MATHEMATICAL METHODS FOR FINANCE

Description

Objectives: 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. Assessment: Written and oral examination. Intermediate tests and/or quizzes during lectures. Teaching Methods: Lectures and tutorials in the classroom. Applications in Excel or MatLab environment. Prerequisites: Content learned in the Mathematics courses taught in the Bachelor's degree. Contents: Part 1: Static optimisation and applications. Part 2: Linear algebra and dynamic systems. Reference Texts: 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. Thesis assignment criteria: Interviews with the teacher. Extended Program And Reference Reading Material: Week 1: Real vector spaces, matrices and linear operators. Invertible matrices and change of basis. Extended Program And Reference Reading Material: Week 10: Unconstrained static optimisation. First- and second-order necessary and sufficient conditions. Extended Program And Reference Reading Material: Week 11: Constrained optimisation. First-order necessary conditions. Qualification of constraints. Concave and quasi-concave functions. Envelope theorem. Extended Program And Reference Reading Material: Week 12: Applications of constrained optimisation. The portfolio selection problem. Extended Program And Reference Reading Material: Week 2: Eigenvalues and eigenvectors. Diagonisable matrices. Extended Program And Reference Reading Material: Week 3: Non-diagonisable matrices. Spectral decomposition and Jordan's canonical form for square matrices. Extended Program And Reference Reading Material: Week 4: Introduction to dynamical systems. Extended Program And Reference Reading Material: Week 5: DE/ODE Linear. Markov matrices. Associated DE and ODE. Applications. Extended Program And Reference Reading Material: Week 6: Non-linear DE/ODE. Extended Program And Reference Reading Material: Week 7: Symmetrical matrices and quadratic forms. Basics of functions of several variables. Extended Program And Reference Reading Material: Week 8: Differential calculus for functions of several variables. Extended Program And Reference Reading Material: Week 9: Implicit function theorem and applications. 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.