## 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 semestre | Oct 1, 2015 | Jan 29, 2016 |

II semestre | Mar 1, 2016 | Jun 10, 2016 |

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

Sessione straordinaria Appelli d'esame | Feb 1, 2016 | Feb 29, 2016 |

Sessione estiva Appelli d'esame | Jun 13, 2016 | Jul 29, 2016 |

Sessione autunnale Appelli d'esame | Sep 1, 2016 | Sep 30, 2016 |

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

Sess. autun. App. di Laurea | Oct 12, 2015 | Oct 12, 2015 |

Sess. invern. App. di Laurea | Mar 15, 2016 | Mar 15, 2016 |

Sess. estiva App. di Laurea | Jul 19, 2016 | Jul 19, 2016 |

Sess. autun. 2016 App. di Laurea | Oct 11, 2016 | Oct 11, 2016 |

Sess. invern. 2017 App. di Laurea | Mar 16, 2017 | Mar 16, 2017 |

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

Festività dell'Immacolata Concezione | Dec 8, 2015 | Dec 8, 2015 |

Vacanze di Natale | Dec 23, 2015 | Jan 6, 2016 |

Vacanze Pasquali | Mar 24, 2016 | Mar 29, 2016 |

Anniversario della Liberazione | Apr 25, 2016 | Apr 25, 2016 |

Festa del S. Patrono S. Zeno | May 21, 2016 | May 21, 2016 |

Festa della Repubblica | Jun 2, 2016 | Jun 2, 2016 |

Vacanze estive | Aug 8, 2016 | Aug 15, 2016 |

## 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.

Should you have any doubts or questions, please check the Enrollment FAQs

## Academic staff

Mazzuoccolo Giuseppe

giuseppe.mazzuoccolo@univr.it +39 0458027838Squassina Marco

marco.squassina@univr.it +39 045 802 7913## 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 |
---|

#### 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 (2015/2016)

Teaching code

4S001109

Credits

6

Language

English

Scientific Disciplinary Sector (SSD)

MAT/06 - PROBABILITY AND STATISTICS

The teaching is organized as follows:

##### Teoria 2

##### Teoria 1

##### Esercitazioni

## Learning outcomes

The Mathematical Finance course for the internationalized Master's Degree (delivered completely in English) aims to introduce the main concepts of stochastic discrete and continuous time part of the modern theory of 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 and / or interest rates determined by stochastic differential equations driven by Brownian motion. Basic ingredients are the foundation of the theory of continuous-time martingale, Girsanov theorems and the Faynman-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.

## Program

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

Binomial model: numerical approaches

•Parameters for the binomial model

•The binomial model implementation

•Hedging for the binomial model

Black and Scholes: review and numerical implementation(s)

•The Itȏ-Döblin formula: a review

•Pricing Hedging and Risk for the Plain Vanilla Options

•Delta-Vega-Gamma Approsimations: Numerical Experiments

•Morte Carlo pricing princples: some easy cases

•The leverage effect of an option

Functionals of the Brownian motion and geometric Brownian motion

•The first hittting time distribution and density

•The occupation time and the Takacs formula

•The Local time definition, related quantities and density

Applications of the BM functional to the option pricing

•Barrier Options

•Digital Options

•Accumulators Options

•Implementations in VBA and MATHCAD

The Asian style options

•Impact of the fixing frequency

•Monte Carlo Evaluations

•Monte Carlo accuracy: how to build a confidence level for the pricer

•The Matching moment method (MMM): principles

•The MMM implementation for the Asian option

Advanced probabilistic tools for exotic options

•The best-of and the worst-of options.

•The worst and the best distribution for uncorrelated basket

•Extension to correlated basket

Credit Risk

•Default definition

•The relevant parameters for the credit risk: PD, EAD, LGD

•The credit portfolio Loss

•The Basel Gordy model

•A Monte Carlo framework for the loss distribution

Paolo Guasoni mini course: the didactic material concerning this part is the subset of the following topics that have been concretely treated during the 8 hours mini course given by Prof. Paolo Guasoni (DCU-Dublin) [ ask to Prof. Di Persio for further infos] :

1. Classical Theory.

The discussion starts with a review of the Merton consumption-investment problem with constant investment

opportunities, and with its asset-pricing counterpart, the Lucas model.

2. Long-run, state variables, and stochastic investment opportunities.

The discussion starts with the general model of a market with several state variables, the objectives of

equivalent safe rate and equivalent annuity, their corresponding HJB equations, and nite-horizon bounds.

Applications to models of return predictability and stochastic volatility conclude. [4].

3. Transaction Costs.

A market with transaction costs and constant investment opportunities is equivalent to another market,

found explicitly, in which investment opportunities are stochastic, but transaction costs absent. [1].

4. Price Impact.

If trading speed aects execution prices, portfolio weights are no longer controls, but state variables. The

optimal trading speed follows an autonomous diusion process, interpreted as trading volume. Short and

levered positions are endogenously banned by this friction. [5]

5. High-water marks and hedge-fund fees.

In a model of hedge fund compensation, the state variable is the ratio between the fund's assets and its

historical maximum. The long-run solution leads to a simple optimal portfolio, which shows the interplay

between fees and risk aversion. [3, 2]

6. Path-dependent Preferences and Shortfall Aversion.

A model in which the marginal utility from increases in consumption above its historical maximum is lower

than the marginal utility of marginal decreases in consumption (shortfall aversion) can explain high asset

prices and low interest rates, as well as smooth consumption with volatile wealth. [6]

References

[1] S. Gerhold, P. Guasoni, J. Muhle-Karbe, and W. Schachermayer. Transaction costs, trading volume, and the

liquidity premium. Finance and Stochastics, 18(1):1{37, 2014.

[2] P. Guasoni and J. Muhle-Karbe. Long horizons, high risk-aversion, and endogenous spreads. Mathematical

Finance, 25(4):724753, 2011.

[3] P. Guasoni and J. Obloj. The incentives of hedge fund fees and high-water-marks. Mathematical Finance,

2015, 2009.

[4] P. Guasoni and S. Robertson. Portfolios and risk premia for the long run. The Annals of Applied Probability,

22(1):239{284, 2012.

[5] P. Guasoni and M. Weber. Dynamic trading volume. Mathematical Finance, 2015. forthcoming.

[6] Paolo Guasoni, Gur Huberman, and Dan Ren. Shortfall aversion. Available at SSRN 2564704, 2015.

## Examination Methods

Final Exam : the exam will consists in an oral session plus a case study developed according with prof. Michele Bonollo with respect to the following list of case studies:

#1: Stress Test of derivatives portfolios

#2: Derivatives portfolio evaluation and management

#3: Credit Portfolio Risk

#4: Exotic Options Pricing

**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 soon 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, 44 KB, 30/08/21 |

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

## Attendance

As stated in the Teaching Regulations for the A.Y. 2022/2023, 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.

## 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 |