# MATHEMATICAL MODELS IN FINANCE

MATHEMATICAL MODELS IN FINANCE mod. 1 (6 credits) | MATHEMATICAL MODELS IN FINANCE mod. 2 (3 credits) |

## Learning outcomes of the course unit

Mod. 1:

- To learn numerical methods for solving financial problems modeled by partial differential equations

- To acquire competence in the numerical and financial analysis of results.

Mod. 2:

At the end of the course the students should know some specific tools in order to properly investigate current research topics in the frame of kinetic equations for socio-economic sciences, and they should be able to present contents in a clear way and with a mathematically correct language.

## Course contents summary

Mod. 1: Description of some types of financial options and of differential models that model their evaluation. Description of numerical methods for differential problems applied to the Black-Scholes equation.

Mod. 2:

Introduction to kinetic equations for a simple market economy.

Investigation (from a modelling and an analytical point of view) of several interaction models for wealth exchange:

- basic deterministic model;

- model with random variables;

- model with taxation and redistribution.

## Recommended readings

Mod. 1:

La maggior parte del programma è basato su:

- P.Wilmott, J. Dewynne and S. Howison, 'Option Pricing', Oxford Financial Press, 1993

- R. Seydel, 'Tools for Computational Finance', Springer, 2009

Mod. 2:

Books or extended reviews:

- B. During, D. Matthes, G. Toscani, "A Boltzmann-type approach to the formation of wealth distribution curves", Riv. Mat. Univ. Parma 1 (2009) 199–261.

- L. Pareschi, G. Toscani, "Interacting multiagent systems. Kinetic equations and Monte Carlo methods", Oxford University Press (2013).

Research papers:

- A. Chakraborti, B.K. Chakrabarti, "Statistical mechanics of money: how saving propensity affects its distributions", Eur. Phys. J. B. 17 (2000), 167-170.

- S. Cordier, L. Pareschi, G. Toscani, "On a kinetic model for a simple market economy", J. Stat. Phys 120 (2005) 253–277.

- D. Matthes, G. Toscani, "On steady distributions of kinetic models of conservative economies", J. Stat. Phys. 130 (2008), 1087-1117.

- M. Bisi, G. Spiga, G. Toscani, "Kinetic models of conservative economies with wealth redistribution", Comm. Math. Sci. 7 (2009) 901–916.

## Teaching methods

Mod. 1: During the lectures the contents of the course will be analyzed, highlighting the difficulties related to the introduced numerical techniques. Moreover, the course will consist of a part of supervised autonomous re-elaboration consisting in the application of the numerical techniques through laboratory programming. This activity will allow students to acquire the ability to deal with "numerical" difficulties, it will allow to evaluate the reliability and consistency of the obtained results and to analyse them from a financial point of view.

Mod. 2: Class lectures.

## Assessment methods and criteria

Oral exam for both modula, to be given at the same time, with possible term paper on the topics of Modulus 1.

## Other informations

The course "Mathematical Models for Finance" is composed by two modula, which have to be simultaneously chosen by the students. The exam of the two parts will give rise to a unique final grade.

N.B.: Read the Syllabus of each single modulus for having additional information on the program of the course or on the exam.