Teaching code

4S001109

Teacher

Luca Di Persio

Coordinator

Luca Di Persio

Credits

6

Language

English

Scientific Disciplinary Sector (SSD)

MAT/06 - PROBABILITY AND STATISTICS

Period

Semester 1 dal Oct 2, 2023 al Jan 26, 2024.

Courses Single

Authorized

Lessons timetable Moodle Seminars 1

Learning objectives

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.

Prerequisites and basic notions

Basic probability theory tools. Basic knowledge of the theory of stochastic processes. Preferential: knowledge of the rudiments of stochastic calculus.

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

Didactic methods

Lectures with sharing of teaching material via slides, lecture notes, scientific articles and specific bibliographic references to deepen particular subjects and (optional) developing ongoing projects.

Learning assessment procedures

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.

Evaluation criteria

Evaluation of the understanding of mathematical tools, with particular reference to the theory of stochastic calculus, necessary for the rigorous definition of the main models of modern mathematical finance.

Criteria for the composition of the final grade

The final grade is the result of the evaluation of the final oral exam, with the possible inclusion of the ongoing assessments relating to projects (optional) carried out by the student .

Exam language

Inglese / English

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