Unit MATHEMATICAL METHODS FOR RISK MANAGEMENT
- Course
- Finance and quantitative methods for economics
- Study-unit Code
- A000199
- Location
- PERUGIA
- Curriculum
- Finanza ed assicurazione
- Teacher
- Marco Nicolosi
- Teachers
-
- Marco Nicolosi
- Hours
- 63 ore - Marco Nicolosi
- CFU
- 9
- Course Regulation
- Coorte 2020
- Offered
- 2020/21
- Learning activities
- Caratterizzante
- Area
- Matematico, statistico, informatico
- Academic discipline
- 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) Two variable functions: the graph, the Taylor's formula, free and constrained optimizations.
3) Linear algebra: linear systems, matrix diagonalization. Principal components analysis.
4) An introduction to the theory of utility.
5) Applications in finance. - 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
4) Utility theory:
"Manuale di Finanza II", G. Castellani, M. De Felice, F. Moriconi, chapters 1, 2 e 3 - 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
- elements of linear algebra
- functions of two variables
The main competence will be:
- to compute probabilities, expected values, conditioned expected values, quantiles
- to analyze and compare financial payoffs under uncertainty condition
- to diagonalize a matrix
- 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 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
- The exam consists of a written test and an oral test (which is optional). The written exam consists in solving 3 or 4 exercises and it lasts for 1,5 hours. As an alternative, the student can take a written midterm exam (during the teaching pause at the beginning of November) and a completion exam (instead of the complete exam on the first exam date in January). The rules for the mid-term and the completion exams are the same as for the complete exam. The written exam has the aim to test the competence acquired during the class. The oral exam is optional, and consists of two questions on the whole program that are valued +/-2 points each. The final score is the sum of the score to the written exam (or of the average of the mid-term and completion exam scores) and to the oral exam. 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) 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) An introduction to the theory of 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.
5)Applications in finance:
Factorial models. PCA of the term structure. OLS estimation as an optimization problem. Portfolio optimization. Computation of the call price in the Black-Scholes model.