Numerical Methods in Finance: A MATLAB-Based Introduction

Contents......Page 8
Preface......Page 14
Part I: Background......Page 18
1 Financial problems and numerical methods......Page 20
1.1 MATLAB environment......Page 21
1.2 Fixed-income securities: analysis and portfolio immunization......Page 23
1.3 Portfolio optimization......Page 44
1.4 Derivatives......Page 58
1.5 Value-at-risk......Page 83
S1.1 Stochastic differential equations and Ito's lemma......Page 87
References......Page 89
Part II: Numerical Methods......Page 92
2 Basics of numerical analysis......Page 94
2.1 Nature of numerical computation......Page 95
2.2 Solving systems of linear equations......Page 103
2.3 Function approximation and interpolation......Page 121
2.4 Solving nonlinear equations......Page 128
2.5 Numerical integration......Page 134
References......Page 138
3.1 Classification of optimization problems......Page 140
3.2 Numerical methods for unconstrained optimization......Page 158
3.3 Methods for constrained optimization......Page 165
3.4 Linear programming......Page 185
3.5 Branch and bound methods for nonconvex optimization......Page 197
3.6 Heuristic methods for nonconvex optimization......Page 206
3.7 L-shaped method for two-stage linear stochastic programming......Page 211
S3.1 Elements of convex analysis......Page 214
4 Principles of Monte Carlo simulation......Page 224
4.1 Monte Carlo integration......Page 225
4.2 Generating pseudorandom variates......Page 227
4.3 Setting the number of replications......Page 235
4.4 Variance reduction techniques......Page 237
4-5 Quasi-Monte Carlo simulation......Page 251
4.6 Integrating simulation and optimization......Page 263
References......Page 266
5 Finite difference methods for partial differential equations......Page 268
5.1 Introduction and classification of PDEs......Page 269
5.2 Numerical solution by finite difference methods......Page 273
5.3 Explicit and implicit methods for second-order PDEs......Page 282
5.4 Convergence, consistency, and stability......Page 292
S5.1 Classification of second-order PDEs and characteristic curves......Page 294
References......Page 296
Part III: Applications to Finance......Page 298
6 Optimization models for portfolio management......Page 300
6.1 Mixed-integer programming models......Page 302
6.2 Multistage stochastic programming models......Page 306
6.3 Fixed-mix model based on global optimization......Page 326
References......Page 329
7 Option valuation by Monte Carlo simulation......Page 332
7.1 Simulating asset price dynamics......Page 333
7.2 Pricing a vanilla European option by Monte Carlo simulation......Page 336
7.3 Introduction to exotic and path-dependent options......Page 343
7.4 Pricing a down-and-out put......Page 349
7.5 Pricing an Asian option......Page 357
References......Page 362
8.1 Applying finite difference methods to the Black- Scholes equation......Page 364
8.2 Pricing a vanilla European option by an explicit method......Page 367
8.3 Pricing a vanilla European option by a fully implicit method......Page 371
8.4 Pricing a barrier option by the Crank-Nicolson method......Page 374
8.5 Dealing with American options......Page 375
References......Page 381
Part IV: Appendices......Page 382
A.1 MATLAB environment......Page 384
A.2 MATLAB graphics......Page 391
A.3 MATLAB programming......Page 392
B.1 Sample space, events, and probability......Page 396
B.2 Random variables, expectation, and variance......Page 398
B.3 Jointly distributed random variables......Page 403
B.4 Independence, covariance, and conditional expectation......Page 405
B.5 Parameter estimation......Page 408
References......Page 412
C......Page 414
E......Page 415
L......Page 416
O......Page 417
P......Page 418
S......Page 419
Z......Page 420