Distance Learning - Mathematics for Quantitative and Computational Finance - Analysis, Algebra and Numerical Methods - (code DL-MQCF)

The goal of this distance learning course is to introduce and elaborate the mathematical concepts, methods and algorithms that lay the foundations for quantitative finance applications. The course contents are applicable to a wide range of real-world domains, including derivatives pricing and risk management applications. It has been created in such a way that there is a clear path from problem formulation to its algorithmic specification and integration with financial models. Finally, we provide source code and you can test your models in C++, C# and Excel, thus reinforcing the learning experience.

The level of this course is similar to what a second year mathematics course would offer.

You can decide how long you need to complete the course but typically this should take between one and two years.

Click here for a video overview of this course.

What do you learn?

• Understand and apply mathematics
• Formulate and solve mathematical problems
• Apply your mathematical knowledge to pricing applications
• Developing algorithmic, numeric (and computational) skills

In short, the goal of this course is to develop the mathematical skills to help you in your work in quantitative and computational finance. The topics included are at a level that you see in the standard and accepted textbooks.

About your trainer

Daniel J. Duffy has BA (Mod), MSc and PhD degrees, all of which in mathematics and numerical analysis. He has been working with numerical methods on finance, industry and engineering since 1979. He has written four books on numerical methods and C++ for quantitative finance and he has developed a number of new schemes for this field as well as more than 20 years of training experience.

Course Contents

1. Analysis

The goal of this part of the course is to introduce the theory of real-valued and complex-valued functions. We also discuss a number of topics related to vector and vector-valued functions. Important topics are defining essential function properties (continuity, series and sequence of functions), differential and integral calculus and some of their applications. The skills in this part will be needed later when we manipulate functions to satisfy various requirements, for example numerical integration of functions.

1.1 Fundamental Analysis

Sets and Relations

• Sets and set operations
• Relations and functions
• Injective, surjective and bijective relations
• Countable and uncountable sets

Numerical Sequences and Series

• Convergent and divergent sequences
• Cauchy sequences
• Series: root and ratio tests
• Power series
• Addition and summation of series

Continuity

• Limit of a function
• Continuous and discontinuous functions
• Monotonic functions
• Uniform continuity
• Lipschitz and Holder continuity
• Some theorems

Limits and Functions

• Limits as x goes to (minus) infinity
• L'Hôpital's rule
• Sequences and series of functions
• Fourier series

1.2 Differentiation and Integration

Differentiation

• The derivative of a real-valued function 
• Continuity of derivatives 
• Higher-order derivatives 
• Fundamental theorems on differentiation

Theorems

• Taylor's theorem 
• Mean value theorems
• Rolle's theorem
• The exponential and logarithmic functions

The Riemann Integral

• Definition of a Riemann integral
• Upper and lower sums
• Special types of Riemann integrable functions
• Theorems on sequences and series
• Improper integrals

The Riemann-Stieltjes Integral  

• Definition and existence 
• Integration and differentiation 
• Functions of bounded variation 
• Mean value theorem

1.3 Advanced Analysis

Functions of Several Variables

• Limits and continuity
• Derivatives
• Composite functions
• Partial differentiation

Further Results in Several Variables

• Taylor's theorem in several variables 
• Implicit function theorem
• Inverse function theorem
• Jacobians

The Lebesgue Theory 

• Measure theory
• Lebesgue integral
• Theorems
• Applications of the Lebesgue integral

1.4 An Introduction to Complex Analysis

Basic Theory

• The complex number system 
• Fundamental operations
• Complex conjugates
• Polynomial equations and complex numbers
• Trigonometric and hyperbolic trigonometric functions

Complex-valued Functions

• Single and multiple-valued functions
• Branch points and branch lines
• Continuity and limits
• Infinity

Complex Differentiation

• Analytic functions
• Cauchy-Riemann equations
• Harmonic functions
• Derivatives of elementary functions
• Curves

Complex Integration

• Complex line integrals
• Change of variables
• Simple and multiply-connected regions
• Cauchy's theorem
• Applications of complex integration

Integral Transforms 

• Fourier transform
• Inverse Fourier transform
• Laplace transform
• Inverse Laplace transform
• Discrete Fourier and Laplace transforms

2. Discrete Mathematics

In this part of the course we discuss a number of concepts and methods that are concerned with variables, functions and transformations defined on finite or infinite discrete sets. In particular, linear algebra will be important because of its role in numerical analysis in general and quantitative finance in particular. We also introduce probability theory and statistics as well as a number of sections on numerical analysis. The latter group is of particular importance when we approximate differential equations using the finite difference method and when we use the Monte Carlo method, for example.

2.1 Linear Algebra

Linear Spaces 

• Linear dependence
• Bases, components and dimension
• Subspaces
• Mappings between linear spaces
• Hyperplanes

Basic Matrix Properties 

• Operations on matrices and vectors 
• Determinants 
• Eigenvectors and eigenvalues 
• Positive-definite matrices 
• Patterned matrices (tridiagonal, banded, sparse, …)

Systems of Linear Equations 

• Rank of a matrix 
• Simultaneous linear equations 
• General solution of a linear system 
• Gram-Schmidt orthogonalisation

2.2 Fundamental Probability and Statistics

Random Variables 

• What is a random variable? 
• Distribution functions 
• Discrete and continuous random variables 
• Multiple random variables 
• Functions of random variables

Measures of Central Tendency and of Dispersion 

• Averages and summation 
• Arithmetic, geometric and harmonic means 
• Root mean squares 
• Quartiles, percentiles and deciles

Other Measures 

• Standard deviation and variance 
• Measures of dispersion 
• Moments of a random variable 
• Kurtosis and skewness

Distributions 

• Uniform 
• Normal and lognormal 
• Poisson and exponential 
• Gamma and beta 
• Chi-squared and noncentral chi-squared 
• Student's t-distribution 
• Weibull

Time Series

• Time series models
• Forecasting
• Fitting time series models
• Kalman filter

2.3 Linear and Nonlinear Data Structures

An Introduction to Graph Theory 

• Graphs and multigraphs 
• Undirected and directed graphs 
• Subgraphs 
• Graph algorithms

Binary Trees 

• Complete and extended binary trees 
• Binary search trees 
• Traversing binary trees 
• Priority queues and heaps

Other Algebraic Structures 

• Lists and stacks 
• Hash tables and maps 
• Polynomials 
• Rational functions

3. Numerical Methods

The goal of this part of the course is to develop robust, efficient and accurate numerical schemes that allow us to produce algorithms in applications. These methods lie at the heart of computational finance and a good understanding of how to use them is vital if you wish to create applications. In general, the methods approximate equations and models defined in a continuous, infinite-dimensional space by models that are defined on a finite-dimensional space.

3.1 Numerical Linear Algebra

Direct Methods for solving linear Equations 

• Triangular systems 
• Gaussian elimination 
• LU (and Cholesky) decomposition
• Positive definite matrices

Iterative Methods 

• Fixed point and contraction mapping theorems 
• Jacobi, Gauss-Seidel and SOR methods 
• PSOR method 
• Conjugate gradient method 
• Iterative methods for special matrices

Matrix Eigenvalue and Eigenvector Calculations 

• Gerschgorin's theorem 
• The Power method 
• Similarity Transformations 
• The Q-R method

Overdetermined Linear Systems 

• Normal equations 
• Orthogonalisation methods 
• Least squares problems with linear constraints

3.2 Solving Nonlinear Equations

Background 

• Vector functions 
• Fixed point theorems 
• Jacobian matrix/gradient vector 
• Convergence and error estimation

First Methods 

• Bisection 
• Secant / Steffensen 
• Regula Falsi 
• Newton-Raphson

Advanced Methods 

• Modified Newton-Raphson 
• Continuation (homotopy) methods 
• Error estimation 
• Ill-conditioned systems

Optimisation

• Constrained and unconstrained optimization
• Deterministic and stochastic optimisation
• Some popular optimisation algorithms
• Applications to finance

3.3 Numerical Integration, Interpolation and Differentiation

Numerical Integration in one Dimension 

• Basic rules (rectangle, Romberg, trapezoidal) 
• Truncation error 
• Newton-Cotes formulae 
• Improper integrals

Advanced Numerical Integration 

• Gaussian quadrature and orthogonal polynomials 
• Multidimensional integrals 
• Gauss-Kronod method

Interpolation and Extrapolation 

• Linear interpolation 
• Cubic spline interpolation 
• Hermite interpolation 
• Inverse interpolation 
• Extrapolation using polynomials and rational functions

Numerical Differentiation

• Approximation of derivatives
• Accuracy and stability
• Applications

3.4 Introduction to the Finite Difference Method (FDM)

Stochastic Differential Equations (SDE)

• The Ito integral
• Modelling SDEs
• SDEs in finance
• Numerical solution of SDEs
• Fundamentals of the Monte Carlo Method

Overview of Equation Categories 

• Initial value problems
• Two-point boundary value problems
• Volterra and Fredholm integral equations
• First-order partial differential equations
• Second-order partial differential equations

An Introduction to the Finite Difference Method

• Approximation of continuous quantities 
• Consistency, stability and convergence 
• Conditional and unconditional stability 
• A taxonomy of finite difference schemes

Mini Projects for Finite Difference 

• Initial value problem 
• Two-point boundary value problem 
• Heat equation in one dimension 
• Laplace equation in two dimensions

• STL
• boost
• Using C++ in finance

• uBlas (matrices, vectors and numerical linear algebra)
• Statistical distributions (discrete and continuous)
• Special functions (orthogonal polynomials, Bessel, elliptic etc.)
• Interval arithmetic
• Random numbers

• Ode/Pde solvers
• Numerical integration and interpolation
• Complex analysis package

Prerequisites

Who should attend?

Course Form

• Schaum's Outline of Discrete Mathematics, 3rd Ed. (Schaum's Outline Series), by Seymour Lipschutz;
• Numerical Methods, by Germund Dahlquist
• A Primer for the Mathematics of Financial Engineering, by Dan Stefanica
• Solutions Manual - A Primer For The Mathematics Of Financial Engineering, by Dan Stefanica
• Calculus: One-variable Calculus, with an Introduction to Linear Algebra v. 1, by TM Apostol
• Finite Difference Methods in Financial Engineering: A Partial Differential Equation Approach, by Daniel J. Duffy

Duration, price, date, locations and registration