# QUANTITATIVE METHODS FOR FINANCIAL MARKETS (2° MODULO)

## Learning outcomes of the course unit

The aim is to provide the basic instruments for the valuation of financial derivatives.

During the course, the students will learn the basic concepts of probability theory, which are employed to construct and analyze models of financial markets under uncertainty.

The student will also learn the basic principles of arbitrage pricing and completeness in the market, notions which will be described and analyzed in detail in an elementary model but can be easily extended to more complicated frameworks. Finally, we will illustrate how to represent preferences for a rational decision maker and how to optimally select of a portfolio, given the returns and covariances of the traded assets.

At the end of the course, the student will be able to construct an elementary model for a financial market under uncertainty, to analyze the properties of this market and compute in this framework prices of derivatives and portfolio strategies.

## Prerequisites

Basic elements of calculus and financial mathematics

## Course contents summary

Introduction to probability theory: the various approaches. The axiomatic approach. conditional probability and Bayes'theorem. Random numbers: the discrete case and the continuous case. Random vectors. Basic notions on financial markets. One-period financial market. Fundamental theorems of asset pricing. Pricing of derivatives. Introduction to expected utility theory. Portfolio selection: Markowitz's model.

## Course contents

Introduction to probability theory. Classical, empirical and subjective approaches. Axiomatic approach: sample space, sigma-algebra and probability measure. Axioms of probability. conditional probability, Bayes theorem. Random numbers, measurability. Distribution function. Discrete random numbers: probability mass function. Continuous random numbers: density function.

Expectation, variance and standard deviation. Moments of a random number.

Random vectors. Independent random numbers. Covariance and correlation.

Introduction to financial market. A 1-period financial market, with zero e non-zero interest rate.

Law of one price. Arbitrage and completeness. State price densities and risk-neutral probabilities. Fundamental theorems of asset pricing. Derivatives: call and put options. Put-call parity. Forward contracts and forward prices.

Introduction to expected utility theory. Von-Neumann-Morgenstern axioms. Expected Utility theorem. Portfolio selection: Mean-variance principle. Markowitz's model.

## Recommended readings

E. CASTAGNOLI, Brevissimo Abbecedario di Matematica Finanziaria, scaricabile dalla sezione "materiali didattici" o disponibile presso il Centro fotocopie della Facoltà.

E. CASTAGNOLI, M. CIGOLA, L. PECCATI, Probability. A Brief Introduction, 2° edizione, Egea, 2009

## Teaching methods

Classes will take place online through the use of the Teams and Elly platforms. They will be both synchronous (via Teams) and asynchronous (uploaded on the Elly page of the course). Asynchronous lessons will be dedicated to a rigourous exposition of the theory, for the acquisition of knowledge.

Synchrounous classes will be mainly dedicated to the presentation of examples and exercises. Students will be encouraged to actively participate to the discussion, to develop the ability to apply knowledge and the acquisition of judgements and learning skills.

For the acquisition of technical language, the meaning of the specific terms used in the course will be illustrated.

At the beginning of the course, one of the reference texts will be uploaded to the Elly platform, as well as exercises and exam topics assigned in previous years.

The teaching materials (slides) used during the lessons will also be uploaded during the course.

To download the material it is necessary to register for the online course.

The links to the videos will be available on Elly until December 31st, 2020.

During the classes, a theoretical exposition of the contents of the course will be given.

In addition a great number of examples and exercises will be discussed,

with a particular focus on the financial examples. The students will be asked to discuss and propose possible solutions to the exercises.

## Assessment methods and criteria

The summative evaluation of the learning will be done through a written test evaluated on a 0-33 scale: the test will be a quiz on Elly, with the use of the Respondus software (alternativley, Teams will be used).

The instructions can be found on the web pages:

http://selma.unipr.it/;

http://selma.unipr.it/wp-content/uploads/Guida-Respondus.pdf;

https://elly2020.sea.unipr.it

The ID will be uploaded on a OneDrive foldere which link will be sent after the deadline for the registration.

The test will consist of 11 multiple choice questions, each of which is worth 3 points. In particular there will be

a problem, structured in 7 questions, aimed at the analysis of an elementary model of

financial market in order to test learning ability, the capacty of applying knowledge to real problems, and the independence of judgment; 2 theoretical questions on the theory of financial markets and 2 theoretical questions on probability theory to ascertain the capacity of

communicate with an appropriate technical language.

In case of return to activity in presence, the summative evaluation of the learning will be done through a written

test evaluated on a 0-33 scale.

During the test, the student is asked to: 1) solve a problem, structured in

4 questions, aimed at the analysis of an elementary model of

financial market (21pt) in order to test learning ability, the capacty of

applying knowledge to real problems, and the independence of

judgment; 2) present the theoretical arguments learned during the

course, by answering two open questions (6pt each) to ascertain the

capacity of

communicate with an appropriate technical language.

Simulation of the test in both forms can be found on Elly.

During the course, an set consisting of 5 exercises on Probability theory will be assigned. These exercises are optional, but their solution will be evaluated on a 0-30 scale.

The students who deliver the solution of the exercises before the first exam are exempt from answering to the probability questions.

A scientific calculator may be used during the test.

The text of the test with its solution will be uploaded to Elly within a week after the test.

The result of the test will be published on Elly within 10 days after the test.

Information about the lenght, the evaluation of the global exam and the awarding of the "lode" can be found in the syllabus of the whole course.

Please note that online registration for the exam is mandatory.

Written exam.

The acquisition of knowledge and understanding will be tested by means of a problem (a) and 2 theoretical questions (b).

To evaluate the learning ability, the capacity of applying the learned concepts to real problems and the independence of judgement, a problem (value: 18 pt.) will be proposed to the student, who will be asked to develop a detailed analysis of an elementary financial market and to price some derivatives in this market.

The acquisition of a technical language will be evaluate through 2 questions (6 pt. each) on theoretical topics covered in the course.