Numerical methods in finance : a MATLAB-based introduction / Paolo Brandimarte.

Format:

Book

Author/Creator:

Brandimarte, Paolo.

Contributor:

Charles R. Anderson Endowment Fund.

Series:

Wiley series in probability and statistics

Language:

English

Subjects (All):

Finance--Statistical methods.

Finance.

Physical Description:

xv, 403 pages : illustrations ; 25 cm.

Place of Publication:

New York : Wiley, [2002]

Contents:

1 Financial problems and numerical methods 3

1.1 MATLAB environment 4

1.1.1 Why MATLAB? 5

1.2 Fixed-income securities: analysis and portfolio immunization 6

1.2.1 Basic valuation of fixed-income securities 7

1.2.2 Interest rate sensitivity and bond portfolio immunization 14

1.2.3 MATLAB functions to deal with fixed-income securities 17

1.3 Portfolio optimization 27

1.3.1 Basics of mean-variance portfolio optimization 30

1.3.2 MATLAB functions to deal with mean-variance portfolio optimization 32

1.4 Derivatives 41

1.4.1 Modeling the dynamics of asset prices 46

1.4.2 Black-Scholes model 52

1.4.3 Black-Scholes model in MATLAB 54

1.4.4 Pricing American options by binomial lattices 57

1.4.5 Option pricing by Monte Carlo simulation 65

1.5 Value-at-risk 66

S1.1 Stochastic differential equations and Ito's lemma 70

Part II Numerical Methods

2 Basics of numerical analysis 77

2.1 Nature of numerical computation 78

2.1.1 Working with a finite precision arithmetic 78

2.1.2 Number representation, rounding, and truncation 82

2.1.3 Error propagation and instability 84

2.1.4 Order of convergence and computational complexity 85

2.2 Solving systems of linear equations 86

2.2.1 Condition number for a matrix 87

2.2.2 Direct methods for solving systems of linear equations 90

2.2.3 Tridiagonal matrices 94

2.2.4 Iterative methods for solving systems of linear equations 95

2.3 Function approximation and interpolation 104

2.4 Solving nonlinear equations 111

2.4.1 Bisection method 112

2.4.2 Newton's method 113

2.4.3 Solving nonlinear equations in MATLAB 114

2.5 Numerical integration 117

3 Optimization methods 123

3.1 Classification of optimization problems 123

3.1.1 Finite- vs. infinite-dimensional problems 124

3.1.2 Unconstrained vs. constrained problems 128

3.1.3 Convex vs. nonconvex problems 129

3.1.4 Linear vs. nonlinear problems 130

3.1.5 Continuous vs. discrete problems 132

3.1.6 Deterministic vs. stochastic problems 136

3.2 Numerical methods for unconstrained optimization 141

3.2.1 Steepest descent method 142

3.2.2 The subgradient method 143

3.2.3 Newton and the trust region methods 143

3.2.4 No-derivatives algorithms: quasi-Newton method and simplex search 144

3.2.5 Unconstrained optimization in MATLAB 146

3.3 Methods for constrained optimization 148

3.3.1 Penalty function approach 149

3.3.2 Kuhn-Tucker conditions 154

3.3.3 Duality theory 159

3.3.4 Kelley's cutting plane algorithm 165

3.3.5 Active set method 166

3.4 Linear programming 168

3.4.1 Geometric and algebraic features of linear programming 170

3.4.2 Simplex method 172

3.4.3 Duality in linear programming 174

3.4.4 Interior point methods 176

3.4.5 Linear programming in MATLAB 179

3.5 Branch and bound methods for nonconvex optimization 180

3.5.1 LP-based branch and bound for MILP models 186

3.6 Heuristic methods for nonconvex optimization 189

3.7 L-shaped method for two-stage linear stochastic programming 194

S3.1 Elements of convex analysis 197

S3.1.1 Convexity in optimization 197

S3.1.2 Convex polyhedra and polytopes 201

4 Principles of Monte Carlo simulation 207

4.1 Monte Carlo integration 208

4.2 Generating pseudorandom variates 210

4.2.1 Generating pseudorandom numbers 210

4.2.2 Inverse transform method 212

4.2.3 Acceptance-rejection method 213

4.2.4 Generating normal variates by the polar approach 215

4.3 Setting the number of replications 218

4.4 Variance reduction techniques 220

4.4.1 Antithetic variates 220

4.4.2 Common random numbers 224

4.4.3 Control variates 226

4.4.4 Variance reduction by conditioning 227

4.4.5 Stratified sampling 227

4.4.6 Importance sampling 229

4.5 Quasi-Monte Carlo simulation 234

4.5.1 Generating Halton's low-discrepancy sequences 235

4.5.2 Generating Sobol's low-discrepancy sequences 241

4.6 Integrating simulation and optimization 246

5 Finite difference methods for partial differential equations 251

5.1 Introduction and classification of PDEs 252

5.2 Numerical solution by finite difference methods 256

5.2.1 Bad example of a finite difference scheme 257

5.2.2 Instability in a finite difference scheme 259

5.3 Explicit and implicit methods for second-order PDEs 265

5.3.1 Solving the heat equation by an explicit method 266

5.3.2 Solving the heat equation by an implicit method 270

5.3.3 Solving the heat equation by the Crank-Nicolson method 274

5.4 Convergence, consistency, and stability 275

S5.1 Classification of second-order PDEs and characteristic curves 277

Part III Applications to Finance

6 Optimization models for portfolio management 283

6.1 Mixed-integer programming models 285

6.2 Multistage stochastic programming models 289

6.2.1 Split-variable formulation 292

6.2.2 Compact formulation 297

6.2.3 Sample asset and liability management model formulation 301

6.2.4 Scenario generation for multistage stochastic programming 303

6.3 Fixed-mix model based on global optimization 309

7 Option valuation by Monte Carlo simulation 315

7.1 Simulating asset price dynamics 316

7.2 Pricing a vanilla European option by Monte Carlo simulation 319

7.2.1 Using antithetic variates to price a vanilla European option 321

7.2.2 Using antithetic variates to price a European option with truncated payoff 322

7.2.3 Using control variates to price a vanilla European option 323

7.2.4 Using Halton low-discrepancy sequences to price a vanilla European option 325

7.3 Introduction to exotic and path-dependent options 326

7.3.1 Barrier options 326

7.3.2 Asian options 330

7.3.3 Lookback options 331

7.4 Pricing a down-and-out put 332

7.5 Pricing an Asian option 340

8 Option valuation by finite difference methods 347

8.1 Applying finite difference methods to the Black-Scholes equation 347

8.2 Pricing a vanilla European option by an explicit method 350

8.2.1 Financial interpretation of the instability of the explicit method 352

8.3 Pricing a vanilla European option by a fully implicit method 354

8.4 Pricing a barrier option by the Crank-Nicolson method 357

8.5 Dealing with American options 358

Appendix A Introduction to MATLAB programming 367

A.1 MATLAB environment 367

A.2 MATLAB graphics 374

A.3 MATLAB programming 375

Appendix B Refresher on probability theory 379

B.1 Sample space, events, and probability 379

B.2 Random variables, expectation, and variance 381

B.2.1 Common continuous random variables 383

B.3 Jointly distributed random variables 386

B.4 Independence, covariance, and conditional expectation 388

B.5 Parameter estimation 391.

Notes:

"A Wiley-Interscience publication."

Includes bibliographical references and index.

Local Notes:

Acquired for the Penn Libraries with assistance from the Charles R. Anderson Endowment Fund.

ISBN:

0471396869

OCLC:

47092043

Online:

Table of Contents