MATHEMATICAL METHODS FOR FINANCE
Instructional goals
Learn the basic techniques for the study of linear algebra, linear and non-linear dynamic systems, and optimisation as essential tools for understanding and developing mathematical models in economics and finance.
Intended learning outcomes
The course aims to provide students with fundamental knowledge and techniques of mathematics applied to economics and finance, with particular reference to linear algebra, dynamic systems, differential calculus for functions of several real variables and optimisation. A further objective is to prepare the students to understand basic mathematical problems and the application of analytical techniques to economics and finance disciplines. Students shall acquire knowledge, skills and competences in order to be able to transfer theoretical information and operational skills to the context of economics and finance. They will have to develop a critical capacity in identifying the appropriate solution to the proposed problems. To this end, examples and case studies will be analysed and students will be encouraged to discuss them. Autonomy of judgement will be tested by means of practical and theoretical tests aimed at assessing the students ability to autonomously and originally elaborate on the issues of mathematical methods applied to economics and finance. Students will be stimulated to develop communication skills through the organisation and preparation of an open-ended theoretical test. They shall acquire not only knowledge to pass the examination, but above all, adequate stimuli and learning methods for the continuous updating and improvement of their skills in the field of mathematics applied to economics and finance.
Course Contents
Part 1: Static optimisation and applications. Part 2: Linear algebra and dynamic systems.
Reference Books
Title: Mathematics for Economics and the Social Sciences, (Volumes 1 and 2). Authors: Carl Simon and Lawrence Blume (transl. it edited by Alberto Zaffaroni) Publisher: Univ. Bocconi. Teaching materials edited by the lecturer will be made regularly available on the course Learn page.
Teaching Methods
Lectures and tutorials in the classroom. Applications in Excel or MatLab environment.
Assessment Method
Written and oral examination. Intermediate tests and/or quizzes during lectures.
Thesis assignment criteria
Interviews with the teacher.
Week 1
Real vector spaces, matrices and linear operators. Invertible matrices and change of basis.
Week 2
Eigenvalues and eigenvectors. Diagonisable matrices.
Week 3
Non-diagonisable matrices. Spectral decomposition and Jordan's canonical form for square matrices.
Week 4
Introduction to dynamical systems.
Week 5
DE/ODE Linear. Markov matrices. Associated DE and ODE. Applications.
Week 6
Non-linear DE/ODE.
Week 7
Symmetrical matrices and quadratic forms. Basics of functions of several variables.
Week 8
Differential calculus for functions of several variables.
Week 9
Implicit function theorem and applications.
Week 10
Unconstrained static optimisation. First- and second-order necessary and sufficient conditions.
Week 11
Constrained optimisation. First-order necessary conditions. Qualification of constraints. Concave and quasi-concave functions. Envelope theorem.
Week 12
Applications of constrained optimisation. The portfolio selection problem.