# Mathematical Methods For Finance MATHS5016

**Academic Session:**2019-20**School:**School of Mathematics and Statistics**Credits:**20**Level:**Level 5 (SCQF level 11)**Typically Offered:**Semester 1**Available to Visiting Students:**No

#### Short Description

The aim of this course is to cover the specialised methods and concepts of Multivariable Calculus and Linear Algebra that are required for the application of Mathematics and Statistics to the theory and practice of Finance.

#### Timetable

Weekly tutorial on Mondays. Lectures on Tuesdays, Wednesdays, Thursdays and Fridays.

#### Requirements of Entry

First class or upper second class first degree in a discipline that included the equivalent of at least two years of university mathematics (Scotland) and at least one year of university mathematics (rest of the UK) or equivalent (other countries).

#### Excluded Courses

None

#### Assessment

Two class tests each contributing 10% of assessment and one degree examination of duration 2 hour 30 mins and contributing 80% of assessment.

**Main Assessment In:** April/May

**Are reassessment opportunities available for all summative assessments?** No

Reassessments are normally available for all courses, except those which contribute to the Honours classification. For non Honours courses, students are offered reassessment in all or any of the components of assessment if the satisfactory (threshold) grade for the overall course is not achieved at the first attempt. This is normally grade D3 for undergraduate students and grade C3 for postgraduate students. Exceptionally it may not be possible to offer reassessment of some coursework items, in which case the mark achieved at the first attempt will be counted towards the final course grade. Any such exceptions for this course are described below.

#### Course Aims

This aim of this course is to cover the specialised methods and concepts of multivariable Calculus and Linear Algebra that are required for the application of Mathematics and Statistics to the theory and practice of Finance.

#### Intended Learning Outcomes of Course

By the end of this programme students will be able to:

■ differentiate functions of many variables including change of variable using the chain rule for partial differentiation;

■ draw three dimensional surfaces using cross-sections and contours;

■ solve simple PDEs by using a given change of variable;

■ perform double and triple integration: calculate the volume under a surface using double integration; obtain limits for a given domain; change the order of integration; change to polar coordinates and calculate the area element; understand the interpretation of the triple integral; use spherical polar coordinates; use general change of variables and the Jacobian;

■ find stationary points for functions of several variables and classify them using first principles and the Hessian criterion;

■ use the method of Lagrange multipliers to solve practical extreme value problems;

■ use the properties of Beta and Gamma functions to evaluate certain integrals;

■ find Fourier series for elementary functions defined on finite intervals;

■ handle fluently problems involving matrices and their entries;

■ recognise vector spaces and subspaces over R and C:

■ test sets of vectors for linear independence and spanning properties, and understand methods for obtaining bases for a specified subspace of a vector space;

■ decide whether or not a map between spaces is linear, describe a linear map in matrix form, and calculate various objects (eg image, kernel) associated with a linear map;

■ evaluate determinants recursively and using elementary row and column operations, factorise algebraic determinants, and apply results about determinants in theoretical problems;

■ find the characteristic polynomial of a square matrix, and use it to determine the eigenvalues and eigenvectors of the matrix, and deal with theoretical problems involving eigenvalues;

■ use diagonalisation of a matrix to solve systems of Ordinary Differential Equations and difference equations;

■ recognise quadratic forms, understand the connection with symmetric matrices and determine their rank and signature; diagonalise quadratic forms;

■ understand the definitions and basic properties of Real and Hermitian inner products, understand the idea of orthogonality and be able to implement the Gram-Schmidt orthogonalisation procedure;

■ establish the basic properties of the eigenvalues and eigenvectors of symmetric, orthogonal, unitary and similar matrices.

Students should be familiar with all definitions and results covered in lectures, should learn and be able to apply formulae used in the course, should understand the proofs of results, should learn to be rigorously logical in their presentation of solutions to problems and should be able to solve problems similar to those given in lectures.

#### Minimum Requirement for Award of Credits

Attendance at the degree (or resit) exam and at least 70% tutorial attendance.