# MATHEMATICAL FINANCE mod.1

## Learning outcomes of the course unit

Skills to be developed and expected learning outcomes:

a) Knowledge and ability to understand.

In the first part of the course the student will acquire the basic quantitative tools to approach quantitative finance. In particular functions in several variables are presented.

Modern finance is today an extremely rich field and often uses complex mathematical tools.

The main purpose of the first part of the course is to present the main topics of quantitative finance in a clear and accessible way with the aim to stimulate intuition without abandoning the aspects of formalization that are now indispensable for anyone wishing to operate on financial markets.

In the second part of the Course the student will study the most recent valuation models of financial stocks and derivatives. Starting from the axiomatic foundations, he will analyze the market with the intention of learning how to formalize some financial phenomena.

Finally, he will study the main methods for the numerical approximation of partial differential equations and stochastic differential equations.

In particular, he will analyze the main differential models for the evaluation of financial securities and derivatives.

He will attend several hours of computer lab, during which the student can experience the main theoretical concepts presented and deepen her/his understanding and use through the development of application programs that use the software Matlab.

b) Ability to apply knowledge and understanding.

Students can apply what they have learned to work on the most common financial products, both from a theoretical point of view (knowledge of derivatives and, more generally, markets), and from a technical-quantitative point of view (knowledge of the most used technicalities in modern finance).

c) Autonomy of judgment.

The student will be able to critically evaluate the different situations and different financial products offered by the markets. It will also be able to calculate the price of a derivative under non-arbitrage conditions.

d) Communication skills.

The student will acquire a wealth of knowledge, methods and skills in solving problems absolutely essential for the training, presentation and communication of a quantitative analyst in the financial markets.

e) Ability to learn.

The student will experiment with a traditional teaching method together with lessons in the laboratory and the possibility of presenting an assignment to be solved by MatLab. He will therefore be stimulated from the point of view of learning on different sides.

In short: at the end of the course, the student:

- will have understood and adopted the main models presented in the course;

- will be able to solve problems of a practical nature (in the form of exercises and IT applications);

- will have achieved a good judgment autonomy;

- will be able to communicate clearly what has been learned.

## Prerequisites

Basic elements of Financial Calculus and Theory of Probabilities.

## Course contents summary

Functions in several variables.

Optimization with and without constraints.

Markets.

Shares, goods, currencies, forward, futures contracts and options.

Options: the binomial model.

The binomial tree. The value of an option. Arbitrage and non-arbitrage.

The drift. Volatility. The Wiener process. Basic knowledge of stochastic calculus. Ito's lemma. Random walks.

The Black and Scholes model.

Towards elimination of risk: hedging.

Elements of stochastic calculus.

Stochastic differential equations. Kolmogorov equation.

Numerical methods for partial differential and stochastic equations. Monte Carlo Method and Finite Difference Method.

Valuation of derivative securities.

Plain vanilla options, path dependent and other exotic options.

For each topic applications are provided.

## Recommended readings

- E. Castagnoli, M. Cigola, L. Peccati, La matematica in azienda 2: complementi di analisi, Egea, Milan, 2010.

- John C. Hull, Opzioni, futures e altri derivati, Pearson, Milan, 2018.

Lecture notes for the second part of the course will be provided by the teacher and made available on Elly.

## Teaching methods

Oral and practical lessons. Exercises in the IT laboratory.

The course provides several hours of computer lab, during which students can experiment with the main theoretical concepts presented and deepen their understanding and use through the development of application programs that use the software Matlab.

Students can create a MatLab dissertation with a group work (2 or 3 students per group) that will be evaluated during the exam.

The Syllabus, the detailed program of the course, the exams already assigned and, for the second part of the course, the lecture notes and some videos of the Matlab classes are published on Elly at the beginning of the course.

## Assessment methods and criteria

Written examination with integration by Matlab programming.

Assessment of the achievement of learning outcomes is conducted mainly through written tests, in the form of open questions and exercises aimed at testing the ability relating to the application of knowledge, the independence of judgment and the ability to communicate with technical language appropriately.

The verification is integrated by means of the implementation (possibly in groups) of a Matlab program in order to check the ability to solve operational problems.

The students may take the examination with a unique test or with two partial tests at the end of the first and the second period of lessons.

In particular, in the first part of the exam there are one theoretical question and one exercise or two theoretical questions. Indicative marks: theoretical questions: 7,5-9/30; exercise 6-7,5/30.

The second part of the test is composed of two theoretical questions. The maximum achievable score is 13,5/30. The student must supplement his vote by presenting a paper implementing in Matlab one of the exercises proposed in class.

The University will send to the students an email message to their University email address with the result of the exam (through Essetre system). The students can reject the result in a week, through an online procedure clearly described in the message.