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.
Type D and Type F activities
This information is intended exclusively for students already enrolled in this course.If you are a new student interested in enrolling, you can find information about the course of study on the course page:
Laurea magistrale in Mathematics - Enrollment from 2025/2026Le attività formative di tipologia D sono a scelta dello studente, quelle di tipologia F sono ulteriori conoscenze utili all’inserimento nel mondo del lavoro (tirocini, competenze trasversali, project works, ecc.). In base al Regolamento Didattico del Corso, alcune attività possono essere scelte e inserite autonomamente a libretto, altre devono essere approvate da apposita commissione per verificarne la coerenza con il piano di studio. Le attività formative di tipologia D o F possono essere ricoperte dalle seguenti attività.
1. Insegnamenti impartiti presso l'Università di Verona
Comprendono gli insegnamenti sotto riportati e/o nel Catalogo degli insegnamenti (che può essere filtrato anche per lingua di erogazione tramite la Ricerca avanzata).
Modalità di inserimento a libretto: se l'insegnamento è compreso tra quelli sottoelencati, lo studente può inserirlo autonomamente durante il periodo in cui il piano di studi è aperto; in caso contrario, lo studente deve fare richiesta alla Segreteria, inviando a carriere.scienze@ateneo.univr.it il modulo nel periodo indicato.
2. Attestato o equipollenza linguistica CLA
Oltre a quelle richieste dal piano di studi, per gli immatricolati dall'A.A. 2021/2022 vengono riconosciute:
- Lingua inglese: vengono riconosciuti 3 CFU per ogni livello di competenza superiore a quello richiesto dal corso di studio (se non già riconosciuto nel ciclo di studi precedente).
- Altre lingue e italiano per stranieri: vengono riconosciuti 3 CFU per ogni livello di competenza a partire da A2 (se non già riconosciuto nel ciclo di studi precedente).
Tali cfu saranno riconosciuti, fino ad un massimo di 6 cfu complessivi, di tipologia F se il piano didattico lo consente, oppure di tipologia D. Ulteriori crediti a scelta per conoscenze linguistiche potranno essere riconosciuti solo se coerenti con il progetto formativo dello studente e se adeguatamente motivati.
Gli immatricolati fino all'A.A. 2020/2021 devono consultare le informazioni che si trovano qui.
Modalità di inserimento a libretto: richiedere l’attestato o l'equipollenza al CLA e inviarlo alla Segreteria Studenti - Carriere per l’inserimento dell’esame in carriera, tramite mail: carriere.scienze@ateneo.univr.it
3. Competenze trasversali
Scopri i percorsi formativi promossi dal TALC - Teaching and learning center dell'Ateneo, destinati agli studenti regolarmente iscritti all'anno accademico di erogazione del corso https://talc.univr.it/it/competenze-trasversali
Modalità di inserimento a libretto: non è previsto l'inserimento dell'insegnamento nel piano di studi. Solo in seguito all'ottenimento dell'Open Badge verranno automaticamente convalidati i CFU a libretto. La registrazione dei CFU in carriera non è istantanea, ma ci saranno da attendere dei tempi tecnici.
4. Periodo di stage/tirocinio
Oltre ai CFU previsti dal piano di studi (verificare attentamente quanto indicato sul Regolamento Didattico): qui informazioni su come attivare lo stage.
Insegnamenti e altre attività che si possono inserire autonomamente a libretto
years | Modules | TAF | Teacher | |
---|---|---|---|---|
1° | Genetics | D |
Massimo Delledonne
(Coordinator)
|
|
1° 2° | Algorithms | D |
Roberto Segala
(Coordinator)
|
years | Modules | TAF | Teacher |
---|---|---|---|
1° 2° | Algorithms | D |
Roberto Segala
(Coordinator)
|
1° 2° | LaTeX Language | D |
Enrico Gregorio
(Coordinator)
|
1° 2° | Organization Studies | D |
Serena Cubico
(Coordinator)
|
1° 2° | History and Didactics of Geology | D |
Guido Gonzato
(Coordinator)
|
years | Modules | TAF | Teacher | |
---|---|---|---|---|
1° 2° | Advanced topics in financial engineering | F | Not yet assigned | |
1° 2° | ECMI modelling week | F | Not yet assigned | |
1° 2° | ESA Summer of code in space (SOCIS) | F | Not yet assigned | |
1° 2° | Google summer of code (GSOC) | F | Not yet assigned | |
1° 2° | Mathematics mini courses |
Sisto Baldo
(Coordinator)
|
||
1° 2° 3° | Python programming language | D |
Giulio Mazzi
(Coordinator)
|
Mathematical finance (2021/2022)
Teaching code
4S001109
Teacher
Coordinator
Credits
6
Language
English
Scientific Disciplinary Sector (SSD)
MAT/06 - PROBABILITY AND STATISTICS
Period
Primo semestre dal Oct 4, 2021 al Jan 28, 2022.
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
Bibliography
Examination Methods
Oral exam with written exercises:
the exam is based on open questions as well as on the resolution of written exercises to be solved during the test itself
and/or on questions and exercises based on specific projects presented during the exam and previously agreed with the professor. 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.