# Unit MATHEMATICAL METHODS FOR RISK MANAGEMENT

Course
Finance and quantitative methods for economics
Study-unit Code
A000199
Curriculum
Data science for finance and insurance
Teacher
Marco Nicolosi
Teachers
• Marco Nicolosi
• Marco Nicolosi
Hours
• 35 ore - Marco Nicolosi
• 28 ore - Marco Nicolosi
CFU
9
Course Regulation
Coorte 2022
Offered
2022/23
Learning activities
Caratterizzante
Area
Matematico, statistico, informatico
SECS-S/06
Type of study-unit
Obbligatorio (Required)
Type of learning activities
Attività formativa monodisciplinare
Language of instruction
English
Contents
1) Continuous and discrete stochastic variables. Probability distributions: density function and cumulative probabilities, moments and quantiles of a distribution. Conditional expectations.
2)Functions of two variables: the graph, the Taylor's formula, free and constrained optimizations.
3) Linear algebra: linear systems, matrix diagonalization. Principal components analysis.
4) Financial Applications.
Reference texts
1) Probability Theory:
“A first course in Probability”, S. Ross
2) Functions of 2 variables:
“Calculus II, Lecture Notes”, R. Tavakol;
“Essential Mathematics for Economic Analysis”, K.Sydsaeter, P. Hammond, A. Strom
3) Linear algebra
“Eigenvalues and Eigenvectors”, P. Dawkins;
“Linear algebra for economists”, F.Aleskerov, H. Ersel, D.Piontkovski
Educational objectives
The main objective of the course is to provide the students with the analytical instruments that are necessary for the management of financial instruments, corporate finance and asset pricing.
All the arguments are first introduced theoretically and then some related case studies are implemented in python.
Prerequisites
In order to be able to understand and apply the majority of the techniques described within the course, you must have successfully passed the following exams:
- matematica generale
- matematica finanziaria
- teoria matematica del portafoglio
- statistica
Teaching methods
face-to-face and practical training
Other information
For further details contact the professor to the email address: marco.nicolosi@unipg.it
Learning verification modality
Written test. The exam consists in solving (either analytically or numerically with Python) some exercises on the main topics of the course. The written exam has the aim to test the competence acquired during the class.
Extended program
1) Probability:
Continuous and discrete stochastic variables. The binomial distribution and the CRR model.
The partition function and the moments of a distribution. Conditional mean. Quantiles. Some distributions: uniform, Pareto, exponential, normal and lognormal. The moment generating function.
2) A short introduction to the study of two variable functions:
The graph. Parallel and vertical sections. Partial derivatives. The gradient vector and the hessian matrix. The Taylor's formula. Stationary points. Free and constrained optimizations. Lagrange multipliers. Quadratic forms.
3) Linear algebra:
Solution of linear system. Eigenvalues and eigenvectors. Diagonalization of a matrix. Spectral theorem for symmetric matrices. Principal components analysis (PCA).

4)Applications in finance:
Factorial models. PCA of the term structure of interest rates. Portfolio optimization: Markowitz model, Black-Litterman model. Bootstrap of the term structure of interest rates. Monte Carlo method.
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