# QUANTITATIVE METHODS FOR FINANCIAL MARKETS

## Learning outcomes of the course unit

SKILLS TO BE DEVELOPED AND LEARNING OUTCOMES EXPECTED

1) Knowledge and understanding.The course aims to provide the basic

tools most suitable for the analysis of some fundamental aspects of

monetary and financial market. In module I, particular attention will be paid to time

series of financial issues: exchange rates, interest rates, prices and

equity returns, prices and yields of derivatives.

During the II module, 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.

2) Ability to apply knowledge and understanding . At the end of the

course, the student will be able to implement in an autonomous way the

techniques described above. The student will have therefore

developed specific skills, they are associated with critical skills for

diagnostic, which are essential ingredients in building a good statistical and probability

model, with the possible assistance of the appropriate level of computer

tools.

3) Making judgments .At the end of the course, the student will be able to

perform independently all the considerations regarding the problems of

analysis of financial time series correctly interpret the results of such analyses, even when made by other

users or experts. The student will also 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.

4) Communication skills . At the end of the course, the student will be

able to use appropriate technical language in communicating with the

operators of financial markets.

5) Learning skills. We want to give the student the opportunity to

assimilate the key results of the statistical and probability theory that

form the basis of building a statistical model of a financial model under uncertainty. At the end of the course,

the student will have acquired the key concepts to be able to accurately

use quantitative tools, if they become necessary in the solution of

concrete problems of a financial nature.

## Prerequisites

Knowledge of basic elements of calculus and financial mathematics, basic descriptive and inferential statistics

## Course contents summary

Elementary theory of stochastic processes for stationary series.

Empirical evidence of the observed time series.

Overview of analysis of the trend of stock market prices and moving averages.

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

Elementary theory of stochastic processes for stationary series:

recalls elements of probability 'for random vectors;

transformation of univariate and multivariate random variables;

gaussian and White Noise processes; brief introduction to non-stationary processes of type Random Walk.

Empirical evidence of the observed time series:

empirical characteristics of the time series of financial returns and

formulas combinations of multi-period returns;

the shape of the distribution of returns; test of symmetry, kurtosis,

and normality; the time dependence (linear and nonlinear) of returns; autocorrelation function and tests of significance 'associates; autoregressive processes for stationary series of returns and

transforms associated with them.

Overview of analysis of the trend of stock market prices and moving averages.

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 utiltiy theory. Von-Neumann-Morgenstern axioms. Expected utility theorem. Portfolio selection: mean-variance principle. Markowitz's model.

## Recommended readings

GOZZI G., Taluni argomenti di Metodi Quantitativi per i Mercati Finanziari, Libreria Medico Scientifica , Parma, Edizione 2014 e materiale didattico integrativo reso disponibile durante il corso caricato sul sito docente.

E. CASTAGNOLI, Brevissimo Abbecedario di Matematica Finanziaria, downloadable from Elly course page) or available at the “Centro fotocopie” of the Department..

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

## Teaching methods

Acquisition of knowledge: a theoretical (oral) exposition of the several topics will be given, supported by a detailed discussion of examples and exercises.During Module 1, it will be used the software Microsoft Excel e di Gretl.

Acquisition of the ability to apply knowledge: tutorials and exercise sessions.

Acquisition of judgements: the students will be asked to choose the right instrument to solve a problem and to discuss strenghts and weaknesses of the several tools.

Acquisition of learing skills: for each topic, several problems will be shown and a critical analysis of the possibile solution will be carried on.

Acquisition of technical language: technical terms will be explained and used during the course.

## Assessment methods and criteria

The exam is in written form.

The student will be required to complete two test, one per module, consisting in exercises and theoretical question.

## Other informations

Elective assignment will be proposed to the students, to be possibly solved in group. The results will be taken into account in the final evaluation.