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

Format:

Book

Author/Creator:

Brandimarte, Paolo.

Contributor:

Brandimarte, Paolo.

Alumni and Friends Memorial Book Fund.

Series:

Statistics in practice

Language:

English

Subjects (All):

Finance--Statistical methods.

Finance.

Economics--Statistical methods.

Economics.

Physical Description:

xxiv, 669 pages : illustrations ; 25 cm.

Edition:

Second edition.

Place of Publication:

Hoboken, N.J. : Wiley-Interscience, [2006]

Summary:

The use of mathematical models and numerical techniques is a practice employed by a growing number of applied mathematicians working on applications in finance. Reflecting this development, Numerical Methods in Finance and Economics: A MATLAB[Registered]-Based Introduction, Second Edition bridges the gap between financial theory and computational practice while showing readers how to utilize MATLAB[Registered]-the powerful numerical computing environment-for financial applications.

The author provides an essential foundation in finance and numerical analysis in addition to background material for students from both engineering and economics perspectives. A wide range of topics is covered, including standard numerical analysis methods, Monte Carlo methods to simulate systems affected by significant uncertainty, and optimization methods to find an optimal set of decisions.

Among this book's most outstanding features is the integration of MATLAB[Registered], which helps students and practitioners solve relevant problems in finance, such as portfolio management and derivatives pricing. This tutorial is useful in connecting theory with practice in the application of classical numerical methods and advanced methods, while illustrating underlying algorithmic concepts in concrete terms.

Newly featured in the Second Edition: In-depth treatment of Monte Carlo methods with due attention paid to variance reduction strategies, New appendix on AMPL[Copyright] in order to better illustrate the optimization models in Chapters 11 and 12, New chapter on binomial and trinomial lattices, Additional treatment of partial differential equations with two space dimensions, Expanded treatment within the chapter on financial theory to provide a more thorough background for engineers not familiar with finance, New coverage of advanced optimization methods and applications later in the text.

Numerical Methods in Finance and Economics: A MATLAB[Registered]-Based Introduction, Second Edition presents basic treatments and more specialized literature, and it also uses algebraic languages, such as AMPL[Copyright], to connect the pencil-and-paper statement of an optimization model with its solution by a software library. Offering computational practice in both financial engineering and economics fields, this book equips practitioners with the necessary techniques to measure and manage risk.

Contents:

1 Motivation 3

1.1 Need for numerical methods 4

1.2 Need for numerical computing environments: why MATLAB? 9

1.3 Need for theory 13

2 Financial Theory 23

2.1 Modeling uncertainty 25

2.2 Basic financial assets and related issues 30

2.2.1 Bonds 30

2.2.2 Stocks 31

2.2.3 Derivatives 33

2.2.4 Asset pricing, portfolio optimization, and risk management 37

2.3 Fixed-income securities: analysis and portfolio immunization 42

2.3.1 Basic theory of interest rates: compounding and present value 42

2.3.2 Basic pricing of fixed-income securities 49

2.3.3 Interest rate sensitivity and bond portfolio immunization 57

2.3.4 MATLAB functions to deal with fixed-income securities 60

2.3.5 Critique 64

2.4 Stock portfolio optimization 65

2.4.1 Utility theory 66

2.4.2 Mean-variance portfolio optimization 73

2.4.3 MATLAB functions to deal with mean-variance portfolio optimization 74

2.4.4 Critical remarks 81

2.4.5 Alternative risk measures: Value at Risk and quantile-based measures 83

2.5 Modeling the dynamics of asset prices 88

2.5.1 From discrete to continuous time 88

2.5.2 Standard Wiener process 91

2.5.3 Stochastic integrals and stochastic differential equations 93

2.5.4 Ito's lemma 96

2.5.5 Generalizations 100

2.6 Derivatives pricing 102

2.6.1 Simple binomial model for option pricing 105

2.6.2 Black-Scholes model 108

2.6.3 Risk-neutral expectation and Feynman-Kac formula 111

2.6.4 Black-Scholes model in MATLAB 113

2.6.5 A few remarks on Black-Scholes formula 116

2.6.6 Pricing American options 117

2.7 Introduction to exotic and path-dependent options 118

2.7.1 Barrier options 119

2.7.2 Asian options 123

2.7.3 Lookback options 123

2.8 An outlook on interest-rate derivatives 124

2.8.1 Modeling interest-rate dynamics 126

2.8.2 Incomplete markets and the market price of risk 127

Part II Numerical Methods

3 Basics of Numerical Analysis 137

3.1 Nature of numerical computation 138

3.1.1 Number representation, rounding, and truncation 138

3.1.2 Error propagation, conditioning, and instability 141

3.1.3 Order of convergence and computational complexity 143

3.2 Solving systems of linear equations 145

3.2.1 Vector and matrix norms 146

3.2.2 Condition number for a matrix 149

3.2.3 Direct methods for solving systems of linear equations 154

3.2.4 Tridiagonal matrices 159

3.2.5 Iterative methods for solving systems of linear equations 160

3.3 Function approximation and interpolation 173

3.3.1 Ad hoc approximation 177

3.3.2 Elementary polynomial interpolation 179

3.3.3 Interpolation by cubic splines 183

3.3.4 Theory of function approximation by least squares 188

3.4 Solving non-linear equations 191

3.4.1 Bisection method 192

3.4.2 Newton's method 195

3.4.3 Optimization-based solution of non-linear equations 198

3.4.4 Putting two things together: solving a functional equation by a collocation method 204

3.4.5 Homotopy continuation methods 204

4 Numerical Integration: Deterministic and Monte Carlo Methods 209

4.1 Deterministic quadrature 211

4.1.1 Classical interpolatory formulas 212

4.1.2 Gaussian quadrature 214

4.1.3 Extensions and product rules 219

4.1.4 Numerical integration in MATLAB 220

4.2 Monte Carlo integration 221

4.3 Generating pseudorandom variates 225

4.3.1 Generating pseudorandom numbers 226

4.3.2 Inverse transform method 230

4.3.3 Acceptance-rejection method 233

4.3.4 Generating normal variates by the polar approach 235

4.4 Setting the number of replications 240

4.5 Variance reduction techniques 244

4.5.1 Antithetic sampling 244

4.5.2 Common random numbers 251

4.5.3 Control variates 252

4.5.4 Variance reduction by conditioning 255

4.5.5 Stratified sampling 260

4.5.6 Importance sampling 261

4.6 Quasi-Monte Carlo simulation 267

4.6.1 Generating Halton low-discrepancy sequences 269

4.6.2 Generating Sobol low-discrepancy sequences 281

5 Finite Difference Methods for Partial Differential Equations 289

5.1 Introduction and classification of PDEs 290

5.2 Numerical solution by finite difference methods 293

5.2.1 Bad example of a finite difference scheme 295

5.2.2 Instability in a finite difference scheme 297

5.3 Explicit and implicit methods for the heat equation 303

5.3.1 Solving the heat equation by an explicit method 304

5.3.2 Solving the heat equation by a fully implicit method 309

5.3.3 Solving the heat equation by the Crank-Nicolson method 313

5.4 Solving the bidimensional heat equation 314

5.5 Convergence, consistency, and stability 320

6 Convex Optimization 327

6.1 Classification of optimization problems 328

6.1.1 Finite- vs. infinite-dimensional problems 328

6.1.2 Unconstrained vs. constrained problems 333

6.1.3 Convex vs. non-convex problems 333

6.1.4 Linear vs. non-linear problems 335

6.1.5 Continuous vs. discrete problems 337

6.1.6 Deterministic vs. stochastic problems 337

6.2 Numerical methods for unconstrained optimization 338

6.2.1 Steepest descent method 339

6.2.2 The subgradient method 340

6.2.3 Newton and the trust region methods 341

6.2.4 No-derivatives algorithms: quasi-Newton method and simplex search 342

6.2.5 Unconstrained optimization in MATLAB 343

6.3 Methods for constrained optimization 346

6.3.1 Penalty function approach 346

6.3.2 Kuhn-Tucker conditions 351

6.3.3 Duality theory 357

6.3.4 Kelley's cutting plane algorithm 363

6.3.5 Active set method 365

6.4 Linear programming 366

6.4.1 Geometric and algebraic features of linear programming 368

6.4.2 Simplex method 370

6.4.3 Duality in linear programming 372

6.4.4 Interior point methods 375

6.5 Constrained optimization in MATLAB 377

6.5.1 Linear programming in MATLAB 378

6.5.2 A trivial LP model for bond portfolio management 380

6.5.3 Using quadratic programming to trace efficient portfolio frontier 383

6.5.4 Non-linear programming in MATLAB 385

6.6 Integrating simulation and optimization 387

S6.1 Elements of convex analysis 389

S6.1.1 Convexity in optimization 389

S6.1.2 Convex polyhedra and polytopes 393

Part III Pricing Equity Options

7 Option Pricing by Binomial and Trinomial Lattices 401

7.1 Pricing by binomial lattices 402

7.1.1 Calibrating a binomial lattice 403

7.1.2 Putting two things together: pricing a pay-later option 410

7.1.3 An improved implementation of binomial lattices 411

7.2 Pricing American options by binomial lattices 414

7.3 Pricing bidimensional options by binomial lattices 417

7.4 Pricing by trinomial lattices 422

8 Option Pricing by Monte Carlo Methods 429

8.1 Path generation 430

8.1.1 Simulating geometric Brownian motion 431

8.1.2 Simulating hedging strategies 435

8.1.3 Brownian bridge 439

8.2 Pricing an exchange option 443

8.3 Pricing a down-and-out put option 446

8.3.1 Crude Monte Carlo 446

8.3.2 Conditional Monte Carlo 447

8.3.3 Importance sampling 450

8.4 Pricing an arithmetic average Asian option 454

8.4.1 Control variates 455

8.4.2 Using Halton sequences 458

8.5 Estimating Greeks by Monte Carlo sampling 468

9 Option Pricing by Finite Difference Methods 475

9.1 Applying finite difference methods to the Black-Scholes equation 475

9.2 Pricing a vanilla European option by an explicit method 478

9.2.1 Financial interpretation of the instability of the explicit method 481

9.3 Pricing a vanilla European option by a fully implicit method 482

9.4 Pricing a barrier option by the Crank-Nicolson method 485

9.5 Dealing with American options 486

Part IV Advanced Optimization Models and Methods

10 Dynamic Programming 495

10.1 The shortest path problem 496

10.2 Sequential decision processes 500

10.2.1 The optimality principle and solving the functional equation 501

10.3 Solving

stochastic decision problems by dynamic programming 504

10.4 American option pricing by Monte Carlo simulation 511

10.4.1 A MATLAB implementation of the least squares approach 517

10.4.2 Some remarks and alternative approaches 519

11 Linear Stochastic Programming Models with Recourse 525

11.1 Linear stochastic programming models 526

11.2 Multistage stochastic programming models for portfolio management 530

11.2.1 Split-variable model formulation 532

11.2.2 Compact model formulation 540

11.2.3 Asset and liability management with transaction costs 544

11.3 Scenario generation for multistage stochastic programming 546

11.3.1 Sampling for scenario tree generation 547

11.3.2 Arbitrage free scenario generation 550

11.4. L-shaped method for two-stage linear stochastic programming 555

11.5 A comparison with dynamic programming 558

12 Non-Convex Optimization 563

12.1 Mixed-integer programming models 564

12.1.1 Modeling with logical variables 565

12.1.2 Mixed-integer portfolio optimization models 571

12.2 Fixed-mix model based on global optimization 576

12.3 Branch and bound methods for non-convex optimization 578

12.3.1 LP-based branch and bound for MILP models 584

12.4 Heuristic methods for non-convex optimization 591

Appendix A Introduction to MATLAB Programming 603

A.1 MATLAB environment 603

A.2 MATLAB graphics 614

A.3 MATLAB programming 616

Appendix B Refresher on Probability Theory and Statistics 623

B.1 Sample space, events, and probability 623

B.2 Random variables, expectation, and variance 625

B.2.1 Common continuous random variables 628

B.3 Jointly distributed random variables 632

B.4 Independence, covariance, and conditional expectation 633

B.5 Parameter estimation 637

B.6 Linear regression 642

Appendix C Introduction to AMPL 647

C.1 Running optimization models in AMPL 648

C.2 Mean variance efficient portfolios in AMPL 649

C.3 The knapsack model in AMPL 652

C.4 Cash flow matching 655.

Notes:

Rev. ed. of: Numerical methods in finance. 2002.

Includes bibliographical references and index.

Local Notes:

Acquired for the Penn Libraries with assistance from the Alumni and Friends Memorial Book Fund.

Related Work:

Brandimarte, Paolo. Numerical methods in finance.

ISBN:

0471745030

OCLC:

70129191

Publisher Number:

9780471745037