## 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 semestre | Oct 1, 2018 | Jan 31, 2019 |

II semestre | Mar 4, 2019 | Jun 14, 2019 |

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

Sessione invernale d'esame | Feb 1, 2019 | Feb 28, 2019 |

Sessione estiva d'esame | Jun 17, 2019 | Jul 31, 2019 |

Sessione autunnale d'esame | Sep 2, 2019 | Sep 30, 2019 |

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

Sessione di laurea estiva | Jul 22, 2019 | Jul 22, 2019 |

Sessione di laurea autunnale | Oct 15, 2019 | Oct 15, 2019 |

Sessione di laurea invernale | Mar 19, 2020 | Mar 19, 2020 |

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

Sospensione attività didattica | Nov 2, 2018 | Nov 3, 2018 |

Vacanze di Natale | Dec 24, 2018 | Jan 6, 2019 |

Vacanze di Pasqua | Apr 19, 2019 | Apr 28, 2019 |

Vacanze estive | Aug 5, 2019 | Aug 18, 2019 |

## 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

## 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 (2018/2019)

Teaching code

4S001109

Credits

6

Coordinatore

Scientific Disciplinary Sector (SSD)

MAT/06 - PROBABILITY AND STATISTICS

Language of instruction

English

The teaching is organized as follows:

##### Parte 1

Credits

2

Period

I semestre

Academic staff

Luca Di Persio

##### Parte 2

Credits

4

Period

I semestre

Academic staff

Luca Di Persio

## Learning outcomes

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

[1] Stochastic analysis: basics

Basics on stochastic processes

Stochastic processes: main examples in discrete and continuous time

Stochastic integration

The Itô-Döblin lemma

SDEs: basics with examples ( e.g.: the linear case, multiplicative noise case)

Solution of SDEs as Markov processes

Feynman-Kac formula

Girsanov theorem

Stochastic control: basics with examples (e.g.: dynamic programming principle, Pontryagin maximum principle)

[2] Discrete time models

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

Fundamental theorems of Asset Pricing (in discrete time)

Binomial trees

Random walk and pricing

Black and Scholes formula ( derived by binomial trees analysis )

[3] Brownian Motion (BM)

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

[4] Continuous time models

Black-Scholes-Merton Equation

Evolution of Portfolio/Option Values

Sensitivity analysis

The Martingale approach

Hedging and replicating strategies

Equity market models

Siegel paradox

Packages and Exotic options

[5] Interest rates models

Markovian Models of the Short Rate

Merton model

Stochastic interest rate for the Black and Scholes model

Hedging portfolio

Change of numeraire ( also under multiple risk sources )

Caps, floors, collars

Interest rates models

Vasicek model

Cox-Ingersoll-Ross model

Forward rates modelling

Arbitrage models for term structure

Heath-Jarrow-Morton framework

The Hull-White extended Vasicek model

[6] Portfolio choice and Asset Pricing

Bachelier and Samuelson models

Utility functions

The Merton problem ( value and static programming approach)

Utility maximization problem

[7] Miscellanea

Valuation of Options in Gaussian Models

Forward LIBORs

Swap rates modelling

Mean Field Games approach to systems of interacting financial agents

Calibration for Interest Rate models

Stochastic control and financial models (e.g.: the Heston model case)

Stochastic volatility models and applications

Polynomial/asyntotic espansions for financial models

SDEs on networks with financial applications

## Examination Methods

Oral exam with written exercise:

the exam is based on open questions as well as on the resolution of written exercises to be solved during the test itself. Questions, open-ended and exercises, aim at verify both the knowledge about arguments developed within the course, the solution of concrete problems belonging to Mathematical Finance, and to the acquired acquaintance of associated tools of stochastic analysis.

## Bibliografia

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

Parte 1 | R. Cont, P. Tankov | Financial Modelling With Jump Processes | Chapman and Hall, CRC Press | 2003 | ||

Parte 1 | A. F. McNeil, R. Frey, P. Embrechts | Quantitative Risk Management:Concepts, Techniques and Tools | Princeton University Press | 2015 | ||

Parte 1 | S. E. Shreve | Stochastic Calculus for Finance II: Continuous-Time Models | Springer, New York | 2004 | ||

Parte 1 | S. E. Shreve | Stochastic Calculus for Finance I: The Binomial Asset Pricing Model | Springer, New York | 2004 | ||

Parte 1 | J. -P. Bouchaud, M. Potter | Theory of Financial Risk - From Statistical Physics to Risk Management | University Press, Cambridge | 2000 | ||

Parte 2 | R. Cont, P. Tankov | Financial Modelling With Jump Processes | Chapman and Hall, CRC Press | 2003 | ||

Parte 2 | A. F. McNeil, R. Frey, P. Embrechts | Quantitative Risk Management:Concepts, Techniques and Tools | Princeton University Press | 2015 | ||

Parte 2 | S. E. Shreve | Stochastic Calculus for Finance II: Continuous-Time Models | Springer, New York | 2004 | ||

Parte 2 | S. E. Shreve | Stochastic Calculus for Finance I: The Binomial Asset Pricing Model | Springer, New York | 2004 | ||

Parte 2 | J. -P. Bouchaud, M. Potter | Theory of Financial Risk - From Statistical Physics to Risk Management | University Press, Cambridge | 2000 |

## 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.).

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## 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.