# MATHEMATICAL FINANCE mod.1

## Learning outcomes of the course unit

At first basic quantitative tools are introduced in order 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.

The second part of the Course aims to provide an overview on the most recent valuation models of financial stocks and derivatives. Starting from the axiomatic foundations, it analyzes the market with the intention of showing students how to formalize some financial phenomena.

The course has as main objective the study of the main methods for the numerical approximation of partial differential equations and stochastic differential equations.

In particular, we will analyze the main differential models for the evaluation of financial securities and derivatives. You will have several hours of computer lab, during which students can experience the main theoretical concepts presented and deepen their understanding and use through the development of application programs that use the software Matlab.

## Course contents summary

Functions in several variables.

Maximum and minimum points 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, 2015.

P. Wilmott, Introduzione alla Finanza quantitativa, Egea, Milan, 2003.

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

## Teaching methods

Oral and practical lessons.

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.

## Assessment methods and criteria

Written examination with possible integration by Matlab programming.

Assessment of the achievement of learning outcomes shall be 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 can be 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 28/30. The student can supplement his vote by presenting a paper implementing in Matlab (optionally) one of the exercises proposed in class.