# Financial Mathematics (second cycle)

**Study program cycle and type:**

Second cycle study program.

**Anticipated title:**

In Slovenian: magister finančne matematike, magistrica finančne matematike, abbreviated to mag. fin. mat.

**Duration:**

2 full years (4 terms) based on 120 ECTS credits

**Basic goals:**

This program is designed for Bachelors in Financial Mathematics (academic) as well as graduates of

other first cycle programs wishing to enhance their knowledge of probability and optimization and

acquire new knowledge of economics and financial mathematics. Graduates of this program can take active part in planning and development in financial institutions and financial departments of large companies or engage in scientific research at economic research institutions.

**Subject specific competences developed by the student:**

- familiarity with classical and modern results in the fields of probability, statistics, optimization, economics, as well as financial and actuarial mathematics,
- core knowledge of computer informatics,
- core knowledge of economics and finance,
- ability of continuous self-education in economics and finance,
- ability of logical thinking and understanding mathematical proofs,
- ability to solve concrete problems with the use of different mathematical methods.

**Employment possibilities:**

Graduates of this program can find employment in major banks, insurance companies, stock brokerage companies, investment firms, pension companies, health benefits companies, public sector and public agencies, logistics sector (optimization), economic research institutes and universities.

## Admission requirements

**Admission to the study program is open to the following:**

- Graduates of the Academic study program in Financial Mathematics.
- Graduates of the Academic study program in Mathematics or Mathematics Education.
- Graduates of the University undergraduate study programs in Applied mathematics, Pure Mathematics, Mathematics Education.
- Graduates of some accredited first cycle study program.

Applicants under 2, 3, or 4 above will have to pass additional exams in order to acquire the minimal competences required for admission to the Master's study program in Financial Mathematics. This can be done either prior to admission or prior to completion of the study program. The detailed conditions follow below.

Applicants under 2 or 3 above have to pass the exams from the financial-economic package (Macroeconomics, Microeconomics, Money and finance, Financial markets and institutions). Furthermore, in addition to the obligatory course Probability theory 2 these applicants also have to complete Random processes 1.

Applicants under 4 above have to pass the exams from the financial-economic package (Macroeconomics, Microeconomics, Money and finance, Financial markets and institutions). Additional requirements necessary to cover the minimal competences for admission are determined on a case-bycase basis by the department study committee.

**The minimal competences can be validated or acquired in one of the following ways:**

**(a)** The applicants who were previously enrolled in an accredited study program may submit the relevant transcripts in order to request validation of the completed courses.

**(b)** For applicants who have acquired the necessary competences by self-study, the department organizes one or more ad hoc examinations where the competences are tested. The examination(s) must be completed prior to admission.

**(c)** Applicants can acquire the requisite competences by attending lectures and problem sessions offered at the mathematics department of the FMF within the first-cycle program in Financial Mathematics; the competences are tested by exams at the end of the course(s).

The mathematics department offers the possibility of admission to the second-cycle study program prior to verification of the requisite competences. Applicants taking advantage of this offer acquire the missing competences by way of (c). They have to complete the corresponding examinations either prior to enrollment in the 2nd year or prior to completion of the study program, subject to decision of the department study committee.

Applicants under (b) above receive the list of minimal mathematical competences at the Student Office.

## Admission limitation measures

Applicants are selected according to their GPA of exams, problem sessions and seminars' grades obtained at their respective first-cycle programs.

## Enrollment requirements

To enroll in the second year, students must earn at least 50 ECTS credits from courses and exams in the first year.

## Re-enrollment requirements

To re-enroll in the current study year, a student needs to earn at least half of all possible credits of the current study year (30 ECTS credits). Re-enrollment is only possible once in the course of the study program; any change in a study program as result of disability of enrollment in the second year is automatically counted as re-enrollment.

## Finishing requirements

To finish the program, students must:

- Successfully complete all exams.
- Prepare and defend the master's thesis.

## Transition from other study programs

Graduates of the second-cycle programs in Mathematics and Mathematics Education must meet the admission requirements. Granted that, they are exempted from retaking courses they have already completed within the respective programs. The exact requirements for enrollment in the second study year and completion of the program are determined by the department study committee.

## Study program description

The study program comprises two full academic years based on 120 ECTS credits. Of these, the master's thesis accounts for 20 ECTS credits. All courses are single-term courses with 30 to 45 hours of lectures and 15 to 30 hours of problem sessions altogether (with a weekly load of 2/2 or 3/1 hours of lectures/problem sessions). Each course is worth 5 ECTS credits. A student's choice of courses has to be approved by the department study committee.

The courses are divided into the following groups:

**M1** – Analysis and mechanics

**M2** – Algebra and discrete mathematics

**M3** – Geometry and topology

**M4** – Numerical mathematics

**M5** – Probability, statistics and financial mathematics

**R1** – Computer mathematics

A minimum of 70 ECTS credits must be earned by completing exams from the following categories:

**Basic:**The obligatory Probability theory 2 and at least five electives from the group M5. Together, these courses account for at least 30 ECTS credits.**Cognate:**Financial courses offered by the Faculty of Economics which account for at least 15 ECTS.**Specific:**Electives from groups M1-M4 and R1 which account for at least 25 ECTS credits. Ideally, at least one course out of each of the groups M1-M4 and R1 should be elected.

The master's thesis accounts for 20 ECTS credits.

Of the other requisite credits a maximum of 10 ECTS credits can be awarded for work experience or a scientific publication. To be awarded credits, work experience must comprise at least 150 working hours as well as an obligatory presentation preparation. Meeting this requirement, 1 ECTS credit is awarded for each 30 working hours.

Of the cognate courses offered by the Faculty of Economics, it is recommended that one of the following clusters of courses at the Department of Money and Finance be elected:

**Macroeconomic cluster: **

- Monetary economics 2
- Quantitative financial economics
- Modelling monetary policies
- Public finance 2

**Entrepreneurial finance cluster: **

- Corporate finance 2
- International business finance
- Financial statement analysis 2
- Empirical corporate finance

**Financial institutions and markets cluster: **

- Financial institutions management 2
- Derivatives or Investment management
- Financial markets 2
- Life and pension insurance

## Curriculum

The spreadsheet data are given for both the winter and the summer term.

Each term comprises 15 weeks of classes.

Abbreviations:

**L** = lectures per week (in hours),

**P** = problem sessions per week (in hours),

**ECTS** = ECTS credits worth,

**TSW** = estimated total student workload (in hours).

#### 1st year

Winter term | Summer term | Total | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

Course | L | P | ECTS | TSW | L | P | ECTS | TSW | ECTS | TSW |

Probability theory 2 | 3 | 1 | 5 | 150 | 0 | 0 | 0 | 0 | 5 | 150 |

Elective from group M5 | 2 | 2 | 5 | 150 | 0 | 0 | 0 | 0 | 5 | 150 |

Basic elective from groups M1-4 and R1 | 2 | 2 | 5 | 150 | 0 | 0 | 0 | 0 | 5 | 150 |

Basic elective from groups M1-4 and R1 | 2 | 2 | 5 | 150 | 0 | 0 | 0 | 0 | 5 | 150 |

Elective at Faculty of Economics | 2 | 2 | 5 | 150 | 0 | 0 | 0 | 0 | 5 | 150 |

General elective | 2 | 2 | 5 | 150 | 0 | 0 | 0 | 0 | 5 | 150 |

Elective from group M5 | 0 | 0 | 0 | 0 | 2 | 2 | 5 | 150 | 5 | 150 |

Elective from group M5 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 150 | 5 | 150 |

Basic elective from groups M1-4 and R1 | 0 | 0 | 0 | 0 | 2 | 2 | 5 | 150 | 5 | 150 |

Basic elective from groups M1-4 and R1 | 0 | 0 | 0 | 0 | 2 | 2 | 5 | 150 | 5 | 150 |

Elective at Faculty of Economics | 0 | 0 | 0 | 0 | 2 | 2 | 5 | 150 | 5 | 150 |

General elective | 0 | 0 | 0 | 0 | 2 | 2 | 5 | 150 | 5 | 150 |

Weekly total | 13 | 11 | 12 | 12 | ||||||

Term total | 195 | 165 | 30 | 900 | 180 | 180 | 30 | 900 | 60 | 1800 |

#### 2nd year

Winter term | Summer term | Total | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

Course | L | P | ECTS | TSW | L | P | ECTS | TSW | ECTS | TSW |

Elective from group M5 | 2 | 2 | 5 | 150 | 0 | 0 | 0 | 0 | 5 | 150 |

Elective from group M5 | 2 | 2 | 5 | 150 | 0 | 0 | 0 | 0 | 5 | 150 |

Basic elective from groups M1-4 and R1 | 2 | 2 | 5 | 150 | 0 | 0 | 0 | 0 | 5 | 150 |

Elective at Faculty of Economics | 2 | 2 | 5 | 150 | 0 | 0 | 0 | 0 | 5 | 150 |

General elective | 2 | 2 | 5 | 150 | 0 | 0 | 0 | 0 | 5 | 150 |

General elective | 2 | 2 | 5 | 150 | 0 | 0 | 0 | 0 | 5 | 150 |

Work experience or project work | 0 | 0 | 0 | 0 | 0 | 0 | 5 | 150 | 5 | 150 |

General elective or work experience or project work | 0 | 0 | 0 | 0 | 0 | 0 | 5 | 150 | 5 | 150 |

Master's thesis | 0 | 0 | 0 | 0 | 0 | 0 | 20 | 600 | 20 | 600 |

Weekly total | 12 | 12 | 0 | 0 | ||||||

Term total | 180 | 180 | 30 | 900 | 0 | 0 | 30 | 900 | 60 | 1800 |

#### M1 Analysis and Mechanics

Course | L | P | ECTS | TSW |
---|---|---|---|---|

Measure theory | 2 | 2 | 5 | 150 |

Introduction to functional analysis | 2 | 2 | 5 | 150 |

Functional analysis | 2 | 2 | 5 | 150 |

Introduction to C* - algebras | 3 | 1 | 5 | 150 |

Operator theory | 3 | 1 | 5 | 150 |

Introduction to harmonic analysis | 3 | 1 | 5 | 150 |

Special functions | 2 | 2 | 5 | 150 |

Partial differential equations | 2 | 2 | 5 | 150 |

Complex analysis | 2 | 2 | 5 | 150 |

Analytical mechanics | 2 | 2 | 5 | 150 |

Continuum mechanics | 2 | 2 | 5 | 150 |

Fluid mechanics | 2 | 2 | 5 | 150 |

Mechanics of deformable bodies | 2 | 2 | 5 | 150 |

Dynamical systems | 2 | 2 | 5 | 150 |

Industrial mechanics | 2 (seminar) | 2 (practice) | 5 | 150 |

#### M2 Algebra and Discrete Mathematics

Course | L | P | ECTS | TSW |
---|---|---|---|---|

Commutative algebra | 3 | 1 | 5 | 150 |

Associative algebra | 3 | 1 | 5 | 150 |

Non-associative algebra | 3 | 1 | 5 | 150 |

Ordered algebraic structures | 3 | 1 | 5 | 150 |

Group and semi-group theory | 3 | 1 | 5 | 150 |

Number theory | 3 | 1 | 5 | 150 |

Combinatorics | 2 | 2 | 5 | 150 |

Graph theory | 2 | 2 | 5 | 150 |

Cardinal arithmetic | 3 | 1 | 5 | 150 |

Topics in discrete mathematics | 2 | 2 | 5 | 150 |

Applied discrete mathematics | 1 | 3 | 5 | 150 |

Logic | 2 | 2 | 5 | 150 |

#### M3 Geometry and Topology

Course | L | P | ECTS | TSW |
---|---|---|---|---|

Analysis on manifolds | 3 | 1 | 5 | 150 |

Introduction to algebraic geometry | 3 | 1 | 5 | 150 |

Convexity | 3 | 1 | 5 | 150 |

Algebraic topology 1 | 2 | 2 | 5 | 150 |

Algebraic topology 2 | 2 | 2 | 5 | 150 |

Differential geometry | 3 | 1 | 5 | 150 |

Lie groups | 3 | 1 | 5 | 150 |

Riemann surfaces | 2 | 2 | 5 | 150 |

#### M4 Numerical Mathematics

Course | L | P | ECTS | TSW |
---|---|---|---|---|

Numerical integration and ordinary differential equations | 2 | 2 | 5 | 150 |

Numerical solutions of partial differential equations | 2 | 2 | 5 | 150 |

Iterative numerical methods in linear algebra | 2 | 2 | 5 | 150 |

Computer aided (geometric) design | 2 | 2 | 5 | 150 |

Numerical approximation and interpolation | 2 | 2 | 5 | 150 |

Numerical methods for linear control systems | 2 | 2 | 5 | 150 |

#### M5 Probability, Statistics and Financial Mathematics

Course | L | P | ECTS | TSW |
---|---|---|---|---|

Probability theory 2 | 3 | 1 | 5 | 150 |

Statistics 2 | 3 | 1 | 5 | 150 |

Financial mathematics 2 | 2 | 2 | 5 | 150 |

Introduction to random processes | 2 | 2 | 5 | 150 |

Econometrics | 3 | 1 | 5 | 150 |

Random processes 2 | 2 | 2 | 5 | 150 |

Actuarial mathematics | 2 | 2 | 5 | 150 |

Modelling with random processes | 2 | 2 | 5 | 150 |

Topics in game theory | 2 | 2 | 5 | 150 |

Topics in financial mathematics | 2 | 2 | 5 | 150 |

Optimization in finance | 2 | 2 | 5 | 150 |

Time series | 2 | 2 | 5 | 150 |

Riesz spaces in mathematical economics | 2 | 2 | 5 | 150 |

#### R1 Computer Mathematics

Course | L | P | ECTS | TSW |
---|---|---|---|---|

Doing mathematics with a computer | 1 | 3 | 5 | 150 |

Theory of computability | 2 | 2 | 5 | 150 |

Computational complexity | 2 | 2 | 5 | 150 |

Topics in computer mathematics | 2 | 2 | 5 | 150 |

Topics in optimization | 2 | 2 | 5 | 150 |

Optimization 2 | 2 | 2 | 5 | 150 |

Data structures and algorithms 3 | 2 | 2 | 5 | 150 |