## Studying at the University of Verona

## Academic calendar

Il calendario accademico riporta le scadenze, gli adempimenti e i periodi rilevanti per la componente studentesca, personale docente e personale dell'Università. Sono inoltre indicate le festività e le chiusure ufficiali dell'Ateneo.

L’anno accademico inizia il 1° ottobre e termina il 30 settembre dell'anno successivo.

## 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

The exam roll calls are centrally administered by the operational unit
Science and Engineering Teaching and Student Services Unit

Exam Session Calendar and Roll call enrolment sistema ESSE3.
If you forget your password to the online services, please contact the technical office in your Faculty or to the service credential recovery.

Per dubbi o domande Read the answers to the more serious and frequent questions - F.A.Q. Examination enrolment

## Academic staff

Barbu Viorel

Pauksztello David

Petrakis Iosif

## 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 enrolment year.

Teachings | Credits | TAF | SSD |
---|

1° Anno

Teachings | Credits | TAF | SSD |
---|

Teachings | 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 (2016/2017)

Teaching code

4S001109

Academic staff

Coordinatore

Credits

6

Scientific Disciplinary Sector (SSD)

MAT/06 - PROBABILITY AND STATISTICS

Language of instruction

English

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

## Bibliografia

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.

## Tipologia di Attività formativa D e F

**Course not yet included**

## Career prospects

## Avvisi degli insegnamenti e del corso di studio

##### Per la comunità studentesca

Se sei già iscritta/o a un corso di studio,
puoi consultare tutti gli avvisi relativi al tuo corso di studi nella tua area riservata MyUnivr.

In questo portale potrai visualizzare informazioni, risorse e servizi utili che riguardano la tua carriera universitaria (libretto online, gestione della carriera Esse3, corsi e-learning, email istituzionale, modulistica di segreteria, procedure amministrative, ecc.).

Entra in MyUnivr con le tue credenziali GIA.

## Graduation

## List of theses and work experience 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 |

Stage | Research area |
---|---|

Internship proposals for students in mathematics | Various topics |

## Double degree

The University of Verona, through a network of agreements with foreign universities, offers international courses that **enable students to gain a Double/Joint degree at the time of graduation. **Indeed, students enrolled in a Double/Joint degree programme will be able to obtain both the degree of the University of Verona and the degree issued by the Partner University abroad - where they are expected to attend part of the programme -, in the time it normally takes to gain a common Master’s degree. The institutions concerned shall ensure that both degrees are **recognised in the two countries.**

**Places on these programmes are limited**, and admissions and any applicable grants are subject to applicants being selected in a specific *Call for applications.*

The latest Call for applications for Double/Joint Degrees at the University of Verona is available now!

## University Language Centre - CLA

## Further services

I servizi e le attività di orientamento sono pensati per fornire alle future matricole gli strumenti e le informazioni che consentano loro di compiere una scelta consapevole del corso di studi universitario.