Numerical Methods in Finance: A Matlab-Based Introduction

by Paolo Brandimarte (Politecnico di Torino, Torino, Italy)

• ISBN13: 9780471396864
• ISBN10: 0471396869
• eBook ISBN(s): 9780471461692
• Format: Hardcover
• Copyright: 2001-10-01
• Publisher: Wiley-Interscience
• More Book Details

Summary

This book integrates the topics of numerical methods, financial problem solving, and MATLAB programming into one balanced treatment. Its tutorial approach features MATLAB examples as a means of illustrating the concepts in practical, every day financial problems.

Author Biography

PAOLO BRANDIMARTE is Professor of Quantitative Methods for Finance and Logistics at Politecnico di Torino in Italy.

Table of Contents

Preface	xiii
Part I Background
Financial problems and numerical methods	3	(74)
MATLAB environment	4	(2)
Why MATLAB?	5	(1)
Fixed-income securities: analysis and portfolio immunization	6	(21)
Basic valuation of fixed-income securities	7	(7)
Interest rate sensitivity and bond portfolio immunization	14	(3)
MATLAB functions to deal with fixed-income securities	17	(9)
Critique	26	(1)
Portfolio optimization	27	(14)
Basics of mean-variance portfolio optimization	30	(2)
MATLAB functions to deal with mean-variance portfolio optimization	32	(7)
Critique	39	(2)
Derivatives	41	(25)
Modeling the dynamics of asset prices	46	(6)
Black-Scholes model	52	(2)
Black-Scholes model in MATLAB	54	(3)
Pricing American options by binomial lattices	57	(8)
Option pricing by Monte Carlo simulation	65	(1)
Value-at-risk	66	(11)
Stochastic differential equations and Ito's lemma	70	(2)
References	72	(5)
Part II Numerical Methods
Basics of numerical analysis	77	(46)
Nature of numerical computation	78	(8)
Working with a finite precision arithmetic	78	(4)
Number representation, rounding, and truncation	82	(2)
Error propagation and instability	84	(1)
Order of convergence and computational complexity	85	(1)
Solving systems of linear equations	86	(18)
Condition number for a matrix	87	(3)
Direct methods for solving systems of linear equations	90	(4)
Tridiagonal matrices	94	(1)
Iterative methods for solving systems of linear equations	95	(9)
Function approximation and interpolation	104	(7)
Solving nonlinear equations	111	(6)
Bisection method	112	(1)
Newton's method	113	(1)
Solving nonlinear equations in MATLAB	114	(3)
Numerical integration	117	(6)
References	121	(2)
Optimization methods	123	(84)
Classification of optimization problems	123	(18)
Finite- vs. infinite-dimensional problems	124	(4)
Unconstrained vs. constrained problems	128	(1)
Convex vs. nonconvex problems	129	(1)
Linear vs. nonlinear problems	130	(2)
Continuous vs. discrete problems	132	(4)
Deterministic vs. stochastic problems	136	(5)
Numerical methods for unconstrained optimization	141	(7)
Steepest descent method	142	(1)
The subgradient method	143	(1)
Newton and the trust region methods	143	(1)
No-derivatives algorithms: quasi-Newton method and simplex search	144	(2)
Unconstrained optimization in MATLAB	146	(2)
Methods for constrained optimization	148	(20)
Penalty function approach	149	(5)
Kuhn-Tucker conditions	154	(5)
Duality theory	159	(6)
Kelley's cutting plane algorithm	165	(1)
Active set method	166	(2)
Linear programming	168	(12)
Geometric and algebraic features of linear programming	170	(2)
Simplex method	172	(2)
Duality in linear programming	174	(2)
Interior point methods	176	(3)
Linear programming in MATLAB	179	(1)
Branch and bound methods for nonconvex optimization	180	(9)
LP-based branch and bound for MILP models	186	(3)
Heuristic methods for nonconvex optimization	189	(5)
L-shaped method for two-stage linear stochastic programming	194	(13)
Elements of convex analysis	197	(1)
Convexity in optimization	197	(4)
Convex polyhedra and polytopes	201	(3)
References	204	(3)
Principles of Monte Carlo simulation	207	(44)
Monte Carlo integration	208	(2)
Generating pseudorandom variates	210	(8)
Generating pseudorandom numbers	210	(2)
Inverse transform method	212	(1)
Acceptance-rejection method	213	(2)
Generating normal variates by the polar approach	215	(3)
Setting the number of replications	218	(2)
Variance reduction techniques	220	(14)
Antithetic variates	220	(4)
Common random numbers	224	(2)
Control variates	226	(1)
Variance reduction by conditioning	227	(1)
Stratified sampling	227	(2)
Importance sampling	229	(5)
Quasi-Monte Carlo simulation	234	(12)
Generating Halton's low-discrepancy sequences	235	(6)
Generating Sobol's low-discrepancy sequences	241	(5)
Integrating simulation and optimization	246	(5)
References	249	(2)
Finite difference methods for partial differential equations	251	(32)
Introduction and classification of PDEs	252	(4)
Numerical solution by finite difference methods	256	(9)
Bad example of a finite difference scheme	257	(2)
Instability in a finite difference scheme	259	(6)
Explicit and implicit methods for second-order PDEs	265	(10)
Solving the heat equation by an explicit method	266	(4)
Solving the heat equation by an implicit method	270	(4)
Solving the heat equation by the Crank-Nicolson method	274	(1)
Convergence, consistency, and stability	275	(8)
Classification of second-order PDEs and characteristic curves	277	(2)
References	279	(4)
Part III Applications to Finance
Optimization models for portfolio management	283	(32)
Mixed-integer programming models	285	(4)
Multistage stochastic programming models	289	(20)
Split-variable formulation	292	(5)
Compact formulation	297	(4)
Sample asset and liability management model formulation	301	(2)
Scenario generation for multistage stochastic programming	303	(6)
Fixed-mix model based on global optimization	309	(6)
References	312	(3)
Option valuation by Monte Carlo simulation	315	(32)
Simulating asset price dynamics	316	(3)
Pricing a vanilla European option by Monte Carlo simulation	319	(7)
Using antithetic variates to price a vanilla European option	321	(1)
Using antithetic variates to price a European option with truncated payoff	322	(1)
Using control variates to price a vanilla European option	323	(2)
Using Halton low-discrepancy sequences to price a vanilla European option	325	(1)
Introduction to exotic and path-dependent options	326	(6)
Barrier options	326	(4)
Asian options	330	(1)
Lookback options	331	(1)
Pricing a down-and-out put	332	(8)
Pricing an Asian option	340	(7)
References	345	(2)
Option valuation by finite difference methods	347	(50)
Applying finite difference methods to the Black-Scholes equation	347	(3)
Pricing a vanilla European option by an explicit method	350	(4)
Financial interpretation of the instability of the explicit method	352	(2)
Pricing a vanilla European option by a fully implicit method	354	(3)
Pricing a barrier option by the Crank-Nicolson method	357	(1)
Dealing with American options	358	(9)
References	364	(3)
Part IV Appendices
Appendix A Introduction to MATLAB programming	367	(12)
A.1 MATLAB environment	367	(7)
A.2 MATLAB graphics	374	(1)
A.3 MATLAB programming	375	(4)
Appendix B Refresher on probability theory	379	(18)
B.1 Sample space, events, and probability	379	(2)
B.2 Random variables, expectation, and variance	381	(2)
B.2.1 Common continuous random variables	383	(3)
B.3 Jointly distributed random variables	386	(2)
B.4 Independence, covariance, and conditional expectation	388	(3)
B.5 Parameter estimation	391	(4)
References	395	(2)
Index	397