Numerical Mathematics for Finance; Recipes, Applications and C++ code - (code NMF)

This two-day course is meant as a good introduction to, and overview of a number of important mathematical and numerical techniques that are used in many areas of Quantitative Finance. These numerical techniques underpin many of the important pricing and hedging algorithms in finance and for this reason it is important that they are well-understood. To consolidate the theory we provide working C++ code to show how algorithms are mapped to software.

Overview of Course

The goal of this course is to prepare you for algorithmic work in derivatives pricing and hedging. Many of the numerical techniques - for example Monte Carlo, Finite Difference Method and quadrature techniques - rely on more fundamental numerical methods and it is here that these are discussed in detail:

• Solving linear systems of equations using direct and iterative methods
• Eigenvalues, eigenvectors and Cholesky decomposition
• Solving non-linear systems of equations
• Approximation (interpolation, extrapolation)
• Numerical integration in one and several dimensions
• Continuous and discrete Fourier analysis
• Linear and non-linear optimisation
• Statistics and modeling of data

We discuss each topic in enough detail so that you can apply it in your daily work.

What do you receive?

In this course you receive the course slides, CD with full source code. This means that you can practice at your own location and in your own time after the course. You can also participate on the author's forum where we have support for this course.

Your trainer is Dr. Daniel J. Duffy .

Please note: If you wish to run the C++ programs related to the course, you need to bring your own laptop computer with a C++ compiler (ideally, Microsoft's Visual Studio). However, it is not mandatory.

Course Contents

Part I: Algebraic Systems

Linear Algebra

• Review of matrices and vectors
• Matrix multiplication
• Some properties of matrices

Categories of matrices Solution of Linear Systems, Part I

• Gaussian elimination
• LU decomposition
• Tridiagonal systems
• Cholesky decomposition

Solution of Linear Systems, Part II

• What are iterative methods?
• Jacobi, Gauss-Seidel and SOR methods
• Accelerating convergence
• Sparse linear systems

Nonlinear Systems

• One-dimensional and multi-dimensional problems
• Secant and Regula Falsi methods
• Newton-Raphson method
• Nonlinear systems of equations
• Continuation (homotopy) methods

Part II: Approximation

Interpolation and Extrapolation

• Polynomial versus rational interpolation
• Piecewise polynomial interpolation (linear, cubic spline)
• Interpolation in two and more dimensions
• Polynomial versus rational extrapolation

Numerical Integration

• Some standard integration formulae
• Gauss-Lobatto
• Adaptive Gauss Kronrod
• Numerical integration in multiple dimensions

Fourier Analysis and Fast Fourier Transform

• Review of complex numbers
• Functions of a complex variable
• Fourier transform and its inverse
• Parseval's formula
• Fast Fourier Transform (FFT)
• Discretely sampled data
• Multiple dimensions

Part III: Optimisation

One-dimensional and multi-dimensional problems

• Brent's method
• Simplex method
• Powell's method
• Conjugate gradient

Part IV: Statistics and Modelling

Statistics

• Moments of a distribution
• Linear correlation
• Nonparametric correlation
• Comparing two distributions

Modelling of Data

• An introduction to least squares
• Nonlinear models
• Confidence intervals

Prerequisites

Who should attend?

Course Form

In this course you learn numerical analysis of each of the above topics in a number of steps:

• Presentation of the fundamental theory
• Several useful, simple examples to show how the theory works
• A more complex example (usually associated with finance)
• Using the delivered C++ code to test some examples
• Many exercises (and answers) that you can work out at home

We also discuss and show by examples in C++ a number of relevant problems:

• Option pricing using Fast Fourier Transform
• Quadrature methods for derivatives pricing
• Interpolation and curve fitting
• Generating correlated random variables
• Calibration and optimization

In short, the goal of this course is to get you up to speed as soon as possible so that you can start on real-life systems without having to reinvent the wheel.

Duration, price, date, locations and registration