Instructional goals
The main goal of the course is to give a detailed overview of basic mathematical subjects such as linear algebra, multivariable calculus, optimization and dynamical systems, so as to provide the students with the key tools for economic and financial modeling.
Prerequisites
Mathematical finance at undergraduate level. In particular, basics in linear algebra (operations between vectors and matrices, linear maps and their matrix representation).
Intended learning outcomes
[Conoscenza e capacità di comprensione] The course aims at giving the student the basic mathematical concepts and techniques that are essential to economic and financial modeling, with a particular focus on linear algebra, real multivariable calculus, optimization and dynamical systems. The student will be able to understand and develop mathematical models for banking and finance.
[Conoscenza e capacità di comprensione applicate] The students will understand how to face and solve basic mathematical problems and models that are widely used in the industry. They will also learn how to use their new mathematical knowledge in economic and financial settings by encoding economic data into predictive mathematical models.
[Autonomia di giudizio]
Independent thinking will be encouraged and developed both during the lectures and during the exercise sessions, by studying a detailed collection of examples and exercises and by discussing the main problem-solving strategies. Not only will the students learn how to deal with mathematical models and how to solve problems, but they will also be able to discuss and understand the meaning and the consequences of the outcomes of the models in the economic and financial settings.
[Abilità comunicative] Not only will the students prove their skills in solving problems during the exam, but they will also be asked to discuss their results and to give a clear and comprehensive exposition of their understanding of the subject.
[Capacità di apprendimento] Aside from learning the main subjects, the students will be also stongly encouraged to reflect on problem-solving strategies, so as to improve their critical and independent thinking. This will help them broaden their mathematical knowledge and their potential ability to deal with future case studies in new frameworks.
Course Contents
Parrt 1: Static optimization and applications.
Part 2: Linear algebra and dynamical systems.
Reference Books
Title: Matematica per l'Economia e le Scienze Sociali, (Volumi 1 e 2). Autori: Carl Simon e Lawrence Blume (trad. it a cura di Alberto Zaffaroni) Editore: Univ. Bocconi.
Slides and supplementary notes will be regularly posted on the Learn page by the instructor.
Teaching Methods
Frontal lectures and TA sessions. Applications in Excel or MATLAB.
Assessment Method
Tests. Final written exam.
Thesis assignment criteria
Interview with the instructor.
Week 1
Multivariable calculus - part I.
Week 2
Multivariable calculus - part II.
Week 3
Quadratic forms and symmetric matrices. Free optimization: necessary and sufficient conditions.
Week 4
Constrained optimization: first-order necessary conditions. Optimization for convex functions.
Week 5
Application of constrained optimization to the portfolio optimization problem.
A brief outline of linear programming.
Week 6
Real vector spaces, matrices and linear operators. Invertible matrices and change of basis.
Week 7
Eigenvalues and eigenvectors. Diagonalizable matrices.
Week 8
Spectral decomposition and PCA with applications to Finance (hints).
Week 9
Difference equations (DE): the linear case. Models and examples.
Week 10
Ordinary differential equations (ODE): the linear case. Euler's equation. The Cauchy problem.
Week 11
Introduction to dynamical systems. Linear dynamical systems. Orbits and equlibria.
Week 12
Linear discrete dynamical systems. Examples and applications.