Optimization methods in finance

Half-title......Page 3
Series-title......Page 4
Title......Page 5
Copyright......Page 6
Dedication......Page 7
Contents......Page 9
Foreword......Page 13
1.1 Optimization problems......Page 15
1.1.2 Quadratic programming......Page 17
1.1.3 Conic optimization......Page 18
1.2 Optimization with data uncertainty......Page 19
1.2.1 Stochastic programming......Page 20
1.2.2 Robust optimization......Page 21
1.3 Financial mathematics......Page 22
1.3.1 Portfolio selection and asset allocation......Page 23
1.3.2 Pricing and hedging of options......Page 25
1.3.3 Risk management......Page 26
1.3.4 Asset/liability management......Page 27
2.1 The linear programming problem......Page 29
2.2 Duality......Page 31
2.3 Optimality conditions......Page 35
2.4 The simplex method......Page 37
2.4.1 Basic solutions......Page 39
2.4.2 Simplex iterations......Page 42
2.4.3 The tableau form of the simplex method......Page 46
2.4.4 Graphical interpretation......Page 50
2.4.5 The dual simplex method......Page 51
2.4.6 Alternatives to the simplex method......Page 53
3.1 Short-term financing......Page 55
Constraints......Page 56
Final Model......Page 57
3.1.2 Solving the model with SOLVER......Page 58
3.1.3 Interpreting the output of SOLVER......Page 60
3.1.4 Modeling languages......Page 62
3.1.5 Features of linear programs......Page 63
3.2 Dedication......Page 64
3.3.1 Short-term financing......Page 67
Sensitivity analysis......Page 70
3.3.2 Dedication......Page 72
3.4 Case study: constructing a dedicated portfolio......Page 74
4.1 Derivative securities and the fundamental theorem of asset pricing......Page 76
4.1.1 Replication......Page 78
4.1.2 Risk-neutral probabilities......Page 79
4.1.3 The fundamental theorem of asset pricing......Page 81
4.2 Arbitrage detection using linear programming......Page 83
4.3 Additional exercises......Page 85
4.4 Case study: tax clientele effects in bond portfolio management......Page 90
Taxes......Page 91
Data......Page 92
5.1 Introduction......Page 94
5.3.1 Binary search......Page 96
Golden section search......Page 99
5.3.2 Newton\'s method......Page 100
5.3.3 Approximate line search......Page 104
5.4 Unconstrained optimization......Page 106
5.4.1 Steepest descent......Page 107
5.4.2 Newton\'s method......Page 110
Optimization version......Page 112
5.5 Constrained optimization......Page 114
5.5.1 The generalized reduced gradient method......Page 117
5.5.2 Sequential quadratic programming......Page 122
5.6 Nonsmooth optimization: subgradient methods......Page 124
6.1 Volatility estimation with GARCH models......Page 126
6.2 Estimating a volatility surface......Page 130
7.1 The quadratic programming problem......Page 135
7.2 Optimality conditions......Page 136
7.3.1 Introduction......Page 138
7.3.2 The central path......Page 141
7.3.3 Path-following algorithms......Page 142
7.3.4 Centered Newton directions......Page 143
7.3.5 Neighborhoods of the central path......Page 146
7.3.6 A long-step path-following algorithm......Page 148
7.4 QP software......Page 149
7.5 Additional exercises......Page 150
8.1 Mean-variance optimization......Page 152
8.1.1 Example......Page 154
8.1.2 Large-scale portfolio optimization......Page 158
Diversification......Page 159
Transaction costs......Page 160
Parameter estimation......Page 161
8.1.3 The Black–Litterman model......Page 162
8.1.4 Mean-absolute deviation to estimate risk......Page 166
8.2 Maximizing the Sharpe ratio......Page 169
8.3 Returns-based style analysis......Page 172
8.4 Recovering risk-neural probabilities from options prices......Page 175
8.5 Additional exercises......Page 179
8.6 Case study: constructing an efficient portfolio......Page 181
9.1 Introduction......Page 182
9.2 Second-order cone programming......Page 183
9.2.1 Ellipsoidal uncertainty for linear constraints......Page 184
9.2.2 Conversion of quadratic constraints into second-order cone constraints......Page 186
9.3 Semidefinite programming......Page 187
9.3.1 Ellipsoidal uncertainty for quadratic constraints......Page 189
9.4 Algorithms and software......Page 191
10.1 Tracking error and volatility constraints......Page 192
10.2 Approximating covariance matrices......Page 195
10.3 Recovering risk-neutral probabilities from options prices......Page 199
10.4 Arbitrage bounds for forward start options......Page 201
11.1 Introduction......Page 206
11.2 Modeling logical conditions......Page 207
Solving the model with SOLVER......Page 208
11.3.1 Linear programming relaxation......Page 210
An example......Page 212
The branch-and-bound algorithm......Page 214
Branching......Page 216
Node selection......Page 217
Heuristics......Page 218
11.3.3 Cutting planes......Page 220
11.3.4 Branch and cut......Page 224
12.1 Combinatorial auctions......Page 226
12.2 The lockbox problem......Page 227
12.3 Constructing an index fund......Page 230
12.3.1 A large-scale deterministic model......Page 231
Data requirements......Page 232
Solution strategy......Page 233
12.3.2 A linear programming model......Page 235
12.4 Portfolio optimization with minimum transaction levels......Page 236
12.5 Additional exercises......Page 237
12.6 Case study: constructing an index fund......Page 238
13.1 Introduction......Page 239
13.1.1 Backward recursion......Page 243
13.1.2 Forward recursion......Page 245
13.2 Abstraction of the dynamic programming approach......Page 247
13.3.1 Dynamic programming formulation......Page 250
13.3.2 An alternative formulation......Page 251
13.4 Stochastic dynamic programming......Page 252
14.1 A model for American options......Page 254
14.2 Binomial lattice......Page 256
14.2.1 Specifying the parameters......Page 257
14.2.2 Option pricing......Page 258
15 DP models: structuring asset-backed securities......Page 262
15.1 Data......Page 264
15.2 Enumerating possible tranches......Page 266
15.3 A dynamic programming approach......Page 267
15.4 Case study: structuring CMOs......Page 268
16.1 Introduction......Page 269
16.2 Two-stage problems with recourse......Page 270
16.3 Multi-stage problems......Page 272
16.4 Decomposition......Page 274
16.5 Scenario generation......Page 277
16.5.1 Autoregressive model......Page 278
16.5.2 Constructing scenario trees......Page 279
Random sampling......Page 280
Tree fitting......Page 281
Arbitrage-free scenario trees......Page 282
17.1 Risk measures......Page 285
17.2 Minimizing CVaR......Page 288
17.3 Example: bond portfolio optimization......Page 290
18.1 Asset/liability management......Page 293
18.1.1 Corporate debt management......Page 296
18.2.1 The model......Page 299
18.2.2 An example......Page 301
18.3 Case study: option pricing with transaction costs......Page 302
18.3.1 The standard problem......Page 303
18.3.2 Transaction costs......Page 304
19.1 Introduction to robust optimization......Page 306
19.2 Uncertainty sets......Page 307
19.3.1 Constraint robustness......Page 309
19.3.2 Objective robustness......Page 310
19.3.3 Relative robustness......Page 312
19.3.4 Adjustable robust optimization......Page 314
19.4.1 Sampling......Page 316
19.4.2 Conic optimization......Page 317
19.4.3 Saddle-point characterizations......Page 319
20.1 Robust multi-period portfolio selection......Page 320
20.2 Robust profit opportunities in risky portfolios......Page 325
20.3 Robust portfolio selection......Page 327
20.4 Relative robustness in portfolio selection......Page 329
20.5 Moment bounds for option prices......Page 331
20.6 Additional exercises......Page 332
Appendix A Convexity......Page 334
Appendix B Cones......Page 336
Appendix C A probability primer......Page 337
Appendix D The revised simplex method......Page 341
An example......Page 345
References......Page 352
Index......Page 356