Instructional goals
The student will learn key tools for understanding and developing models in finance. These tools include linear algebra, dynamical systems and optimisation techniques.
Prerequisites
Mathematical finance at an 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] Understanding and developing models for banking and finance.
[Conoscenza e capacità di comprensione applicate] See the above; mastering models and concepts, which are widely used in the industry, represent an invaluable asset to the successful student, especially in finding a job in the field.
[Autonomia di giudizio]
An independent thinking will be developed both during lectures and during ad hoc exercises assigned during the course. Particular attention is put on the hypotheses/consequences of the various models, so to stimulate a critic attitude and pave the way for future model design.
[Abilità comunicative] Students will be asked to give a techincal exposée in front of their peers.
[Capacità di apprendimento] The skills developed above, in particular critical and independent thinking, will help the student in their future career. Reflecting on the importance of each step in model developing will help them to understand and potentially develop new frameworks according to future case studies.
Course Contents
Parrt 1: Static optimisation and applications.
Part 2: Linear algebra, 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 distributed by the professor (posted on Learn)
Teaching Methods
Frontal lectures and TA sessions. Applications in Excel or Matlab.
Assessment Method
Written exam (midterm + endterm), projects during the year.
Dates: 12th October I1st project); 26th October (mid-term), 23rd November (project 2); with weights: 5%, 40% , 5%. The final exam will weigh
the residual 50%.
Thesis assignment criteria
Meetings with the professor.
Week 1 Contenuto sessioni on line e on campus
Differential calculus in several variable, part I.
Week 2 Contenuto sessioni on line e on campus
Differential calculus in several variables, part II.
Week 3 Contenuto sessioni on line e on campus
Quadratic forms and symmetric matrices. Free optimisation, n/s first and second order conditions.
Week 4 Contenuto sessioni on line e on campus
Constrained optimization. Necessary first order conditions. Concave and quasi concave function have special properties.
Week 5 Contenuto sessioni on line e on campus
Applications of constrained optimization to portfolio selection/utility maximization.
Week 6 Contenuto sessioni on line e on campus
Matrices and linear operators. Invertible matrices and basis change in R^n
Week 7 Contenuto sessioni on line e on campus
Eigenvalues and eigenvectors, diagonalizable matrices.
Week 8 Contenuto sessioni on line e on campus
Spectral decomposition and PCA, with applications to Finance (hints).
Week 9 Contenuto sessioni on line e on campus
Intro to discrete dynamical systems. The linear, homogenous case.
Week 10 Contenuto sessioni on line e on campus
Dynamical systems, the affine case. Orbits and equilibria.
Week 11 Contenuto sessioni on line e on campus
Difference equations. Hints to ODE. Euler equation.
Week 12 Contenuto sessioni on line e on campus
Examples and further insights.