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 tools of probability calculus and stochastic process theory, e.g., definitions and use of probability spaces and multidimensional random variables, concepts of dependence/independence, marginal distributions, Markov chains, and limit theorems. Knowledge of the rudiments of stochastic calculus, e.g., definitions of filtering, the martingale process, and continuous-time random dynamics, is preferred, but not required.

Program

[1] Rudiments of Stochastic Analysis
Basic notions of stochastic processes
Stochastic processes: main discrete-time and continuous-time examples
Definition of stochastic integral and Itô-Döblin lemma Stochastic Differential Equations (SDEs): introduction and examples, e.g., the case with multiplicative Gaussian noise and linear coefficients
Solution of SDEs as Markov processes
Feynman-Kac formula
Girsanov's theorem
Stochastic control: introduction and examples, e.g., dynamic programming principle, Pontryagin's maximum principle
[2] Discrete-time models
Options, value process, hedging strategies, completeness, arbitrage
Fundamental theorems of Asset Pricing (discrete-time)
Binomial trees
Random walk and Black-Scholes pricing formula, with its derivation from binomial tree analysis
[3] Brownian motion (BM)
Main properties of the BM, e.g., filtering generated by the BM, martingale property, quadratic variation, volatility, and reflection properties
[4] Continuous-time models
Black-Scholes-Merton equation
Stochastic dynamics for portfolio evolution
Sensitivity analysis (Greeks)
The Martingale approach
Hedging and replication strategies
Stock market models
Siegel paradox
Dynamics of exotic options
[5] Interest rate models
Markov models for short-term rates
Merton model
Stochastic interest rate for the Black and Scholes model Hedging portfolio
Changing numeraire, even in the presence of multiple sources of risk
Caps, floors, collars
Models for the dynamics of interest rates
Vasicek model
Cox-Ingersoll-Ross model
Modeling of forward rates
Arbitrage models for term structure
Heath-Jarrow-Morton framework
Hull-White extension of the Vasicek model
[6] Portfolio choice and asset prices
Bachelier and Samuelson models
Utility functions
Static programming approach and Merton problem
Expected utility maximization
[7] Miscellaneous
Option valuation in Gaussian models
Forward LIBORs
Swap rate modeling
Mean Field Games approach to systems of interacting financial agents
Calibration for interest rate models
Stochastic control and financial models
Stochastic volatility models and applications
Expansions, e.g., polynomial, asymptotic, for financial models SDEs on networks with applications in finance

Didactic methods

Lectures will be held, with the subsequent sharing of the teaching materials produced during the individual lessons. To complement the work completed in class, further readings, e.g., scientific articles, lecture notes, and specific bibliographical references for further study and the development of ongoing projects, may be shared.

Learning assessment procedures

The exam will be an oral one, during which questions will be asked, the answers of which may require the student to write down definitions, theorems, and their proofs, analytical derivations, etc.
The exam will be based on the syllabus actually covered during lectures and will consist of open-ended questions.
Alternatively, individual students may arrange with the instructor to prepare a seminar, focusing on specific topics in depth.
During the seminar, the professor may ask questions both about the material presented and about concepts at the intersection between it and the scientific corpus characterizing the entire course.

Evaluation criteria

Assessment of understanding of the 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 will be the result of the evaluation of the final oral exam, with possible inclusion of evaluations based on the discussion of optional in-depth projects that the individual student wishes to agree with the instructor.

Exam language

Inglese