Financial Mathematics

Code:

M4056

Acronym:

M4056

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2023/2024 - 2S

Active?	Yes
Responsible unit:	Department of Mathematics
Course/CS Responsible:	Master in Mathematical Engineering

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
M:ENM	14	Official Study Plan since 2023/2024	1	-	6	48	162
			2
M:M	2	Plano Oficial do ano letivo 2021	1	-	6	48	162

Teaching language

Suitable for English-speaking students

Objectives

The main objective of the course is to introduce rigorously the main concepts of Mathematical Finance in discrete and continuous time. Those concepts and the relevant mathematical tools to their analysis will be considered in the course.

Learning outcomes and competences

In the first part of the course, some preliminary topics about Probability theory, Brownian Motion and Stochastic Processes will be presented. The main core of the course is divided in two parts: in the first part the course will focus in single period models for financial markets, and the simplicity of these models will facilitate the introduction of the main concepts of mathematical finance. The final part will be devoted to the study of multi period discrete time mathematical models for financial market as well as continuous time models, studying the differences and relations between all the considered models.

Working method

Presencial

Pre-requirements (prior knowledge) and co-requirements (common knowledge)

Linear Algebra and Probability.

Program

Probability theory: random variable and filtrations; brownian motion; stochastic differential processes; optimum control. Single-stage financial markets model: model description; arbitrage; risk neutral measures; contingent claims evaluation, complete and incomplete markets; discrete multi-stage financial markets model; model description; stochastic processes and filtrations; return and dividend; martingales; binomial model; Markov model. Continuous time financial markets model: model description; relevant stochastic processes: portfolio; wins and wealth; arbitrage; complete financial markets; financial markets with infinite horizon; contingent claims evaluation in complete markets.

Mandatory literature

S Pliska; Introduction to Mathematical Finances, Blackwell Publishers
H Follmer and A Schied; Stochastic Finance
M Musiela and M Rutkowski; Martingale Methods in Financial Modelling
R. A. Dana and M. Jeanblanc; Financial Markets in Continuous Time

Complementary Bibliography

Philip Protter; Stochastic integration and differential equations
I. Karatzas and S. E. Shreve; Brownian motion and Stochastic Calculus, Springer, 1988
B. Oksendal; Stochastic Differential Equations, Springer, 2002
R. J. Elliott and P. E. Kopp; Mathematics of Financial Markets, Springer, 2005
T. Bjork; Arbitrage Theory in Continuous Time, Oxford University Press, 1998
Huyên Pham; Continuous-time Stochastic Control and Optimization with Financial Applications, Springer, 2009

Teaching methods and learning activities

Presentation of the topics of the course and their discussion with the students.

Evaluation Type

Distributed evaluation without final exam

Assessment Components

designation	Weight (%)
Apresentação/discussão de um trabalho científico	20,00
Participação presencial	80,00
Total:	100,00

Amount of time allocated to each course unit

designation	Time (hours)
Apresentação/discussão de um trabalho científico	3,00
Estudo autónomo	114,00
Frequência das aulas	45,00
Total:	162,00

Eligibility for exams

No applicable

Calculation formula of final grade

20 % - Written work and its presentation.

80% - Participation in classes.

Special assessment (TE, DA, ...)

Additional presentation of a written work (100%).

Classification improvement

Additional presentation of a written work (100%).