Mathematics in Finance

Year
1
Academic year
2023-2024
Code
02002089
Subject Area
Mathematics
Language of Instruction
Portuguese
Mode of Delivery
Face-to-face
Duration
SEMESTRIAL
ECTS Credits
6.0
Type
Compulsory
Level
2nd Cycle Studies - Mestrado

Recommended Prerequisites

Basic courses in (Differential and Integral) Calculus, Linear Algebra, Differential Equations and Probability and Statistics (in particular Stochastic Processes and Calculus).

Teaching Methods

The classes are essentially of expository style and include examples and exercises to apply the material being taught.

Extensive tutorial time is offered to the students to support the solution of the homework assignments and preparation for the various exams.

Learning Outcomes

The main goal is teaching the basic principles of financial derivatives modeling and the underlying mathematical tools. One aims also at showing how to collect, to process and to analyze financial data and how to use numerical techniques to solve problems within the scope of this modeling.

The course aims at developing the following skills: knowledge of mathematical results; ability to formulate and solve problems; building and using mathematical models for real world situations. On the personal level it also allows to develop self-learning skills and independent thinking.

Work Placement(s)

No

Syllabus

(1st Part – Basics) Stochastic and stochastic differential equations modeling of financial assets. Arbitrage.

The Black-Scholes equation and formula for European options. Risk neutrality. Implied volatility and implied density. Hedging (put-call parity and dynamical hedging). The binomial method.

(2nd Part – Extensions) Extensions of the Black-Scholes model (options on assets paying dividends, options on futures). Exotic options. American options. Options dependent on the asset trajectory. Bonds and term structure models. Options on bonds and other interest rate products. Volatility estimation.

Head Lecturer(s)

Dmitry Vorotnikov

Assessment Methods

Continuous assessment
During the semester there are two mid-term exams (50-75% of the final grade) and a set of homework assignments (50-25% of the final grade) given every two or three other weeks and handed in individually: 100.0%

Final assessment
The exercises in the homework assignments are mathematical problems or short numerical tasks. The final exam option consists of a single exam (100% of the final grade).: 100.0%

Bibliography

.N. Vicente, Introdução à Matemática Financeira, Departamento de Matemática da FCTUC, 2006/2007.

L.D. Abreu, Modelação de Preços de Derivados Financeiros, Departamento de Matemática da FCTUC, 2011.

T. Björk, Arbitrage Theory in Continuous Time, Oxford University Press, 1998.

D.J. Higham, An Introduction to Financial Option Valuation: Mathematics, Stochastics and Computation, Cambridge University Press, 2004.

J.C. Hull, Options, Futures, and Other Derivatives, Prentice-Hall, 2003.

B. Øksendal, Stochastic Differential Equations - An Introduction with Applications, quinta edição, Springer-Verlag, 2000.

P. Wilmott, S. Howison, J. Dewynne, The Mathematics of Financial Derivatives, Cambridge University Press, 1995.