Numerical Methods in Finance A MATLAB Based Introduction 1st Edition by Paolo Brandimarte ISBN 0471461695 9780471461692

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ISBN 10: 0471461695 
ISBN 13: 9780471461692
Author: Paolo Brandimarte

Balanced coverage of the methodology and theory of numerical methods in finance

Numerical Methods in Finance bridges the gap between financial theory and computational practice while helping students and practitioners exploit MATLAB for financial applications.

Paolo Brandimarte covers the basics of finance and numerical analysis and provides background material that suits the needs of students from both financial engineering and economics perspectives. Classical numerical analysis methods; optimization, including less familiar topics such as stochastic and integer programming; simulation, including low discrepancy sequences; and partial differential equations are covered in detail. Extensive illustrative examples of the application of all of these methodologies are also provided.

The text is primarily focused on MATLAB-based application, but also includes descriptions of other readily available toolboxes that are relevant to finance. Helpful appendices on the basics of MATLAB and probability theory round out this balanced coverage. Accessible for students-yet still a useful reference for practitioners-Numerical Methods in Finance offers an expert introduction to powerful tools in finance.

Numerical Methods in Finance A MATLAB Based Introduction 1st Table of contents:

Part I: Mathematical and Computational Foundations

• Chapter 1: Introduction to Financial Modeling and Numerical Methods
  • What are numerical methods in finance?
  • Why are they necessary? (e.g., analytical solutions are rare)
  • Overview of key financial problems requiring numerical solutions.
  • Introduction to MATLAB for numerical computing.
• Chapter 2: Review of Basic Probability and Statistics for Finance
  • Random variables and probability distributions (Normal, Lognormal, Binomial)
  • Stochastic processes (brief introduction)
  • Descriptive statistics, moments
  • Central Limit Theorem
• Chapter 3: Foundations of MATLAB for Numerical Finance
  • MATLAB environment and syntax basics
  • Vectors, matrices, and array operations
  • Functions, scripts, and programming constructs (loops, conditionals)
  • Plotting and data visualization
  • Introduction to toolboxes relevant for finance (e.g., Financial Toolbox)
• Chapter 4: Linear Algebra and Matrix Computations
  • Solving systems of linear equations
  • Eigenvalue problems
  • Matrix decompositions (LU, Cholesky, QR)
  • Applications in portfolio optimization and risk management.

Part II: Core Numerical Methods in Finance

• Chapter 5: Root Finding and Optimization
  • Bisection method, Newton-Raphson, Secant method
  • Numerical optimization (unconstrained, constrained)
  • Applications: Implied volatility, yield to maturity, optimal portfolio weights.
• Chapter 6: Interpolation and Approximation
  • Polynomial interpolation (Lagrange, Newton)
  • Spline interpolation
  • Approximation of functions
  • Applications: Interpolating yield curves, option pricing surfaces.
• Chapter 7: Numerical Integration
  • Trapezoidal rule, Simpson’s rule
  • Gaussian quadrature
  • Monte Carlo integration (introduction)
  • Applications: Pricing path-dependent options.
• Chapter 8: Numerical Differentiation
  • Finite difference approximations
  • Forward, backward, and central differences
  • Applications: Computing option Greeks (delta, gamma, vega).

Part III: Stochastic Processes and Simulation in Finance

• Chapter 9: Introduction to Stochastic Calculus for Finance
  • Brief review of Itô’s Lemma
  • Stochastic Differential Equations (SDEs)
  • Geometric Brownian Motion
• Chapter 10: Monte Carlo Simulation
  • Generating random numbers (uniform, normal)
  • Simulating paths of stochastic processes (e.g., stock prices)
  • Pricing European options via Monte Carlo
  • Variance reduction techniques (antithetic variates, control variates)
  • Applications: Pricing complex derivatives, risk management (VaR).
• Chapter 11: Simulating Discontinuous Processes (Jump-Diffusion)
  • Modeling jumps in asset prices
  • Simulation of Poisson processes and jump components
  • Pricing options under jump-diffusion models.

Part IV: Numerical Solutions for Derivatives Pricing

• Chapter 12: Finite Difference Methods for PDEs in Finance
  • Deriving the Black-Scholes PDE
  • Explicit, Implicit, and Crank-Nicolson schemes
  • Boundary conditions and stability
  • Applications: Pricing European and American options.
• Chapter 13: Binomial and Trinomial Trees
  • Cox-Ross-Rubinstein (CRR) binomial model
  • Pricing American options on binomial trees
  • Extension to trinomial trees
  • Relationship to finite difference methods.
• Chapter 14: Lattice Methods for Path-Dependent Options
  • Pricing options with early exercise features (e.g., American options)
  • Asian options and barrier options on lattices.

Part V: Advanced Topics and Applications (Selection based on typical coverage)

• Chapter 15: Introduction to Interest Rate Modeling
  • Vasicek, CIR, and Hull-White models
  • Lattice methods for interest rate derivatives.
• Chapter 16: Numerical Methods in Portfolio Optimization
  • Mean-Variance Optimization revisited
  • Non-linear portfolio optimization
  • Applications of quadratic programming.
• Chapter 17: Numerical Methods in Risk Management
  • Value at Risk (VaR) and Conditional VaR (CVaR)
  • Monte Carlo for VaR calculation
  • Stress testing and scenario analysis.
• Chapter 18: Calibration of Models
  • Fitting model parameters to market data
  • Least squares, maximum likelihood estimation
  • Numerical challenges in calibration.

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Tags: Paolo Brandimarte, Methods, MATLAB