## Learning objectives

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 unit content

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.

## Full programme

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.

## Bibliography

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

Oral lectures.

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

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.

## Other information

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