101
Price Diffusion: Bachelier......Page 103
The Ruin Problem in Insurance: Lundberg......Page 105
The Principles of Investment: Markowitz......Page 106
Understanding Value: Modigliani and Miller......Page 108
Modigliani-Miller Irrelevance Theorems and the Absence of Arbitrage......Page 109
Efficient Markets: Fama and Samuelson......Page 110
Capital Asset Pricing Model: Sharpe, Lintner, and Mossin......Page 111
The Multifactor CAPM: Merton......Page 112
Arbitrage Pricing Theory: Ross......Page 113
Arbitrage, Hedging, and Option Theory: Black, Scholes, and Merton......Page 114
Summary......Page 115
4 Principles of Calculus......Page 116
Proper Subsets......Page 118
Intersection of Sets......Page 120
Distances and Quantities......Page 121
n-tuples......Page 122
Distance......Page 123
Density of Points......Page 124
Functions......Page 125
Variables......Page 126
Limits......Page 127
Continuity......Page 128
Total Variation......Page 130
Differentiation......Page 131
Commonly Used Rules for Computing Derivatives......Page 132
Higher Order Derivatives......Page 136
Application to Bond Analysis......Page 137
Taylor Series Expansion......Page 146
Application to Bond Analysis......Page 147
Riemann Integrals......Page 152
Properties of Riemann Integrals......Page 154
Lebesque-Stieltjes Integrals......Page 155
Indefinite and Improper Integrals......Page 156
The Fundamental Theorem of Calculus......Page 157
Laplace Transform......Page 159
Fourier Transforms......Page 162
Calculus in More than One Variable......Page 163
Summary......Page 164
Vectors......Page 166
Matrices......Page 169
Diagonals and Antidiagonals......Page 170
Diagonal Matrix......Page 171
Determinants......Page 173
Systems of Linear Equations......Page 174
Linear Independence and Rank......Page 176
Hankel Matrix......Page 177
Vector Operations......Page 178
Matrix Operations......Page 181
Eigenvalues and Eigenvectors......Page 185
Diagonalization and Similarity......Page 186
Singular Value Decomposition......Page 187
Summary......Page 188
Representing Uncertainty with Mathematics......Page 190
Probability in a Nutshell......Page 192
Outcomes and Events......Page 194
Probability......Page 195
Measure......Page 196
Integrals......Page 197
Distributions and Distribution Functions......Page 199
Random Vectors......Page 200
Stochastic Processes......Page 203
Probabilistic Representation of Financial Markets......Page 205
Information Structures......Page 206
Filtration......Page 207
Conditional Probability and Conditional Expectation......Page 209
Moments and Correlation......Page 211
Copula Functions......Page 213
Sequences of Random Variables......Page 214
Sum of Variables......Page 216
Gaussian Variables......Page 219
Linear Regression......Page 222
Summary......Page 224
7 Optimization......Page 226
Maxima and Minima......Page 227
Lagrange Multipliers......Page 229
Linear Programming......Page 231
Quadratic Programming......Page 236
Calculus of Variations and Optimal Control Theory......Page 237
Stochastic Programming......Page 239
Summary......Page 241
8 Stochastic Integrals......Page 242
The Intuition Behind Stochastic Integrals......Page 244
Brownian Motion Defined......Page 250
Properties of Brownian Motion......Page 255
Stochastic Integrals Defined......Page 257
Some Properties of Itô Stochastic Integrals......Page 261
Summary......Page 262
9 Differential Equations and Difference Equations......Page 264
Ordinary Differential Equations......Page 265
Solution to an ODE......Page 266
Systems of Ordinary Differential Equations......Page 268
Closed-Form Solutions of Ordinary Differential Equations......Page 271
Linear Differential Equation......Page 272
The Finite Difference Method......Page 274
Nonlinear Dynamics and Chaos......Page 281
Fractals......Page 283
Diffusion Equation......Page 284
Solution of the Diffusion Equation......Page 286
Numerical Solution of PDEs......Page 288
Summary......Page 290
10 Stochastic Differential Equations......Page 292
The Intuition Behind Stochastic Differential Equations......Page 293
Itô Processes......Page 296
The 1-Dimensional Itô Formula......Page 297
Stochastic Differential Equations......Page 299
Generalization to Several Dimensions......Page 301
Solution of Stochastic Differential Equations......Page 303
The Ornstein-Uhlenbeck Process......Page 305
The Geometric Brownian Motion......Page 306
Summary......Page 307
11 Financial Econometrics: Time Series Concepts, Representations, and Models......Page 308
Concepts of Time Series......Page 309
Stylized Facts of Financial Time Series......Page 311
Univariate Stationary Series......Page 313
The Lag Operator L......Page 314
Stationary Univariate Moving Average......Page 317
Multivariate Stationary Series......Page 318
Nonstationary Series......Page 320
Stationary Univariate ARMA Models......Page 322
Nonstationary Univariate ARMA Models......Page 325
Stationary Multivariate ARMA Models......Page 326
Markov Coefficients and ARMA Models......Page 329
State-Space Representation......Page 330
Equivalence of State-Space and ARMA Representations......Page 333
Integrated Series and Trends......Page 334
Summary......Page 338
Model Selection......Page 340
Learning and Model Complexity......Page 342
Maximum Likelihood Estimate......Page 344
Random Walk Models......Page 349
Correlation......Page 352
Random Matrices......Page 354
Multifactor Models......Page 357
CAPM......Page 359
PCA and Factor Models......Page 360
Vector Autoregressive Models......Page 363
Cointegration......Page 364
State-Space Modeling and Cointegration......Page 367
Empirical Evidence of Cointegration in Equity Prices......Page 368
Nonstationary Models of Financial Time Series......Page 370
The ARCH/GARCH Family of Models......Page 371
Markov Switching Models......Page 372
Summary......Page 374
13 Fat Tails, Scaling, and Stable Laws......Page 376
Fat Tails......Page 377
The Class of Fat-Tailed Distributions......Page 378
The Law of Large Numbers and the Central Limit Theorem......Page 383
Stable Distributions......Page 385
Maxima......Page 387
Generalized Extreme Value Distributions......Page 393
Order Statistics......Page 394
Point Process of Exceedances or Peaks over Threshold......Page 396
Estimation......Page 398
Eliminating the Assumption of IID Sequences......Page 403
Heavy-Tailed ARMA Processes......Page 406
ARCH/GARCH Processes......Page 407
Subordinated Processes......Page 408
Estimation......Page 409
Scaling and Self-Similarity......Page 410
Evidence of Fat Tails in Financial Variables......Page 413
On the Applicability of Extreme Value Theory in Finance......Page 416
Summary......Page 417
The Arbitrage Principle......Page 418
Arbitrage Pricing in a One-Period Setting......Page 420
State Prices......Page 422
Risk-Neutral Probabilities......Page 423
Complete Markets......Page 424
Propagation of Information......Page 427
Trading Strategies......Page 428
State-Price Deflator......Page 429
Pricing Relationships......Page 430
Equivalent Martingale Measures......Page 439
Risk-Neutral Probabilities......Page 441
The Binomial Model......Page 448
Risk-Neutral Probabilities for the Binomial Model......Page 451
Valuation of European Simple Derivatives......Page 452
Valuation of American Options......Page 454
Arbitrage Pricing in a Discrete-Time, Continuous-State Setting......Page 455
APT Models......Page 460
Testing APT......Page 461
Summary......Page 464
The Arbitrage Principle in Continuous Time......Page 466
Trading Strategies and Trading Gains......Page 468
Arbitrage Pricing in Continuous-State, Continuous-Time......Page 470
Stock Price Processes......Page 472
Hedging......Page 473
The Black-Scholes Option Pricing Formula......Page 474
Generalizing the Pricing of European Options......Page 477
State-Price Deflators......Page 479
Equivalent Martingale Measures......Page 482
Equivalent Martingale Measures and Girsanov’s Theorem......Page 484
The Diffusion Invariance Principle......Page 486
Application of Girsanov’s Theorem to Black-Scholes Option Pricing Formula......Page 487
Equivalent Martingale Measures and Complete Markets......Page 488
Equivalent Martingale Measures and State Prices......Page 489
Arbitrage Pricing with a Payoff Rate......Page 491
Implications of the Absence of Arbitrage......Page 492
Summary......Page 493
16 Portfolio Selection Using Mean-Variance Analysis......Page 496
Diversification as a Central Theme in Finance......Page 497
Markowitz’s Mean-Variance Analysis......Page 499
Capital Market Line......Page 502
Deriving the Capital Market Line......Page 503
What is Portfolio M?......Page 506
Utility Functions and Indifference Curves......Page 507
Selection of the Optimal Portfolio......Page 509
Extension of the Markowitz Mean-Variance Model to Inequality Constraints......Page 510
The Return Forecast......Page 512
The Utility Function......Page 513
A Global Probabilistic Framework for Portfolio Selection......Page 515
Relaxing the Assumption of Normality......Page 516
Multiperiod Stochastic Optimization......Page 517
Application to the Asset Allocation Decision......Page 519
The Inputs......Page 520
Portfolio Selection: An Example......Page 525
Inclusion of More Asset Classes......Page 528
Extensions of the Basic Asset Allocation Model......Page 532
Summary......Page 534
17 Capital Asset Pricing Model......Page 536
CAPM Assumptions......Page 537
Systematic and Nonsystematic Risk......Page 538
Security Market Line......Page 541
Deriving the Empirical Analogue of the CML......Page 543
Empricial Implications......Page 544
A Critique of Tests of the CAPM......Page 545
Merton and Black Modifications of the CAPM......Page 546
CAPM and Random Matrices......Page 547
The Conditional CAPM......Page 548
Beta, Beta Everywhere......Page 549
The Role of the CAPM in Investment Management Applications......Page 550
Summary......Page 551
18 Multifactor Models and Common Trends for Common Stocks......Page 554
Multifactor Models......Page 555
Determination of Factors......Page 557
Dynamic Market Models of Returns......Page 562
Dynamic Models for Prices......Page 563
Estimation and Testing of Cointegrated Systems......Page 568
Cointegration and Financial Time Series......Page 569
Nonlinear Dynamic Models for Prices and Returns......Page 571
Summary......Page 574
Integrating the Equity Portfolio Management Process......Page 576
Active versus Passive Portfolio Management......Page 577
Tracking Error......Page 578
Backward-Looking versus Forward-Looking Tracking Error......Page 580
The Impact of Portfolio Size, Benchmark Volatility, and Portfolio Beta on Tracking Error......Page 581
Types of Equity Styles......Page 585
Style Classification Systems......Page 587
Constructing an Indexed Portfolio......Page 589
Index Tracking and Cointegration......Page 590
Top-Down Approaches to Active Investing......Page 591
Bottom-Up Approaches to Active Investing......Page 592
Fundamental Law of Active Management......Page 593
Strategies Based on Technical Analysis......Page 596
Nonlinear Dynamic Models and Chaos......Page 598
Technical Analysis and Statistical Nonlinear Pattern Recognition......Page 599
Market-Neutral Strategies and Statistical Arbitrage......Page 600
Risk Decomposition......Page 602
Portfolio Construction and Risk Control......Page 607
Assessing the Exposure of a Portfolio......Page 608
Tilting a Portfolio......Page 612
Summary......Page 614
20 Term Structure Modeling and Valuation of Bonds and Bond Options......Page 618
Basic Principles of Valuation of Debt Instruments......Page 619
Yield-to-Maturity Measure......Page 621
Reinvestment of Cash Flow and Yield......Page 623
The Term Structure of the Interest Rates and the Yield Curve......Page 624
Limitations of Using the Yield to Value a Bond......Page 627
Obtaining Spot Rates from the Treasury Yield Curve......Page 628
The Discount Function......Page 631
Forward Rates......Page 632
Swap Curve......Page 633
Classical Economic Theories About the Determinants of the Shape of the Term Structure......Page 637
Expectations Theories......Page 638
Bond Valuation Formulas in Continuous Time......Page 643
The Term Structure of Interest Rates in Continuous Time......Page 648
Spot Rates: Continuous Case......Page 649
Forward Rates: Continuous Case......Page 650
Relationships for Bond and Option Valuation......Page 651
The Feynman-Kac Formula......Page 652
Multifactor Term Structure Model......Page 657
Arbitrage-Free Models versus Equilibrium Models......Page 659
Examples of One-Factor Term Structure Models......Page 660
Pricing of Interest-Rate Derivatives......Page 663
The Heath-Jarrow-Morton Model of the Term Structure......Page 665
The Brace-Gatarek-Musiela Model......Page 668
Discretization of Itô Processes......Page 669
Summary......Page 671
Management versus a Bond Market Index......Page 674
Tracking Error and Bond Portfolio Strategies......Page 676
Risk Factors and Portfolio Management Strategies......Page 677
Illustration of the Multifactor Risk Model......Page 679
Liability-Funding Strategies......Page 686
Cash Flow Matching......Page 689
Portfolio Immunization......Page 692
Scenario Optimization......Page 697
Stochastic Programming......Page 698
Summary......Page 702
Credit Default Swaps......Page 704
Single-Name Credit Default Swaps......Page 705
Basket Default Swaps......Page 706
Credit Risk Modeling: Structural Models......Page 708
The Black-Scholes-Merton Model......Page 710
Geske Compound Option Model......Page 715
Barrier Structural Models......Page 719
Credit Risk Modeling: Reduced Form Models......Page 721
The Poisson Process......Page 722
The Jarrow-Turnbull Model......Page 723
Transition Matrix......Page 728
The Duffie-Singleton Model......Page 731
Pricing Single-Name Credit Default Swaps......Page 735
General Framework......Page 736
Survival Probability and Forward Default Probability: A Recap......Page 737
Credit Default Swap Value......Page 738
Delivery Option in Default Swaps......Page 741
Default Swaps with Counterparty Risk......Page 742
The Pricing Model......Page 743
How to Model Correlated Default Processes......Page 747
Summary......Page 759
23 Risk Management......Page 762
Market Completeness......Page 763
The Mathematics of Market Completeness......Page 764
The Economics of Market Completeness......Page 767
Why Manage Risk?......Page 769
Market Risk......Page 770
Operational Risk......Page 771
Risk Measures......Page 772
Risk Management in Asset and Portfolio Management......Page 776
Risk Measurement in Practice......Page 777
Getting Down to the Lowest Level......Page 778
Regulatory Implications of Risk Measurement......Page 779
Summary......Page 780
INDEX......Page 782