Teaching code

4S001142

Teacher

Cecilia Mancini

Coordinator

Cecilia Mancini

Credits

9

Language

Italian

Scientific Disciplinary Sector (SSD)

SECS-S/06 - MATHEMATICAL METHODS OF ECONOMICS, FINANCE AND ACTUARIAL SCIENCES

Period

Secondo semestre (lauree magistrali) dal Feb 26, 2024 al May 24, 2024.

Courses Single

Authorized

Lessons timetable Moodle Seminars 0

Learning objectives

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

Prerequisites and basic notions

Preparatory courses: Mathematics, Financial Mathematics, Stochastic processes
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)

Program

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

Bibliography

Didactic methods

Organization of teaching activities: lessons, exercises, quizzes during the course
Useful material on the moodle page of the course: slides of the lessons, link to the notes on OneNote, exercises
Skills necessary for successful learning: willingness and ability to conduct logical reasoning in a rigorous way, and to motivate each step and the conclusions

Learning assessment procedures

The exam consists of a written test and a series of 3 quizzes during the course. Also an oral examination could be compulsory, in case the teacher needs for specific insights
The final 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 tests is forbidden
The exam is not passed if the mark in the final written test is less than 18/30.
Students not attending the lectures: the examination methods are not differentiated between attending and non-attending students

Evaluation criteria

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, connections among different parts of the program may be required and exercises which are different form the ones solved during the lessons can be proposed.
The concise but comprehensive exposure, the rigor, the direct pointing towards the core of the matter will be particularly appreciated. Vague, inaccurate, poorly justified or incorrect answers will be penalized

Criteria for the composition of the final grade

The final exam mark is awarded considering the outcome of the final written test and adding the possible bonus cumulated with the quizzes.
In case the teacher call also to an oral exam, the mark may become insufficient if inconsistencies are found with what is written in the final test. 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.

Exam language

italiana