Carnegie Mellon University
Masters program in Computational Finance
MBA programat the Tepper School of Business
权威教程，经典出品。
1 Introduction 9
1.1 Optimization Problems . . . . . . . . . . . . . . . . . . . . . . 9
1.1.1 Linear and Nonlinear Programming . . . . . . . . . . 10
1.1.2 Quadratic Programming . . . . . . . . . . . . . . . . . 11
1.1.3 Conic Optimization . . . . . . . . . . . . . . . . . . . 12
1.1.4 Integer Programming . . . . . . . . . . . . . . . . . . 12
1.1.5 Dynamic Programming . . . . . . . . . . . . . . . . . 13
1.2 Optimization with Data Uncertainty . . . . . . . . . . . . . . 13
1.2.1 Stochastic Programming . . . . . . . . . . . . . . . . . 13
1.2.2 Robust Optimization . . . . . . . . . . . . . . . . . . . 14
1.3 Financial Mathematics . . . . . . . . . . . . . . . . . . . . . . 16
1.3.1 Portfolio Selection and Asset Allocation . . . . . . . . 16
1.3.2 Pricing and Hedging of Options . . . . . . . . . . . . . 18
1.3.3 Risk Management . . . . . . . . . . . . . . . . . . . . 19
1.3.4 Asset/Liability Management . . . . . . . . . . . . . . 20
2 Linear Programming: Theory and Algorithms 23
2.1 The Linear Programming Problem . . . . . . . . . . . . . . . 23
2.2 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.3 Optimality Conditions . . . . . . . . . . . . . . . . . . . . . . 28
2.4 The Simplex Method . . . . . . . . . . . . . . . . . . . . . . . 31
2.4.1 Basic Solutions . . . . . . . . . . . . . . . . . . . . . . 32
2.4.2 Simplex Iterations . . . . . . . . . . . . . . . . . . . . 35
2.4.3 The Tableau Form of the Simplex Method . . . . . . . 39
2.4.4 Graphical Interpretation . . . . . . . . . . . . . . . . . 42
2.4.5 The Dual Simplex Method . . . . . . . . . . . . . . . 43
2.4.6 Alternatives to the Simplex Method . . . . . . . . . . 45
3 LP Models: Asset/Liability Cash Flow Matching 47
3.1 Short Term Financing . . . . . . . . . . . . . . . . . . . . . . 47
3.1.1 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.1.2 Solving the Model with SOLVER . . . . . . . . . . . . 50
3.1.3 Interpreting the output of SOLVER . . . . . . . . . . 53
3.1.4 Modeling Languages . . . . . . . . . . . . . . . . . . . 54
3.1.5 Features of Linear Programs . . . . . . . . . . . . . . 55
3.2 Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.3 Sensitivity Analysis for Linear Programming . . . . . . . . . 58
34 CONTENTS
3.3.1 Short Term Financing . . . . . . . . . . . . . . . . . . 58
3.3.2 Dedication . . . . . . . . . . . . . . . . . . . . . . . . 63
3.4 Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4 LP Models: Asset Pricing and Arbitrage 69
4.1 The Fundamental Theorem of Asset Pricing . . . . . . . . . . 69
4.1.1 Replication . . . . . . . . . . . . . . . . . . . . . . . . 71
4.1.2 Risk-Neutral Probabilities . . . . . . . . . . . . . . . . 72
4.1.3 The Fundamental Theorem of Asset Pricing . . . . . . 74
4.2 Arbitrage Detection Using Linear Programming . . . . . . . . 75
4.3 Additional Exercises . . . . . . . . . . . . . . . . . . . . . . . 78
4.4 Case Study: Tax Clientele Eects in Bond Portfolio Manage-
ment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5 Nonlinear Programming: Theory and Algorithms 85
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.2 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.3 Univariate Optimization . . . . . . . . . . . . . . . . . . . . . 88
5.3.1 Binary search . . . . . . . . . . . . . . . . . . . . . . . 88
5.3.2 Newton's Method . . . . . . . . . . . . . . . . . . . . . 92
5.3.3 Approximate Line Search . . . . . . . . . . . . . . . . 95
5.4 Unconstrained Optimization . . . . . . . . . . . . . . . . . . . 97
5.4.1 Steepest Descent . . . . . . . . . . . . . . . . . . . . . 97
5.4.2 Newton's Method . . . . . . . . . . . . . . . . . . . . . 101
5.5 Constrained Optimization . . . . . . . . . . . . . . . . . . . . 104
5.5.1 The generalized reduced gradient method . . . . . . . 107
5.5.2 Sequential Quadratic Programming . . . . . . . . . . . 112
5.6 Nonsmooth Optimization: Subgradient Methods . . . . . . . 113
6 NLP Models: Volatility Estimation 115
6.1 Volatility Estimation with GARCH Models . . . . . . . . . . 115
6.2 Estimating a Volatility Surface . . . . . . . . . . . . . . . . . 119
7 Quadratic Programming: Theory and Algorithms 125
7.1 The Quadratic Programming Problem . . . . . . . . . . . . . 125
7.2 Optimality Conditions . . . . . . . . . . . . . . . . . . . . . . 126
7.3 Interior-Point Methods . . . . . . . . . . . . . . . . . . . . . . 128
7.4 The Central Path . . . . . . . . . . . . . . . . . . . . . . . . . 131
7.5 Interior-Point Methods . . . . . . . . . . . . . . . . . . . . . . 132
7.5.1 Path-Following Algorithms . . . . . . . . . . . . . . . 132
7.5.2 Centered Newton directions . . . . . . . . . . . . . . . 133
7.5.3 Neighborhoods of the Central Path . . . . . . . . . . . 135
7.5.4 A Long-Step Path-Following Algorithm . . . . . . . . 138
7.5.5 Starting from an Infeasible Point . . . . . . . . . . . . 138
7.6 QP software . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
7.7 Additional Exercises . . . . . . . . . . . . . . . . . . . . . . . 139CONTENTS 5
8 QP Models: Portfolio Optimization 141
8.1 Mean-Variance Optimization . . . . . . . . . . . . . . . . . . 141
8.1.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 143
8.1.2 Large-Scale Portfolio Optimization . . . . . . . . . . . 148
8.1.3 The Black-Litterman Model . . . . . . . . . . . . . . . 151
8.1.4 Mean-Absolute Deviation to Estimate Risk . . . . . . 155
8.2 Maximizing the Sharpe Ratio . . . . . . . . . . . . . . . . . . 158
8.3 Returns-Based Style Analysis . . . . . . . . . . . . . . . . . . 160
8.4 Recovering Risk-Neural Probabilities from Options Prices . . 162
8.5 Additional Exercises . . . . . . . . . . . . . . . . . . . . . . . 166
8.6 Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
9 Conic Optimization Tools 171
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
9.2 Second-order cone programming: . . . . . . . . . . . . . . . . 171
9.2.1 Ellipsoidal Uncertainty for Linear Constraints . . . . . 173
9.2.2 Conversion of quadratic constraints into second-order
cone constraints . . . . . . . . . . . . . . . . . . . . . 175
9.3 Semidenite programming: . . . . . . . . . . . . . . . . . . . 176
9.3.1 Ellipsoidal Uncertainty for Quadratic Constraints . . . 178
9.4 Algorithms and Software . . . . . . . . . . . . . . . . . . . . . 179
10 Conic Optimization Models in Finance 181
10.1 Tracking Error and Volatility Constraints . . . . . . . . . . . 181
10.2 Approximating Covariance Matrices . . . . . . . . . . . . . . 184
10.3 Recovering Risk-Neural Probabilities from Options Prices . . 187
10.4 Arbitrage Bounds for Forward Start Options . . . . . . . . . 189
10.4.1 A Semi-Static Hedge . . . . . . . . . . . . . . . . . . . 190