Study-unit MATHEMATICAL METHODS FOR RISK MANAGEMENT

Course name Finance and quantitative methods for economics
Study-unit Code A000199
Location PERUGIA
Curriculum Statistics for finance and economics
Lecturer Marco Nicolosi
Lecturers
  • Marco Nicolosi
Hours
  • 63 ore - Marco Nicolosi
CFU 9
Course Regulation Coorte 2018
Supplied 2018/19
Learning activities Caratterizzante
Area Matematico, statistico, informatico
Sector SECS-S/06
Type of study-unit Obbligatorio (Required)
Type of learning activities Attività formativa monodisciplinare
Language of instruction English
Contents Continuous and discrete stochastic variables: the partition function, the moments and the quantiles of a distribution. Conditional expectations.

Utility theory

Linear algebra: linear systems, matrix diagonalization.

Functions of two variables: the graph, the Taylor's approximation, free and constrained optimizations.
Reference texts 1) Probability:
"A First Course in Probability", Sheldon Ross

2)The theory of the expected utility:

"Utility theory for decision making", Peter C. Fishburn

3) Two variable functions

"Calculus II, Lecture Notes" R. Tavakol (section. 2)

4) Linear algebra

"Eigenvalues and Eigenvectors" P. Dawkins
Educational objectives The main objective of the course is to provide the students with some analytical instruments that are necessary in quantitative finance

The knowledge acquired are:

- elements of probability theory

- utility theory

- elements of linear algebra

- functions of two variables

The main competences will be:

- computation of probabilities, expected values, conditioned expected values, quantiles

- to analyze and compare financial payoffs under uncertainty condition

- matrix diagonalization

- to solve free and constrained optimization problems
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 matematica generale exam

- the matematica finanziaria exam

- the teoria matematica del portafoglio exam

- statistica exam
Teaching methods face-to-face and practical training
Learning verification modality The exam consists of a written test and an oral test.

The written exam consists in solving 3 or 4 exercises. The written exam has the aim to test the competences acquired during the class.

The oral exam consists in a discussion of the written exam and of theoretical questions on the whole program. The oral exam has the aim to test also the student communication skills.
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)The theory of the expected utility:

Decision theory under uncertainty. The criterion of the expected value. Utility function. The criterion of expected utility. The certainty equivalent and the indifference risk premium. Quadratic and exponential utility. HARA utility. The indifference curves. An insurance example.

3) A short introduction to the study of functions of two variables:

The graph. Parallel and vertical sections. Partial derivatives. The gradient vector and the hessian matrix. Taylor's approximation. Stationary points. Free and constrained optimizations. Lagrange multipliers. Quadratic forms.

4) Linear algebra:
Eigenvalues and eigenvectors. Diagonalization of a matrix. Spectral theorem for symmetric matrices.

5)Applications in finance:
Factorial models. Principal component analysis (PCA). PCA of the term structure. OLS estimation as an optimization problem. Portfolio optimization. Computation of the price of a call in the Black-Scholes model.