Mathematical finance (2020/2021)

Course code
Name of lecturer
Cecilia Mancini
Cecilia Mancini
Number of ECTS credits allocated
Academic sector
Language of instruction
secondo semestre (lauree magistrali) dal Mar 1, 2021 al Jun 1, 2021.

Lesson timetable

Go to lesson schedule

Learning outcomes

The course offers an introduction to arbitrage theory and its applications to financial derivatives pricing in discrete and continuous time.


1. Discrete market models

Uniperiod models: binomial and general. Multiperiod models: binomial and general.
Financial portfolios, the principle of non-arbitrage.
Derivatives: definition, examples, properties.
Absence of arbitrage.
Discrete-time martingale processes
Equivalent martingale measures and risk neutrality.
Replicable securities and valuation of derivatives
Completeness of the markets
The return of risky securities
The two fundamental asset pricing theorems

2. Market models in continuous time

Transition from discret to continuous times.
Geometric Brownian motion and modeling of empirical data
Risk quantification with a model
Ito Integral Ito, quadratic variation / covariation,
stochastic differential equations, characterization of martingales
Ito Lemma
Market model with n + 1 assets and m Brownian motions
Self-financing portfolios
Absence of arbitrage
Girsanov's theorem
Equivalent martingale measure
Replicability and pricing of derivatives
Completeness and EDP for the price function of a derivative
Delta hedging
Black and Scholes model
Formula for the price of call and put options

Useful material on the moodle page of the course: slides of the lessons, link to the notes on OneNote, exercises

Important knowledge for a successful learning: matrix calculations, linear systems, real functions of one or more real variables (in particular: continuous functions, composition of functions, partial derivatives), basic concepts of financial mathematics (interest rate, return of an investment, difference between bonds and shares of a firm), fundamental concepts of probability theory (sigma algebra, random variables, expected values, covariance, space L ^ 2 of rv, independence, conditional probabilities and expected values, equivalent probability measures, probability density, distribution function, Gaussian law, convergence in distribution, in probability, in L ^ 2, almost certain equality), basic concepts on stochastic processes (martingale, Brownian motion)

Preparatory courses: Mathematics, Financial Mathematics, Stochastic processes

Skills necessary for successful learning: willingness and ability to conduct logical reasoning in a rigorous way, and to motivate each step and the conclusions

Organization of teaching activities: lessons, exercises

Reference books
Author Title Publisher Year ISBN Note
T. Bjork Arbitrage theory in continuous time (Edizione 3) Oxford University Press 2009 978-0-199-57474-2
F. Menoncin Mercati finanziari e gestione del rischio Isedi 2006

Assessment methods and criteria

The exam consists of a written test. Also an oral examination could be compulsory, in case the teacher needs for specific insights

The written test consists of practical exercises and theoretical questions, and can cover the whole programme of the course. Using notes or books or similar material during the test is forbidden

The exam is not passed if the mark in the written test is less than 18/30.

In case of oral exam, the mark may become insufficient if inconsistencies are found with what is written. The mark score may increase if parts of exercises have not been evaluated for doubt of interpretation. Requests for further questions to increase the score are not accepted. The adequacy of requesting the necessary clarifications will be established only by the teacher.

Characteristics of the expected performance. The student is required to demonstrate a critical and in-depth knowledge of the topics covered in the course. The concepts must not be exposed mechanically but in a reasoned way, the student is expected to be able to recognize when a formula obtained for a specific example is not appropriate for the case she has to deal with. Connections among different parts of the program may be required and advanced level exercises can be (marginally) proposed.
The concise but comprehensive exposure, the rigor, the direct pointing towards the core of the matter will be particularly appreciated. Vague, inaccurate, poorly detailed or incorrect answers will be penalized

Students not attending the lectures: the examination methods are not differentiated between attending and non-attending students

The exam will be organized either as remote quiz or as written test in class. Since for the whole academic year 2020/21 the remote modality must be guaranteed for all students who request it, students wishing to adopt the remote modality are therefore strongly encouraged to notify as soon as possible.
At the closure of the exam registration list, the participants will be notified if the exam will be in presence or not.

The next exam (13-1-2021) will be in the form of a quiz on the web: 4-5 exercises, exactly as for a usual written exam.

For further details about the exam see the forum at the moodle page