## Studying at the University of Verona

Here you can find information on the organisational aspects of the Programme, lecture timetables, learning activities and useful contact details for your time at the University, from enrolment to graduation.

## Academic calendar

The academic calendar shows the deadlines and scheduled events that are relevant to students, teaching and technical-administrative staff of the University. Public holidays and University closures are also indicated. The academic year normally begins on 1 October each year and ends on 30 September of the following year.

## Course calendar

The Academic Calendar sets out the degree programme lecture and exam timetables, as well as the relevant university closure dates..

Period | From | To |
---|---|---|

I sem. | Oct 3, 2016 | Jan 31, 2017 |

II sem. | Mar 1, 2017 | Jun 9, 2017 |

Session | From | To |
---|---|---|

Sessione invernale Appelli d'esame | Feb 1, 2017 | Feb 28, 2017 |

Sessione estiva Appelli d'esame | Jun 12, 2017 | Jul 31, 2017 |

Sessione autunnale Appelli d'esame | Sep 1, 2017 | Sep 29, 2017 |

Session | From | To |
---|---|---|

Sessione estiva Appelli di Laurea | Jul 20, 2017 | Jul 20, 2017 |

Sessione autunnale Appelli di laurea | Oct 17, 2017 | Oct 17, 2017 |

Sessione invernale Appelli di laurea | Mar 22, 2018 | Mar 22, 2018 |

Period | From | To |
---|---|---|

Festa di Ognissanti | Nov 1, 2016 | Nov 1, 2016 |

Festa dell'Immacolata Concezione | Dec 8, 2016 | Dec 8, 2016 |

Vacanze di Natale | Dec 23, 2016 | Jan 8, 2017 |

Vacanze di Pasqua | Apr 14, 2017 | Apr 18, 2017 |

Anniversario della Liberazione | Apr 25, 2017 | Apr 25, 2017 |

Festa del Lavoro | May 1, 2017 | May 1, 2017 |

Festa della Repubblica | Jun 2, 2017 | Jun 2, 2017 |

Vacanze estive | Aug 8, 2017 | Aug 20, 2017 |

## Exam calendar

Exam dates and rounds are managed by the relevant Science and Engineering Teaching and Student Services Unit.

To view all the exam sessions available, please use the Exam dashboard on ESSE3.

If you forgot your login details or have problems logging in, please contact the relevant IT HelpDesk, or check the login details recovery web page.

## Academic staff

Barbu Viorel

Pauksztello David

## Study Plan

The Study Plan includes all modules, teaching and learning activities that each student will need to undertake during their time at the University.**Please select your Study Plan based on your enrollment year.**

1° Year

Modules | Credits | TAF | SSD |
---|

Modules | Credits | TAF | SSD |
---|

Modules | Credits | TAF | SSD |
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#### Legend | Type of training activity (TTA)

TAF (Type of Educational Activity) All courses and activities are classified into different types of educational activities, indicated by a letter.

### Mathematical finance (2016/2017)

Teaching code

4S001109

Academic staff

Coordinator

Credits

6

Language

English

Scientific Disciplinary Sector (SSD)

MAT/06 - PROBABILITY AND STATISTICS

Period

I sem. dal Oct 3, 2016 al Jan 31, 2017.

## Learning outcomes

Mathematical Finance

Academic Year 2016/2017

The Mathematical Finance course for the internationalized Master's Degree ( completely taught in English) aims to introduce the main concepts of discrete as well as continuous time, stochastic approach to the theory of modern financial markets.

In particular, the fundamental purpose of the course is to provide the mathematical tools characterizing the setting of Itȏ stochastic calculus for the determination, the study and the analysis of models for options, interest rates models, financial derivatives, etc., determined by stochastic differential equations driven by Brownian motion and/or impulsive random noises.

Basic ingredients are the foundation of the theory of continuous-time martingale, Girsanov theorems and the Feynman–Kac theorem and their applications to the theory of option pricing with specific examples in equities, also considering path-dependent options, and within the framework of interest rates models.

Great attention will be also given to the practical study and realisation of concrete models characterising the modern approach to both the risk managment and option pricing frameworks, also by mean of numerical computations and computer oriented lessons.

It is important to emphasize how the Stochastic Systems course is organized in such a way that students can concretely complete and further develop their own:

°ability to establish profound connections with non-mathematical disciplines, both in terms of motivation of mathematical research and of the application of the results of such surveys;

° capacity of analysis, synthesis and abstraction;

° specific computational and computer skills;

° ability to understand texts, even advanced, of Mathematics in general and Applied Mathematics in particular;

• ability to develop mathematical models for physical and natural sciences, while being able to analyze its limits and actual applicability, even from a computational point of view;

° skills concerning how to develop mathematical and statistical models for the economy and financial markets;

° capacity to extract qualitative information from quantitative data;

° knowledge of programming languages or specific software.

## Program

Mathematical Finance

2016/2017

The MathFin course will be enriched by the contributions of Michele Bonollo e Luca Spadafora, for the details of their respective parts, please see below.

[ Luca Di Persio ]

Discrete time models

• Contingent claims, value process, hedging strategies, completeness, arbitrage

• Fundamental theorems of Asset Pricing (in discrete time)

The Binomial model for Assset Pricing

• One period / multiperiod Binomial model

• A Random Walk (RW) interlude (scaled RW, symmetric RW, martingale property and quadratic variation of the symmetric RW, limiting distribution)

• Derivation of the Black-Scholes formula (continuous-time limit)

Brownian Motion (BM)

• review of the main properties of the BM: filtration generated by BM, martingale property, quadratic variation, volatility, reflection properties, etc.

Stochastic Calculus

• Itȏ integral

• Itȏ-Döblin formula

• Black-Scholes-Merton Equation

• Evolution of Portfolio/Option Values

• Solution to the Black-Scholes-Merton Equation

• Sensitivity analysis

Risk-Neutral Pricing

• Risk-Neutral Measure and Girsanov's Theorem

• Pricing under the Risk-Neutral Measure

• Fundamental Theorems of Asset Pricing

• Existence/uniqueness of the Risk-Neutral Measure

• Dividend/continuously-Paying

• Forwards and Futures

[ Luca Spadfora ]

***Statistics

*Theory Review: distributions, the moments of a distribution, statistical estimators, Central Limit Theorem (CLT), mean, variance and empirical distributions.

*Elements of Extreme Value Theory: what is the distribution of the maximum?

Numerical studies: statistical error of the sample mean, CLT at work, distributions of extreme values.

***Risk Modelling

*How can we measure risk? Main risk measures: VaR and Expected Shorfall

*How to model risk: historical, parametric and Montecarlo methods

*We have a risk model: does it works? The backtesting methodology

*Empirical studies a) empirical behavior and stylized facts of historical series

*Empirical studies b) Implementation of risk models

*Empirical studies c) Implementation of risk models backtesting

[ Miche Bonollo ]

*** Tools for derivatives pricing

* Functionals of brownian motions: fist hitting time, occupation time, local time, min-MAX distribution review

* Application 1: range accrual payoff

* Application 2: worst of and Rainbow payoff

*** Credit portfolio models

* The general framework. The credit portfolio data

* Gaussian Creidit Metrics - Merton model

* The quantile estimation problem with Montecarl approach. L-Estimators, Harrel-Davis

Bibliography:

A. F. McNeil, R. Frey, P. Embrechts, Quantitative Risk Management:Concepts, Techniques and Tools, Princeton University Press, 2015.

J. -P. Bouchaud, M. Potter, Theory of Financial Risk - From Statistical Physics to Risk Management, University Press, Cambridge, 2000.

R. Cont, P. Tankov, Financial Modelling With Jump Processes, Chapman and Hall, CRC Press, 2003.

E. J. Gumbel, Statistics of Extremes, Dover Publications, Mineola (NY), 2004.

M.Yor et al, "Exponential Functionals of Brownian Motion and related Processes", Springer.

Shreve, Steven , Stochastic Calculus for Finance II: Continuous-Time Models

Shreve, Steven , Stochastic Calculus for Finance I: The Binomial Asset Pricing Model

Author | Title | Publishing house | Year | ISBN | Notes |
---|---|---|---|---|---|

M.Yor et al | Exponential Functionals of Brownian Motion and related Processes | Springer | 2010 | ||

R. Cont, P. Tankov | Financial Modelling With Jump Processes | Chapman and Hall, CRC Press | 2003 | ||

A. F. McNeil, R. Frey, P. Embrechts | Quantitative Risk Management:Concepts, Techniques and Tools | Princeton University Press | 2015 | ||

E. J. Gumbel | Statistics of Extremes | Dover Publications, Mineola (NY) | 2004 | ||

S. E. Shreve | Stochastic Calculus for Finance II: Continuous-Time Models | Springer, New York | 2004 | ||

S. E. Shreve | Stochastic Calculus for Finance I: The Binomial Asset Pricing Model | Springer, New York | 2004 | ||

J. -P. Bouchaud, M. Potter | Theory of Financial Risk - From Statistical Physics to Risk Management | University Press, Cambridge | 2000 |

## Examination Methods

Mathematical Finance

Academic Year 2016/2017

Final Exam : the exam will consists in an oral session, to be given with prof. L. Di Persio, which will be targeted on the theory behind all the arguments treated in the whole course, hence including the parts developed by M. Bonollo and L. Spadafora.

Moreover each student will be called to develop a case study within a list of projects proposed by both M. Bonollo and L. Spadafora, according with the notions that will have been addressed during their respective parts [ see the Course Program section ].

The final vote is expressed out of 30: in particular:

° The doctors Bonollo and Spadafora will communicate to prof. Of Persio a report on the goodness of the project presented by the single student;

° professor. Di Persio will use the previous report, along with the outcome of the oral examination he conducted, to decide on a final grade expressed out of 30.

It is important to emphasize how the skills acquired by students at the end of the course will enable them to:

- carry out high-profile technical and / or professional tasks, both mathematically oriented and of

computational type, both in laboratories and / or research organizations, in the fields of finance, insurance, services, and public administration, both individually and in groups;

° read and understand advanced texts of math and applied sciences, even at the level of advanced research;

• to use high-tech computing and computing tools with the utmost ease of implementation algorithms and models illustrated in the course, as well as to acquire further information;

- to know in depth the demonstration techniques used during the course in order to be able to exploit them to solve problems in different mathematical fields, also by taking the necessary tools and methods, from seemingly distant contexts, thus mathematically formalizing problems expressed in languages of other scientific disciplines as well as economical ones, using, adapting and developing advanced models.

**Students with disabilities or specific learning disorders (SLD), who intend to request the adaptation of the exam, must follow the instructions given HERE**

## Type D and Type F activities

**Modules not yet included**

## Career prospects

## Module/Programme news

##### News for students

There you will find information, resources and services useful during your time at the University (Student’s exam record, your study plan on ESSE3, Distance Learning courses, university email account, office forms, administrative procedures, etc.). You can log into MyUnivr with your GIA login details: only in this way will you be able to receive notification of all the notices from your teachers and your secretariat via email and also via the Univr app.

## Alternative learning activities

In order to make the study path more flexible, it is possible to request the substitution of some modules with others of the same course of study in Mathematics at the University of Verona (if the educational objectives of the modules to be substituted have already been achieved in the previous career), or with others of the course of study in Mathematics at the University of Trento.## Documents

Title | Info File |
---|---|

1. Convenzione | Learning Agreement UNITN - UNIVR | pdf, it, 167 KB, 27/08/21 |

2. Sostituzione insegnamenti a UNITN - Courses replacement at UNITN | pdf, it, 97 KB, 29/07/24 |

3. Sostituzione insegnamenti a UNIVR - Courses replacement at UNIVR | pdf, it, 113 KB, 30/08/21 |

## Attendance modes and venues

As stated in the Teaching Regulations , except for specific practical or lab activities, attendance is not mandatory. Regarding these activities, please see the web page of each module for information on the number of hours that must be attended on-site.

Part-time enrolment is permitted. Find out more on the Part-time enrolment possibilities page.

The course's teaching activities take place in the Science and Engineering area, which consists of the buildings of Ca‘ Vignal 1, Ca’ Vignal 2, Ca' Vignal 3 and Piramide, located in the Borgo Roma campus.

Lectures are held in the classrooms of Ca‘ Vignal 1, Ca’ Vignal 2 and Ca' Vignal 3, while practical exercises take place in the teaching laboratories dedicated to the various activities.

## Career management

## Student login and resources

## Graduation

#### Deadlines and administrative fulfilments

For deadlines, administrative fulfilments and **notices on graduation sessions**, please refer to the Graduation Sessions - Science and Engineering service.

#### Need to activate a thesis internship

For** thesis-related internships**, it is not always necessary to activate an internship through the Internship Office. For further information, please consult the dedicated document, which can be found in the 'Documents' section of the Internships and work orientation - Science e Engineering service.

#### Final examination regulations

Upon completion of the Master’s degree dissertation students are awarded 32 CFU. The final examination consists of a written dissertation on a specific topic agreed with a supervising professor and presented to a commission (Dissertation Committee).

The dissertation can be high-level theoretical or experimental (in the latter case, it may focus on either basic or applied research), it can deal with a theoretical topic or propose the resolution of a specific problem, or description of a work project, and may be carried out at universities, research institutions, schools, laboratories and companies in the framework of internships, traineeships, study stays in Italy and abroad. The dissertation must be original and written by the student under the guidance of a Supervisor. At the request of the student, the dissertation may be written and presented in Italian.

Professors belonging to the Mathematics Teaching Committee, the Department of Computer Science, and any associated departments may be appointed as Supervisors, as well as any professors from the University of Verona whose area of interest (SSD - Scientific-disciplinary Sector) is included in the teaching regulations of the degree programme.

Students may take the final exam only if meeting all requirements set by the School of Sciences and Engineering.

The Master's degree in Mathematics is obtained by successfully passing the final examination and thus earning the 120 CFU included in the study plan.

The material submitted by the student for the final examination will be examined by the Dissertation Committee, which comprises three professors, possibly including the Supervisor, and appointed by the President of the Teaching Committee. The final examination will be assessed based on the following criteria: the student’s performance during the entire study programme, the knowledge acquired during the dissertation work, their understanding of the topic and autonomy of judgment, their ability to apply such knowledge, and communicate effectively and fully all the outcomes of the work and the main results obtained.

The final examination and the degree ceremony will be carried out, in one of the four graduation sessions throughout the academic year, by the Final Examination Committee appointed by the President of the Teaching Committee, and made up of a president and at least four members chosen from among the professors of the University.

For further information, please refer to the Final examination regulations.

## Documents

Title | Info File |
---|---|

1. Come scrivere una tesi | pdf, it, 31 KB, 02/11/22 |

2. How to write a thesis | pdf, en, 31 KB, 02/11/22 |

5. Regolamento tesi | pdf, it, 171 KB, 20/03/24 |

## List of thesis proposals

theses proposals | Research area |
---|---|

Controllo di sistemi multiagente | Calculus of variations and optimal control; optimization - Hamilton-Jacobi theories, including dynamic programming |

Controllo di sistemi multiagente | Calculus of variations and optimal control; optimization - Manifolds |

Controllo di sistemi multiagente | Calculus of variations and optimal control; optimization - Optimality conditions |

Formule di rappresentazione per gradienti generalizzati | Mathematics - Analysis |

Formule di rappresentazione per gradienti generalizzati | Mathematics - Mathematics |

Mathematics Bachelor and Master thesis titles | Various topics |