## 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, 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

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

Squassina 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 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 (2015/2016)

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:

##### Esercitazioni

Credits

1

Period

I semestre

Academic staff

Michele Bonollo

##### Teoria 1

Credits

2

Period

I semestre

Academic staff

Luca Di Persio

##### Teoria 2

Credits

3

Period

I semestre

Academic staff

Michele Bonollo

### ???OrarioLezioni???

## 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

## 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.