Numerical Methods in Finance and Economics A MATLAB-Based Introduction

by Brandimarte, Paolo

• ISBN13: 9780471745037
• ISBN10: 0471745030
• Edition: 2nd
• Format: Hardcover
• Copyright: 2006-10-06
• Publisher: Wiley-Interscience

Summary

A state-of-the-art introduction to the powerful mathematical and statistical tools used in the field of finance 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?-Based Introduction, Second Edition bridges the gap between financial theory and computational practice while showing readers how to utilize MATLAB?-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?, 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? 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?-Based Introduction, Second Edition presents basic treatments and more specialized literature, and it also uses algebraic languages, such as AMPL?, 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.

Author Biography

PAOLO BRANDIMARTE is Professor of Quantitative Methods for Finance and Logistics at Politecnico di Torino in Italy. He is the author of several publications, including five books, on the application of optimization and simulation to diverse areas such as production management, telecommunications, and finance. Dr. Brandimarte has extensive teaching experience in engineering and economics faculties, including master's and PhD-level courses.

Table of Contents

Preface to the Second Edition	xvii
From the Preface to the First Edition	xxiii
Part I Background
1 Motivation	3	(20)
1.1 Need for numerical methods	4	(5)
1.2 Need for numerical computing environments: why MATLAB?	9	(4)
1.3 Need for theory	13	(7)
For further reading	20	(1)
References	21	(2)
2 Financial Theory	23	(114)
2.1 Modeling uncertainty	25	(5)
2.2 Basic financial assets and related issues	30	(12)
2.2.1 Bonds	30	(1)
2.2.2 Stocks	31	(2)
2.2.3 Derivatives	33	(4)
2.2.4 Asset pricing, portfolio optimization, and risk management	37	(5)
2.3 Fixed-income securities: analysis and portfolio immunization	42	(23)
2.3.1 Basic theory of interest rates: compounding and present value	42	(7)
2.3.2 Basic pricing of fixed-income securities	49	(8)
2.3.3 Interest rate sensitivity and bond portfolio immunization	57	(3)
2.3.4 MATLAB functions to deal with fixed-income securities	60	(4)
2.3.5 Critique	64	(1)
2.4 Stock portfolio optimization	65	(23)
2.4.1 Utility theory	66	(7)
2.4.2 Mean-variance portfolio optimization	73	(1)
2.4.3 MATLAB functions to deal with mean-variance portfolio optimization	74	(7)
2.4.4 Critical remarks	81	(2)
2.4.5 Alternative risk measures: Value at Risk and quantile-based measures	83	(5)
2.5 Modeling the dynamics of asset prices	88	(14)
2.5.1 From discrete to continuous time	88	(3)
2.5.2 Standard Wiener process	91	(2)
2.5.3 Stochastic integrals and stochastic differential equations	93	(3)
2.5.4 Ito's lemma	96	(4)
2.5.5 Generalizations	100	(2)
2.6 Derivatives pricing	102	(16)
2.6.1 Simple binomial model for option pricing	105	(3)
2.6.2 Black-Scholes model	108	(3)
2.6.3 Risk-neutral expectation and Feynman-Kac formula	111	(2)
2.6.4 Black-Scholes model in MATLAB	113	(3)
2.6.5 A few remarks on Black-Scholes formula	116	(1)
2.6.6 Pricing American options	117	(1)
2.7 Introduction to exotic and path-dependent options	118	(6)
2.7.1 Barrier options	119	(4)
2.7.2 Asian options	123	(1)
2.7.3 Lookback options	123	(1)
2.8 An outlook on interest-rate derivatives	124	(6)
2.8.1 Modeling interest-rate dynamics	126	(1)
2.8.2 Incomplete markets and the market price of risk	127	(3)
For further reading	130	(1)
References	131	(6)
Part II Numerical Methods
3 Basics of Numerical Analysis	137	(72)
3.1 Nature of numerical computation	138	(7)
3.1.1 Number representation, rounding, and truncation	138	(3)
3.1.2 Error propagation, conditioning, and instability	141	(2)
3.1.3 Order of convergence and computational complexity	143	(2)
3.2 Solving systems of linear equations	145	(28)
3.2.1 Vector and matrix norms	146	(3)
3.2.2 Condition number for a matrix	149	(5)
3.2.3 Direct methods for solving systems of linear equations	154	(5)
3.2.4 Tridiagonal matrices	159	(1)
3.2.5 Iterative methods for solving systems of linear equations	160	(13)
3.3 Function approximation and interpolation	173	(18)
3.3.1 Ad hoc approximation	177	(2)
3.3.2 Elementary polynomial interpolation	179	(4)
3.3.3 Interpolation by cubic splines	183	(5)
3.3.4 Theory of function approximation by least squares	188	(3)
3.4 Solving non-linear equations	191	(15)
3.4.1 Bisection method	192	(3)
3.4.2 Newton's method	195	(3)
3.4.3 Optimization-based solution of non-linear equations	198	(6)
3.4.4 Putting two things together: solving a functional equation by a collocation method	204	(1)
3.4.5 Homotopy continuation methods	204	(2)
For further reading	206	(1)
References	207	(2)
4 Numerical Integration: Deterministic and Monte Carlo Methods	209	(80)
4.1 Deterministic quadrature	211	(10)
4.1.1 Classical interpolatory formulas	212	(2)
4.1.2 Gaussian quadrature	214	(5)
4.1.3 Extesions and product rules	219	(1)
4.1.4 Numerical integration in MATLAB	220	(1)
4.2 Monte Carlo integration	221	(4)
4.3 Generating pseudorandom variates	225	(15)
4.3.1 Generating pseudorandom numbers	226	(4)
4.3.2 Inverse transform method	230	(3)
4.3.3 Acceptance–rejection method	233	(2)
4.3.4 Generating normal variates by the polar approach	235	(5)
4.4 Setting the number of replications	240	(4)
4.5 Variance reduction techniques	244	(23)
4.5.1 Antithetic sampling	244	(7)
4.5.2 Common random numbers	251	(1)
4.5.3 Control variates	252	(3)
4.5.4 Variance reduction by conditioning	255	(5)
4.5.5 Stratified sampling	260	(1)
4.5.6 Importance sampling	261	(6)
4.6 Quasi-Monte Carlo simulation	267	(19)
4.6.1 Generating Halton low-discrepancy sequences	269	(12)
4.6.2 Generating Sobol low-discrepancy sequences	281	(5)
For further reading	286	(1)
References	287	(2)
5 Finite Difference Methods for Partial Differential Equations	289	(38)
5.1 Introduction and classification of PDEs	290	(3)
5.2 Numerical solution by finite difference methods	293	(10)
5.2.1 Bad example of a finite difference scheme	295	(2)
5.2.2 Instability in a finite difference scheme	297	(6)
5.3 Explicit and implicit methods for the heat equation	303	(11)
5.3.1 Solving the heat equation by an explicit method	304	(5)
5.3.2 Solving the heat equation by a fully implicit method	309	(4)
5.3.3 Solving the heat equation by the Crank-Nicolson method	313	(1)
5.4 Solving the bidimensional heat equation	314	(6)
5.5 Convergence, consistency, and stability	320	(4)
For further reading	324	(1)
References	324	(3)
6 Convex Optimization	327	(74)
6.1 Classification of optimization problems	328	(10)
6.1.1 Finite- vs. infinite-dimensional problems	328	(5)
6.1.2 Unconstrained vs. constrained problems	333	(1)
6.1.3 Convex vs. non-convex problems	333	(2)
6.1.4 Linear vs. non-linear problems	335	(2)
6.1.5 Continuous vs. discrete problems	337	(1)
6.1.6 Deterministic vs. stochastic problems	337	(1)
6.2 Numerical methods for unconstrained optimization	338	(8)
6.2.1 Steepest descent method	339	(1)
6.2.2 The subgradient method	340	(1)
6.2.3 Newton and the trust region methods	341	(1)
6.2.4 No-derivatives algorithms: quasi-Newton method and simplex search	342	(1)
6.2.5 Unconstrained optimization in MATLAB	343	(3)
6.3 Methods for constrained optimization	346	(20)
6.3.1 Penalty function approach	346	(5)
6.3.2 Kuhn-Tucker conditions	351	(6)
6.3.3 Duality theory	357	(6)
6.3.4 Kelley's cutting plane algorithm	363	(2)
6.3.5 Active set method	365	(1)
6.4 Linear programming	366	(11)
6.4.1 Geometric and algebraic features of linear programming	368	(2)
6.4.2 Simplex method	370	(2)
6.4.3 Duality in linear programming	372	(3)
6.4.4 Interior point methods	375	(2)
6.5 Constrained optimization in MATLAB	377	(10)
6.5.1 Linear programming in MATLAB	378	(2)
6.5.2 A trivial LP model for bond portfolio management	380	(3)
6.5.3 Using quadratic programming to trace efficient portfolio frontier	383	(2)
6.5.4 Non-linear programming in MATLAB	385	(2)
6.6 Integrating simulation and optimization	387	(9)
S6.1 Elements of convex analysis	389	(13)
S6.1.1 Convexity in optimization	389	(4)
S6.1.2 Convex polyhedra and polytopes	393	(3)
For further reading	396	(1)
References	397	(4)
Part III Pricing Equity Options
7 Option Pricing by Binomial and Trinomial Lattices	401	(28)
7.1 Pricing by binomial lattices	402	(12)
7.1.1 Calibrating a binomial lattice	403	(7)
7.1.2 Putting two things together: pricing a pay-later option	410	(1)
7.1.3 An improved implementation of binomial lattices	411	(3)
7.2 Pricing American options by binomial lattices	414	(3)
7.3 Pricing bidimensional options by binomial lattices	417	(5)
7.4 Pricing by trinomial lattices	422	(3)
7.5 Summary	425	(1)
For further reading	426	(1)
References	426	(3)
8 Option Pricing bp Monte Carlo Methods	429	(46)
8.1 Path generation	430	(13)
8.1.1 Simulating geometric Brownian motion	431	(4)
8.1.2 Simulating hedging strategies	435	(4)
8.1.3 Brownian bridge	439	(4)
8.2 Pricing an exchange option	443	(3)
8.3 Pricing a down-and-out put option	446	(8)
8.3.1 Crude Monte Carlo	446	(1)
8.3.2 Conditional Monte Carlo	447	(3)
8.3.3 Importance sampling	450	(4)
8.4 Pricing an arithmetic average Asian option	454	(14)
8.4.1 Control variates	455	(3)
8.4.2 Using Halton sequences	458	(10)
8.5 Estimating Greeks by Monte Carlo sampling	468	(4)
For further reading	472	(1)
References	473	(2)
9 Option Pricing by Finite Difference Methods	475
9.1 Applying finite difference methods to the Black-Scholes equation	475	(3)
9.2 Pricing a vanilla European option by an explicit method	478	(4)
9.2.1 Financial interpretation of the instability of the explicit method	481	(1)
9.3 Pricing a vanilla European option by a fully implicit method	482	(3)
9.4 Pricing a barrier option by the Crank-Nicolson method	485	(1)
9.5 Dealing with American options	486	(5)
For further reading	491	(1)
References	491	(4)
Part IV Advanced Optimization Models and Methods
10 Dynamic Programming	495	(108)
10.1 The shortest path problem	496	(4)
10.2 Sequential decision processes	500	(4)
10.2.1 The optimality principle and solving the functional equation	501	(3)
10.3 Solving stochastic decision problems by dynamic programming	504	(7)
10.4 American option pricing by Monte Carlo simulation	511	(10)
10.4.1 A MATLAB implementation of the least squares approach	517	(2)
10.4.2 Some remarks and alternative approaches	519	(2)
For further reading	521	(1)
References	522	(3)
11 Linear Stochastic Programming Models with Recourse	525	(1)
11.1 Linear stochastic programming models	526	(4)
11.2 Multistage stochastic programming models for portfolio management	530	(16)
11.2.1 Split-variable model formulation	532	(8)
11.2.2 Compact model formulation	540	(4)
11.2.3 Asset and liability management with transaction costs	544	(2)
11.3 Scenario generation for multistage stochastic programming	546	(9)
11.3.1 Sampling for scenario tree generation	547	(3)
11.3.2 Arbitrage free scenario generation	550	(5)
11.4 L-shaped method for two-stage linear stochastic programming	555	(3)
11.5 A comparison with dynamic programming	558	(1)
For further reading	559	(1)
References	560	(3)
12 Non-Convex Optimization	563	(1)
12.1 Mixed-integer programming models	564	(12)
12.1.1 Modeling with logical variables	565	(6)
12.1.2 Mixed-integer portfolio optimization models	571	(5)
12.2 Fixed-mix model based on global optimization	576	(2)
12.3 Branch and bound methods for non-convex optimization	578	(13)
12.3.1 LP-based branch and bound for MILP models	584	(7)
12.4 Heuristic methods for non-convex optimization	591	(6)
For further reading	597	(1)
References	598	(5)
Part V Appendices
Appendix A Introduction to MATLAB Programming	603	(54)
A.1 MATLAB environment	603	(11)
A.2 MATLAB graphics	614	(2)
A.3 MATLAB programming	616	(7)
Appendix B Refresher on Probability Theory and Statistics	623	(1)
B.1 Sample space, events, and probability	623	(2)
B.2 Random variables, expectation, and variance	625	(7)
B.2.1 Common continuous random variables	628	(4)
B.3 Jointly distributed random variables	632	(1)
B.4 Independence, covariance, and conditional expectation	633	(4)
B.5 Parameter estimation	637	(5)
B.6 Linear regression	642	(3)
For further reading	645	(1)
References	645	(2)
Appendix C Introduction to AMPL	647	(1)
C.1 Running optimization models in AMPL	648	(1)
C.2 Mean variance efficient portfolios in AMPL	649	(3)
C.3 The knapsack model in AMPL	652	(3)
C.4 Cash flow matching	655	(1)
For further reading	655	(1)
References	656	(1)
Index	657