# 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:

- Knowledge of language and technical procedures typical of Mathematical Finance. Ability to understand numerical methods for the resolution of differential problems arising from derivatives evaluation in Finance.

- Ability to apply knowledge and understanding in the critical analysis of the numerical results obtained giving a financial interpretation.

- Autonomy of judgment in evaluating the approximation algorithms and the obtained results also through comparison with one's peers.

- Ability to communicate clearly the acquired concepts and to discuss the obtained results.

- Ability to learn limits and advantages of models and methods of resolution and to apply them in different working and scientific contexts.

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.

Specifically, these are the expected skills after the course:

- Knowledge and understanding: students will know in detail and will autonomously use mathematical tools in the frame of mathematical models for simple market economies; furthermore they will acquire a good level of understanding of the most recent mathematical contents and theories on the course topics, which enables them to read and understand advanced texts and research articles, and to elaborate then original ideas in specific research contexts.

- Applying knowledge and understanding: students will be able to produce rigorous proofs of (even new) mathematical results and to face new problems in the frame of equations of kinetic or Black-Scholes type in socio-economic sciences, proposing new models and studying their properties owing also to appropriate numerical algorithms.

- Making judgements: students will be able to build up non trivial logical proofs, and will be capable to recognize correct or rather fallacious steps of the proofs.

- Communication: students will have to present the main topics of the course in a clear and mathematically correct way.

- Lifelong learning skills: the course will help students to form a flexible mentality allowing them to be employed in work environments that require the ability to face ever-new problems.

## 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.