Numerical Methods in Finance and Economics A MATLAB-Based Introduction

ISBN-10: 0471745030
ISBN-13: 9780471745037
Edition: 2nd 2006 (Revised)
List price: $182.00 Buy it from $65.83
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Description: 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  More...

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Book details

List price: $182.00
Edition: 2nd
Copyright year: 2006
Publisher: John Wiley & Sons, Incorporated
Publication date: 10/6/2006
Binding: Hardcover
Pages: 696
Size: 6.50" wide x 9.50" long x 1.75" tall
Weight: 3.432
Language: English

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(R)-Based Introduction, Second Edition bridges the gap between financial theory and computational practice while showing readers how to utilize MATLAB(R)--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(R), 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(c) in order to better illustrate the optimization models in Chapters 11 and 12 New chapter on binomial and trinomial lattices Additional treatment of partialdifferential 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(R)-Based Introduction, Second Edition presents basic treatments and more specialized literature, and it also uses algebraic languages, such as AMPL(c), 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.

Preface to the Second Edition
From the Preface to the First Edition
Background
Motivation
Need for numerical methods
Need for numerical computing environments: why MATLAB?
Need for theory
For further reading
References
Financial Theory
Modeling uncertainty
Basic financial assets and related issues
Bonds
Stocks
Derivatives
Asset pricing, portfolio optimization, and risk management
Fixed-income securities: analysis and portfolio immunization
Basic theory of interest rates: compounding and present value
Basic pricing of fixed-income securities
Interest rate sensitivity and bond portfolio immunization
MATLAB functions to deal with fixed-income securities
Critique
Stock portfolio optimization
Utility theory
Mean-variance portfolio optimization
MATLAB functions to deal with mean-variance portfolio optimization
Critical remarks
Alternative risk measures: Value at Risk and quantile-based measures
Modeling the dynamics of asset prices
From discrete to continuous time
Standard Wiener process
Stochastic integrals and stochastic differential equations
Ito's lemma
Generalizations
Derivatives pricing
Simple binomial model for option pricing
Black-Scholes model
Risk-neutral expectation and Feynman-Kac formula
Black-Scholes model in MATLAB
A few remarks on Black-Scholes formula
Pricing American options
Introduction to exotic and path-dependent options
Barrier options
Asian options
Lookback options
An outlook on interest-rate derivatives
Modeling interest-rate dynamics
Incomplete markets and the market price of risk
For further reading
References
Numerical Methods
Basics of Numerical Analysis
Nature of numerical computation
Number representation, rounding, and truncation
Error propagation, conditioning, and instability
Order of convergence and computational complexity
Solving systems of linear equations
Vector and matrix norms
Condition number for a matrix
Direct methods for solving systems of linear equations
Tridiagonal matrices
Iterative methods for solving systems of linear equations
Function approximation and interpolation
Ad hoc approximation
Elementary polynomial interpolation
Interpolation by cubic splines
Theory of function approximation by least squares
Solving non-linear equations
Bisection method
Newton's method
Optimization-based solution of non-linear equations
Putting two things together: solving a functional equation by a collocation method
Homotopy continuation methods
For further reading
References
Numerical Integration: Deterministic and Monte Carlo Methods
Deterministic quadrature
Classical interpolatory formulas
Gaussian quadrature
Extensions and product rules
Numerical integration in MATLAB
Monte Carlo integration
Generating pseudorandom variates
Generating pseudorandom numbers
Inverse transform method
Acceptance-rejection method
Generating normal variates by the polar approach
Setting the number of replications
Variance reduction techniques
Antithetic sampling
Common random numbers
Control variates
Variance reduction by conditioning
Stratified sampling
Importance sampling
Quasi-Monte Carlo simulation
Generating Halton low-discrepancy sequences
Generating Sobol low-discrepancy sequences
For further reading
References
Finite Difference Methods for Partial Differential Equations
Introduction and classification of PDEs
Numerical solution by finite difference methods
Bad example of a finite difference scheme
Instability in a finite difference scheme
Explicit and implicit methods for the heat equation
Solving the heat equation by an explicit method
Solving the heat equation by a fully implicit method
Solving the heat equation by the Crank-Nicolson method
Solving the bidimensional heat equation
Convergence, consistency, and stability
For further reading
References
Convex Optimization
Classification of optimization problems
Finite- vs. infinite-dimensional problems
Unconstrained vs. constrained problems
Convex vs. non-convex problems
Linear vs. non-linear problems
Continuous vs. discrete problems
Deterministic vs. stochastic problems
Numerical methods for unconstrained optimization
Steepest descent method
The subgradient method
Newton and the trust region methods
No-derivatives algorithms: quasi-Newton method and simplex search
Unconstrained optimization in MATLAB
Methods for constrained optimization
Penalty function approach
Kuhn-Tucker conditions
Duality theory
Kelley's cutting plane algorithm
Active set method
Linear programming
Geometric and algebraic features of linear programming
Simplex method
Duality in linear programming
Interior point methods
Constrained optimization in MATLAB
Linear programming in MATLAB
A trivial LP model for bond portfolio management
Using quadratic programming to trace efficient portfolio frontier
Non-linear programming in MATLAB
Integrating simulation and optimization
Elements of convex analysis
Convexity in optimization
Convex polyhedra and polytopes
For further reading
References
Pricing Equity Options
Option Pricing by Binomial and Trinomial Lattices
Pricing by binomial lattices
Calibrating a binomial lattice
Putting two things together: pricing a pay-later option
An improved implementation of binomial lattices
Pricing American options by binomial lattices
Pricing bidimensional options by binomial lattices
Pricing by trinomial lattices
Summary
For further reading
References
Option Pricing by Monte Carlo Methods
Path generation
Simulating geometric Brownian motion
Simulating hedging strategies
Brownian bridge
Pricing an exchange option
Pricing a down-and-out put option
Crude Monte Carlo
Conditional Monte Carlo
Importance sampling
Pricing an arithmetic average Asian option
Control variates
Using Halton sequences
Estimating Greeks by Monte Carlo sampling
For further reading
References
Option Pricing by Finite Difference Methods
Applying finite difference methods to the Black-Scholes equation
Pricing a vanilla European option by an explicit method
Financial interpretation of the instability of the explicit method
Pricing a vanilla European option by a fully implicit method
Pricing a barrier option by the Crank-Nicolson method
Dealing with American options
For further reading
References
Advanced Optimization Models and Methods
Dynamic Programming
The shortest path problem
Sequential decision processes
The optimality principle and solving the functional equation
Solving stochastic decision problems by dynamic programming
American option pricing by Monte Carlo simulation
A MATLAB implementation of the least squares approach
Some remarks and alternative approaches
For further reading
References
Linear Stochastic Programming Models with Recourse
Linear stochastic programming models
Multistage stochastic programming models for portfolio management
Split-variable model formulation
Compact model formulation
Asset and liability management with transaction costs
Scenario generation for multistage stochastic programming
Sampling for scenario tree generation
Arbitrage free scenario generation
L-shaped method for two-stage linear stochastic programming
A comparison with dynamic programming
For further reading
References
Non-Convex Optimization
Mixed-integer programming models
Modeling with logical variables
Mixed-integer portfolio optimization models
Fixed-mix model based on global optimization
Branch and bound methods for non-convex optimization
LP-based branch and bound for MILP models
Heuristic methods for non-convex optimization
For further reading
References
Appendices
Introduction to MATLAB Programming
MATLAB environment
MATLAB graphics
MATLAB programming
Refresher on Probability Theory and Statistics
Sample space, events, and probability
Random variables, expectation, and variance
Common continuous random variables
Jointly distributed random variables
Independence, covariance, and conditional expectation
Parameter estimation
Linear regression
For further reading
References
Introduction to AMPL
Running optimization models in AMPL
Mean variance efficient portfolios in AMPL
The knapsack model in AMPL
Cash flow matching
For further reading
References
Index

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