Mathematical Finance 2

A.Y. 2020/2021
6
Max ECTS
42
Overall hours
SSD
SECS-S/06
Language
Italian
Learning objectives
Aim of this course is to cover some of the most important topics of Mathematical Finance in continuous time involving techniques related to Stochastic Calculus and dynamical optimization.
Expected learning outcomes
Pricing and hedging using probabilistic/analytic methods, of financial derivatives in complete/incomplete markets, described by diffusion time-continuous processes.
Resolution of some problems concerning dynamic optimization, using optimal control/stopping methods.
Course syllabus and organization

Single session

Responsible
Lesson period
Second semester
Streaming lectures via MS Teams.
Course syllabus
The course will focus on continuous-time mathematical models for financial markets and it consists of three main parts.

1) Continuous-time modeling
The Black and Scholes model and the derivation of the valuation formula, local volatility models and the derivation of the Dupire's formula, stochastic volatility models.

2) Optimization in continuous time models
The Merton Problem and some of its variations, utility maximization in complete markets, martingale methods for investment-consumptions problems.

3) Model ambiguity
Introduction to the problem of model risk, Skorohod Embedding Problem (SEP) and some of its solutions, introduction to Martingale Optimal Transport in continuous time and connections to SEP, applications to model-independent super-replication and option prices bounds.
Prerequisites for admission
It is highly recommended some knowledge of the foundations of mathematical finance, the theory of probability and stochastic processes.
Teaching methods
Frontal lectures
Teaching Resources
Website:
https://sites.unimi.it/burzonim/

Some reference books:

1. I. Karatzas, S. Shreve: "Methods of Mathematical Finance", Springer.
2. S. Shreve: "Stochastic Calculus for Finance II", Springer.
3. L.C.G. Rogers: "Optimal Investment", Springer.
4. D. Hobson: "The Skorokhod Embedding Problem and Model-Independent Bounds for Option Prices" in Paris-Princeton Lecture Notes.
5. A. Pascucci: "Calcolo stocastico per la finanza" Springer.
Assessment methods and Criteria
The exam consists of an oral discussion in which students will be asked to illustrate some results of the proposed program. Moreover some problems about pricing of financial instruments or dynamic optimization will be proposed, in order to evaluate the knowledge and understanding of the topics covered, as well as the ability to apply them in real finanical models.
The vote ranges out of thirty and will be communicated immediately at the end of the oral test.
SECS-S/06 - MATHEMATICAL METHODS OF ECONOMICS, FINANCE AND ACTUARIAL SCIENCES - University credits: 6
Lessons: 42 hours
Professor(s)
Reception:
on appointment
Office 1043, first floor, Math. Dept., Via Saldini 50.