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Numerical Methods and Optimization in Finance

Book Description

This book describes computational finance tools. It covers fundamental numerical analysis and computational techniques, such as option pricing, and gives special attention to simulation and optimization. Many chapters are organized as case studies around portfolio insurance and risk estimation problems.  In particular, several chapters explain optimization heuristics and how to use them for portfolio selection and in calibration of estimation and option pricing models. Such practical examples allow readers to learn the steps for solving specific problems and apply these steps to others. At the same time, the applications are relevant enough to make the book a useful reference. Matlab and R sample code is provided in the text and can be downloaded from the book's website.



  • Shows ways to build and implement tools that help test ideas
  • Focuses on the application of heuristics; standard methods receive limited attention
  • Presents as separate chapters problems from portfolio optimization, estimation of econometric models, and calibration of option pricing models

Table of Contents

  1. Cover image
  2. Title page
  3. Table of Contents
  4. Copyright
  5. List of Algorithms
  6. Acknowledgements
  7. Chapter One. Introduction
    1. Publisher Summary
    2. 1.1 About this book
    3. 1.2 Principles
    4. 1.3 On software
    5. 1.4 On approximations and accuracy
    6. 1.5 Summary: the theme of the book
  8. Part One: Fundamentals
    1. Chapter Two. Numerical Analysis in a Nutshell
      1. Publisher Summary
      2. 2.1 Computer Arithmetic
      3. 2.2 Measuring Errors
      4. 2.3 Approximating Derivatives with Finite Differences
      5. 2.4 Numerical Instability and Ill-Conditioning
      6. 2.5 Condition Number of a Matrix
      7. 2.6 A Primer on Algorithmic and Computational Complexity
      8. 2.A Operation Count for Basic Linear Algebra Operations
    2. Chapter Three. Linear Equations and Least Squares Problems
      1. Publisher Summary
      2. 3.1 Direct Methods
      3. 3.2 Iterative Methods
      4. 3.3 Sparse Linear Systems
      5. 3.4 The Least Squares Problem
    3. Chapter Four. Finite Difference Methods
      1. Publisher Summary
      2. 4.1 An example of a numerical solution
      3. 4.2 Classification of differential equations
      4. 4.3 The Black–Scholes equation
      5. 4.4 American options
      6. 4.A A note on Matlab's function <span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" class="inlinecode">spdiags</span>
    4. Chapter Five. Binomial Trees
      1. Publisher Summary
      2. 5.1 Motivation
      3. 5.2 Growing the Tree
      4. 5.3 Early Exercise
      5. 5.4 Dividends
      6. 5.5 The Greeks
  9. Part Two: Simulation
    1. Chapter Six. Generating Random Numbers
      1. Publisher Summary
      2. 6.1 Monte Carlo Methods and Sampling
      3. 6.2 Uniform Random Number Generators
      4. 6.3 Nonuniform Distributions
      5. 6.4 Specialized Methods for Selected Distributions
      6. 6.5 Sampling from a Discrete Set
      7. 6.6 Sampling Errors—and How to Reduce them
      8. 6.7 Drawing from Empirical Distributions
      9. 6.8 Controlled Experiments and Experimental Design
    2. Chapter Seven. Modeling Dependencies
      1. Publisher Summary
      2. 7.1 Transformation Methods
      3. 7.2 Markov Chains
      4. 7.3 Copula Models
    3. Chapter Eight. A Gentle Introduction to Financial Simulation
      1. Publisher Summary
      2. 8.1 Setting the Stage
      3. 8.2 Single-Period Simulations
      4. 8.3 Simple Price Processes
      5. 8.4 Processes with Memory in the Levels of Returns
      6. 8.5 Time-Varying Volatility
      7. 8.6 Adaptive expectations and patterns in Price Processes
      8. 8.7 Historical Simulation
      9. 8.8 Agent-based Models and Complexity
    4. Chapter Nine. Financial Simulation at Work: Some Case Studies
      1. Publisher Summary
      2. 9.1 Constant proportion portfolio insurance (CPPI)
      3. 9.2 VaR estimation with Extreme Value Theory
      4. 9.3 Option pricing
  10. Part Three: Optimization
    1. Chapter Ten. Optimization Problems in Finance
      1. Publisher Summary
      2. 10.1 What to optimize?
      3. 10.2 Solving the model
      4. 10.3 Evaluating solutions
      5. 10.4 Examples
      6. 10.5 Summary
    2. Chapter Eleven. Basic Methods
      1. Publisher Summary
      2. 11.1 Finding the Roots of <em xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops">f</em>((<em xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops">x</em>) = 0) = 0
      3. 11.2 Classical Unconstrained Optimization
      4. 11.3 Unconstrained Optimization in One Dimension
      5. 11.4 Unconstrained Optimization in Multiple Dimensions
      6. 11.5 Nonlinear Least Squares
      7. 11.6 Solving Systems of Nonlinear Equations <em xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops">F(x)</em> = 0 = 0
      8. 11.7 Synoptic View of Solution Methods
    3. Chapter Twelve. Heuristic Methods in a Nutshell
      1. Publisher Summary
      2. 12.1 Heuristics
      3. 12.2 Trajectory Methods
      4. 12.3 Population-Based Methods
      5. 12.4 Hybrids
      6. 12.5 Constraints
      7. 12.6 The Stochastics of Heuristic Search
      8. 12.7 General Considerations
      9. 12.8 Summary
      10. 12.A Implementing Heuristic Methods with Matlab
    4. Chapter Thirteen. Portfolio Optimization
      1. Publisher Summary
      2. 13.1 The Investment Problem
      3. 13.2 The Classical Case: Mean–Variance Optimization
      4. 13.3 Heuristic Optimization of One-Period Models
      5. 13.A More Implementation Issues in R
    5. Chapter Fourteen. Econometric Models
      1. Publisher Summary
      2. 14.1 Term Structure Models
      3. 14.2 Robust and Resistant Regression
      4. 14.A Maximizing the Sharpe Ratio
    6. Chapter Fifteen. Calibrating Option Pricing Models
      1. Publisher Summary
      2. 15.1 Implied volatility with Black–Scholes
      3. 15.2 Pricing with the characteristic function
      4. 15.3 Calibration
      5. 15.4 Final remarks
      6. 15.A Quadrature rules for infinity
  11. Bibliography
  12. Index