Mathematical Finance 2
A.Y. 2024/2025
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.
Resolution of some problems concerning dynamic optimization, using optimal control/stopping methods.
Lesson period: Second semester
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course cannot be attended as a single course. Please check our list of single courses to find the ones available for enrolment.
Course syllabus and organization
Single session
Responsible
Lesson period
Second semester
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. Pricing of Asian/American options.
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 Martingale Optimal Transport in discrete/continuous time; applications to model-independent super-replication and option prices bounds.
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. Pricing of Asian/American options.
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 Martingale Optimal Transport in discrete/continuous time; 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://mmaggisfm2.ariel.ctu.unimi.it/
Some reference books:
1. I. Karatzas, S. Shreve: "Methods of Mathematical Finance", Springer.
2. S. Shreve: "Stochastic Calculus for Finance II", Springer.
3. Bijork: "Arbitrage Theory in Continuous Time", Oxford University Press.
4. A. Pascucci: "Calcolo stocastico per la finanza" Springer.
https://mmaggisfm2.ariel.ctu.unimi.it/
Some reference books:
1. I. Karatzas, S. Shreve: "Methods of Mathematical Finance", Springer.
2. S. Shreve: "Stochastic Calculus for Finance II", Springer.
3. Bijork: "Arbitrage Theory in Continuous Time", Oxford University Press.
4. 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.
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
Professors:
Frittelli Marco, Maggis Marco
Professor(s)
Reception:
on appointment
Office 1043, first floor, Math. Dept., Via Saldini 50.