Accreditations
Check here the detailed study plan
The curricular part of the Master is divided into four academic quarters, which make up 78 ECTS, that is, it starts in September of the year 2024 and ends only in December of the year 2025.
The last stage of the Master's degree is dedicated to the completion of the dissertation (42 ECTS), and it is necessary to complete 120 ECTS to obtain the Master's Degree.
Students' academic path is only differentiated in the 1st quarter according to their area of origin: Mathematics (Path A) versus Economics (Path B).
There is the possibility of issuing a Diploma of Postgraduate Studies of the 2nd Cycle for the completion with the use of 78 academic credits  curricular part of the master's degree in Financial Mathematics.
The curricular units will be taught in Portuguese, except for those that include foreign students or teachers, which will be taught in English.
Schedule
Fridays: 5.30 p.m. to 9.15 p.m.
Saturdays: 9 a.m. to 12.45 p.m.
Programme Structure for 2024/2025
Curricular Courses  Credits  

Financial Investments
6.0 ECTS

Parte Escolar > Mandatory Courses  6.0 
Measure Theory
4.0 ECTS

Parte Escolar > Mandatory Courses  4.0 
Stochastic Calculus in Finance I
7.0 ECTS

Parte Escolar > Mandatory Courses  7.0 
Stochastic Calculus in Finance II
7.0 ECTS

Parte Escolar > Mandatory Courses  7.0 
Partial Differential Equations in Finance
7.0 ECTS

Parte Escolar > Mandatory Courses  7.0 
Exotic Options
7.0 ECTS

Parte Escolar > Mandatory Courses  7.0 
Optimization
3.0 ECTS

Parte Escolar > Mandatory Courses  3.0 
Programming
3.0 ECTS

Parte Escolar > Mandatory Courses  3.0 
Credit Risk
3.0 ECTS

Parte Escolar > Mandatory Courses  3.0 
Market Risk
3.0 ECTS

Parte Escolar > Mandatory Courses  3.0 
Risk Theory for NonLife Insurance
6.0 ECTS

Parte Escolar > Paths > 1st Cycle in Economics Or Related  6.0 
Topics of Real Analysis
4.0 ECTS

Parte Escolar > Paths > 1st Cycle in Economics Or Related  4.0 
Derivatives and Risk Management
6.0 ECTS

Parte Escolar > Paths > 1st Cycle in Mathematics Or Related  6.0 
Fundamentals of Economics
2.0 ECTS

Parte Escolar > Paths > 1st Cycle in Mathematics Or Related  2.0 
Financial Markets
2.0 ECTS

Parte Escolar > Paths > 1st Cycle in Mathematics Or Related  2.0 
Dissertation in Mathematical Finance
42.0 ECTS

Parte Escolar > Mandatory Courses  42.0 
Econometrics of Financial Markets
6.0 ECTS

Parte Escolar > Mandatory Courses  6.0 
Numerical Methods
6.0 ECTS

Parte Escolar > Mandatory Courses  6.0 
Models of the Term Structure of Interest Rates
6.0 ECTS

Parte Escolar > Mandatory Courses  6.0 
Financial Investments
At the end of this learning unit?s term, the student must be able:
1. To explain the concept of utility function, marginal utility and risk aversion;
2. To explain the choice theory and the investor's cannonical problem;
3. To characterize the portfolio choise theories and determine the efficient portfolios;
4. To characterize the main asset pricing models.
1. Individual Choice Theory
2. Individual Portfolio Decision
3. Capital Asset Pricing Model
4. Arbitrage Pricing Theory and Factor Models
5. Pricing in Complete Markets
The final grade is computed as follows:
 Final Exam: 70%
 Quizzes, Homework, Class participation: 30%
The exams are closedbook and closednotes. However, you may use a formula sheet.
Title: 3. Pires, Cesaltina, 2011, Mercados e Investimentos Financeiros, Escolar Editora.
2. Cochrane, J.H., 2005, Asset Pricing, Princeton University Press.
1. Danthine, JP and J. Donaldson, 2014, Intermediate Financial Theory, 3rd edition, Elsevier Academic Press.
Authors:
Reference:
Year:
Title: 3. Bodie, Kane, and Marcus, 2021, Investments, 12th Edition, McGrawHill.
2. Huang, Cf and R. H. Litzenberger, 1988, Foundations for Financial Economics, Prentice Hall.
1. Ingersoll, J.E., 1987, Theory of Financial Decision Making, Rowman & Littlefield.
Authors:
Reference:
Year:
Measure Theory
At the end of this learning unit, the student must be able to:
1. understand concept meaning.
2. argue and calculate based on assumptions.
3. concieve proofs in problem solving.
1. Sigmaalgebras. Mesurable spaces and functions.
2. Finite and sigmafinite measures. Measure properties. Measure and probability spaces.
3. Integral of a function on a measure space. Integral properties. Integrability.
4. Lebesgue integral on the real line.
5. Comparison with Riemann integral.
6. Product measures and Fubini's theorem.
7. Density functions and associated measures.
8. Theorem of RadonNikodym.
9. Change of variables. The spaces L1 and L2.
10. Sequence of functions convergence.
Regular grading system:
 One individual exam (100%)
It is considered that a student has course approval if he has a grade equal or above 9.5 points.
Title:  Outros textos de apoio teórico/práticos a facultar pelo docente durante o trimestre;
 M. Ramos, Teoria da Medida, Texto de Apoio às Aulas, 2005;
Authors:
Reference:
Year:
Title:  D. Williams, Probability with Martingales, Cambridge Mathematical Textbooks, 1995 (quarta edição).
 Seán Dineen, Probability Theory in Finance, Graduate Studies in Mathematics, Volume 70, AMS, 2005.
 M. Capinski, E. Kopp, Measure, Integral and Probability, SpringerVerlag, 2004 (segunda edição).
Authors:
Reference:
Year:
Stochastic Calculus in Finance I
At the end of this learning unit, the student must be able to:
1. Explain clearly concepts from advanced probability theory and stochastic calculus.
2. Give valid proofs of certain theoretical results.
3. Apply stochastic calculus to problems in finance.
1. Basic notions of Probability Theory.
2. Conditional Expectation.
3. Discretetime Martingales.
4. Continuous time stochastic processes.
5. Brownian motion.
6. Itô´s stochastic integral.
7. Itô formula.
8. Martingale representation theorem.
9. Stochastic differential equations.
10. Girsanov theorem.
11. FeynmanKac formula.
Regular grading system:
 One individual exam
Students that fail or want to improve their grade in the regular grading system have one additional moment to pass: a resit exam, that is worth 100% of the final grade.
In any of the evaluation systems (regular or resit exam) it is considered that a student has course approval if he has a grade equal or above 9.5 points.
Title: Isabel Simão, Cálculo Estocástico em Finanças I, Texto de apoio às aulas, 2006.
Authors:
Reference:
Year:
Title: B. Oksendal, Stochastic Differential Equations and Applications, SpringerVerlag, 5a edição, 1998.
T. Mikosch, Elementary Stochastic Calculus with Finance in View, World Scientific, 1998.
D. Lamberton and B. Lapeyre, Stochastic Calculus Applied to Finance, Chapman and Hall/CRC, 1996.
Authors:
Reference:
Year:
Stochastic Calculus in Finance II
At the end of this learning unit, the student must be able to:
1. Understand the role of martingales in the theory of derivative pricing.
2. Calculate the value of European and American options using the binomial model .
3. Calculate the value of European options using the BlackScholes model.
1. Discretetime models.
2. The CoxRossRubinstein model.
3. Optimal stopping and American options.
4. The BlackScholes model.
Regular grading system:
 One individual exam
Students that fail or want to improve their grade in the regular grading system have one additional moment to pass: a resit exam, that is worth 100% of the final grade.
In any of the evaluation systems (regular or resit exam) it is considered that a student has course approval if he has a grade equal or above 9.5 points.
Title:  Isabel Simão, Cálculo Estocástico em Finanças II, Texto de apoio às aulas, 2006.
Authors:
Reference:
Year:
Title:  M. Musiela e M. Rutkowski, Martingale Methods in Financial Modelling, SpringerVerlag, 1998.
 A. Etheridge, A Course in Financial Calculus, Cambridge University Press, 2002.
T. Björk, Arbitrage Theory in Continuous Time, Oxford University Press, 1998.
D. Lamberton and B. Lapeyre, Stochastic Calculus Applied to Finance, Chapman and Hall/CRC, 1996.
Authors:
Reference:
Year:
Partial Differential Equations in Finance
At the end of this learning unit, the student must be able to:
1. recognize various types of equations and problems.
2. Solve certain simple problems, using the method of characteristics and the method of separation of variables in the heat equation.
I. Ordinary Differential Equations:
First order equations: separation of variables and linear equations.
Second order equations: initial conditions and boundary value problems.
II. First order Partial Differential Equations (two variables):
Example: transport equation.
Planar vector fields and integral curves.
Method of characteristics.
III. Second order linear Partial Differential Equations (two variables):
Examples: heat equation, wave equation, Laplace equation.
Other examples: ReactionDiffusion equations; BlackScholes equation.
Classification: characteristics and canonical forms.
Boundary and initial conditions.
Method of separation of variables.
Fourier series.
Solution of the heat equation in a bounded interval.
Fourier integral.
Solution of the heat equation in an unbounded interval.
Solution of the BlackSholes equation for a European option.
Notion of free boundary and the price of an American option.
Regular grading system:
 One written exam with a worth of 100
Students that fail or want to improve their grade in the regular grading system have one additional moment to pass: a resit exam, which is worth 100% of the final grade.
In any of the evaluation systems (regular or resit exam) it is considered that a student has course approval if he has a grade equal or above 9.5 points.
Title: gineers, Dover (1993)
Farlow, S.J.  Partial Differential Equations for Scientists and En
Value Problems , McGrawHill, 7a ed. (2006)
Brown, J.W. ; Churchill, R.  Fourier Series and Boundary
tions, International Press (2003)
Bleecker, D. ; Csordas, G.  Basic Partial Differential Equa
Authors:
Reference:
Year:
Title: ferential Equations with Applications, Dover (1986)
Zachmanoglou, C.C. ; Thoe, D.W.  Introduction to Partial Dif
University Press (1995)
of Financial Derivatives: A Student Introduction, Cambridge
Wilmott, P. ; Howison, S. ; Dewynne, J.  The Mathematics
nance, Nova Science (2007)
Basov, S.  Partial Differential Equations in Economics and Fi
Authors:
Reference:
Year:
Exotic Options
At the end of this learning unit, the student must be able to:
1. Understand and price structured products.
2. Implement hedging strategies for exotic options.
3. Decompose a structured product into simpler financial assets.
1. Basic concepts
2. Structured products
3. Merton?s model: recap
4. Compound options
4.1. Bivariate normal
4.2. Pricing of European options
5. Chooser options: simple and complex
6. Barrier options
6.1. Reflection principle
6.2. Deterministic time change
6.3. Knockins e knockouts
6.4. Rebates
7. Lookback options
8. Asian options
9. Forwardstart options
10. Correlation dependent options
The student should acquire analytical, information gathering, written and oral communication skills, through the following learning methodologies (LM):
1. Expositional, to the presentation of the theoretical reference frames
2. Participative, with analysis and resolution of application exercises
3. Active, with the realization of individual works
4. Selfstudy, related with autonomous work by the student, as is contemplated in the Class Planning.

Regular grading system:
 One individual exam (80%)
 Individual assessment cases, attendance and active participation (20%)
Students that fail or want to improve their grade in the regular grading system have one additional moment to pass: a resit exam, that is worth 100% of the final grade.
In any of the evaluation systems (regular or resit exam) it is considered that a student has course approval if he has a grade equal or above 9.5 points.
Title:  Textos de Apoio teórico/práticos a facultar pela equipa docente durante o trimestre;
 Artigos científicos a facultar pela equipa docente durante o trimestre.
Authors:
Reference:
Year:
Title: Briys, E., M. Bellalah, H. M. Mai and F. De Varenne, Options, Futures, and Exotic Derivatives, Wiley, 1998.
Hull, John C., Options, Futures, and Other Derivative Securities, Prentice Hall, 11th edition, 2022.
Zhang, P., Exotic Options: A Guide to Second Generation Options, World Scientific, 1998, 2nd edition.
Authors:
Reference:
Year:
Optimization
LG1. Analytically solve constrained and unconstrained optimization problems.
LG2. Use MATLAB to determine approximate solutions to the optimization problems. Interpret the results mathematically and computationally and ascertain the applicability of the model.
PC1. Optimization of functions of one variable
PC2. Introduction to MATLAB.
PC3. Unconstrained optimization of functions of several variables:
(a) Necessary and sufficient conditions for the existence of optima.
(b) Steeppest descent and Newton methods.
(c) Optimization in MATLAB.
PC3. Constrained optimization of functions of several variables:
(a) Equality constraints  Necessary and sufficient conditions for the existence of optima.
(b) Inequality constraints  KKT conditions.
(c) Optimization in MATLAB.
Periodic assessment:
· Assignments (A)
· Final exam (E).
The final grade will be calculated according to the following formula::
Final grade = max(0,20 x [Grade for A] + 0,80 x [Grade for E], Grade for E)
Observações:
I) Minimum grade for the final exam = 9,5 pts.
II) Passing grade for the course: Final grade >= 10 pts.
III) Assignment (A) and Final Exam (E) grades are rounded to the nearest decimal grade point; the Final grade is rounded up or down to the nearest whole number (when the decimal grade point is less than 0.5, the grade is to be rounded down to the nearest whole number and when the decimal grade point is greater than or equal to 0.5, the grade is to be rounded up to the nearest whole number.)
Title: . Izmailov, A. e Solodov, M. "Otimização" vols. 1 e 2 IMPA (2014)
. Bonnans, J.F et al, "Numerical Optimization: Theoretical and Practical Aspects" Springer Verlag (2006)
· Nocedal, J. and Wright, St. "Numerical optimization", Springer Verlag (1999)
Authors:
Reference:
Year:
Title: Interscience (2001).
· Brandimarte, P. "Numerical Methods in Finance: A MATLABBased Introduction", Wiley
. Cornu éjols, G. et al. "Optimization in Finance" Cambridge University Press (2007)
Authors:
Reference:
Year:
Programming
At the end of this learning unit, the student must be able to:
1. Understand classes as an abstraction mechanism.
2.Design and implement small programs using his/her own classes and/or classes developed by others.
The key word in this course is ABSTRACTION. In this course, the topic of Programming with Classes is addressed, using the programming language C++.
We deal with the notion of a Class, how to create and use classes, with emphasis on the perspective of a userprogrammer: often one does not need to create a whole solution from scratch, but can take advantage of the functionality of existing classes, done by others, which are often available free of charge in the internet.
Regular grading system:
 One individual exam, comprising a written part and a computer programming part
Students that fail or want to improve their grade in the regular grading system have one additional moment to pass: a resit exam, that is worth 100% of the final grade.
In any of the evaluation systems (regular or resit exam) it is considered that a student has course approval if he has a grade equal or above 9.5 points.
Title: Pedro Guerreiro, Programação com Classes em C++, 3ª edição, FCA, 2003
Complementar (máx. 50 títulos)
 Documentação online da linguagem C++: http://www.cplusplus.com/doc/tutorial/
 Textos de apoio das aulas, facultados pelo docente
Authors:
Reference:
Year:
Credit Risk
At the end of the unit, the student should be able to:
1. Determine the default probability for each obligor using the most appropriate models.
2. Determine the credit risk of portfolios.
3. Use and evaluate credit risk derivatives.
1. Foundations of credit risk
2. Probability of default estimation
2.1. Agency credit ratings
2.2. Credit scoring and internal rating models
3. Structural approach to credit risk modelling
3.1. The Merton`s model
3.2. Extensions to the Merton`s model
3.3. The Moody`s KMV model
3.4. The CreditGrades model
4. Reduced form models
5. Credit risk portfolio models
6. Credit derivatives
7. Unified creditequity models
Periodic grading system:
a) One group work (maximum of 3 elements) with a weight of 30%;
b) One final exam (1st chance) with a weight of 70% in the final grade. It requires a minimum grade of 7.5.
The pass grade for the final valuation is 10.
Exam grading system: Students can do the 1st chance final exam with a weight of 100%. Students who fail approval under the periodic grading period or the 1st chance exam, can apply to the 2nd chance exam with a weight of 100%.
Title:  Artigos científicos a facultar pela equipa docente durante o trimestre.
 Textos de apoio teórico/práticos a facultar pela equipa docente durante o trimestre;
Authors:
Reference:
Year:
Title: ? http://www.bis.org/  for Basel documents.
? http://www.riskmetrics.com/techdoc.html  for CreditMetrics documents.
? http://www.moodyskmv.com/  for KMV documents.
? http://www.defaultrisk.com/  for research papers.
 Useful websites:
 Schönbucher, P. J. (2003). Credit Derivatives Pricing Models: Models, Pricing and Implementation, Wiley.
 Saunders, A. and Cornett, M. M. (2008). Financial Institutions Management: A Risk Management Approach, 6th edition, McGrawHill (Chapters 7, 11, and 12).
 Saunders, A. and Allen L. (2010). Credit Risk Measurement In and Out of the Financial Crises: New Appraches to Value at Risk and Other Paradigms, 3rd edition, Wiley.
 Löffler, G. and Posch, P. N. (2011). Credit Risk Modeling Using Excel and VBA, 2nd edition, Wiley.
 Lando, D. (2004). Credit Risk Modeling: Theory and Applications, Princeton University Press.
 Hull, J. C. (2008), Options, Futures and Other Derivatives, 7th edition, Prentice Hall (Chapters 22 and 23).
Authors:
Reference:
Year:
Market Risk
At the end of this learning unit, the student must be able to:
1. To know the main financial risks and the reasons for risk management;
2. To estimate VaR for a parametric distribution and for a completely general distribution;
3. To compute the VaR of each financial instrument.
1. FINANCIAL RISK
1.1. Types of risks
1.2. The need for risk management
2. III. VALUE at RISK (VaR)
2.1. VaR for general distributions
2.2. VaR for parametric distributions
3. IV. VaR OF FINANCIAL INSTRUMENTS
3.1. Equities and currencies
3.2. Bonds: mapping and bucketing
3.3. Derivatives
Regular grading system:
 One individual exam (minimum 90%)
 Individual assessment cases, attendance and active participation (maximum 10%)
resit exam which is worth 100% of the final grade.
In any of the evaluation systems (regular or resit exam) it is considered that a student has course approval if he has a grade equal or above 9.5 points.
Title:  Textos de Apoio teórico/práticos a facultar pela equipa docente durante o trimestre.
 Alexander, Carol, 2008, Market Risk Analysis  Vol. IV  ValueAtRisk Models, Wiley.
 Hull, John C., 2023, Risk Management and Financial Institutions, 6th Ed, Wiley.
Authors:
Reference:
Year:
Title:  Allen, Steven, 2013, Financial Risk Management: A Practitioner?s Guide to Managing Market and Credit Risk, 2nd Ed., Wiley.
 Dowd, Kevin, 2007, Measuring Market Risk, 2nd Ed., Wiley.
 Jorion, Philippe, 2007, Value at Risk: The New Benchmark for Managing Financial Risk, 3rd Ed., McGrawHill Companies.
 Jorion, Philippe, 2011, Financial Risk Manager Handbook, 6th Ed., Wiley.
Authors:
Reference:
Year:
Risk Theory for NonLife Insurance
At the end of this learning unit, the student must be able to:
1. Understand the concepts concerning principles for the calculation of premiums and types of partial coverage and reinsurance contracts;
2. Understand the risk Models associated with portfolios of policies, either individual or collective model, basically the Classic Model of risk of CramérLundberg;
3. Understand the notion of Ruin, and his relationship with the Maximum Aggregate Loss; approximate calculation of the probability of ruin.
1.Introduction. Review of basic concepts involving random variables, with probabilistic models associated with discrete, continuous, mixed or mixing. The models of Bernoulli, Binomial, Poisson, geometric Negative Binomial, Normal, gamma, Beta, Pareto, etc. Review of Stochastic Processes. Review of asymptotic results; sums of random variables and the Central Limit Theorem.
2.Some concepts in insurance under a perspective of utility. Risk aversion.
3.Individual risk Models in the near future. Approximations and notion of VaR.
4.Collective Risk models for a single period. The Aggregate Claims: compound Poisson models and Negative Binomial.
5.Collective Risk models for a generic period. Notion of Ruin. The processes associated with claims  the process the number of Claims and the Aggregate claim Process. The adjustment Coefficient and its relationship with the probability of Ruin. Applications of Risk Theory to insurance problems.
Theoretical and TheoricalPractical LESSONS, implemented with :
slides,
blackboard,
and students also called to solve questions.
5 questions are proposed for individual written work, at home.
CONTINUOUS EVALUATION (CE)
+
FINAL WRITTEN EXAM (FWE)
MARK=MAX( (FWE), 85%(EEF)+15%(CE) )
Title: (**) Manuais sugeridos para revisão do ?background?
(*) Manuais recomendados na área de Teoria do Risco.
3.(**)(*)Tse YiuKuen (2009). Nonlife Actuarial Models. Cambridge University Press .
2. N. L. Bowers Jr, H. U. Gerber, J. C. Hickman, D. Jones e C. J. Nesbitt, Actuarial Mathematics, Society of Actuaries, Chicago, 1986. (*)
1. (*)M. I. Fraga Alves, Teoria do Risco, Texto de apoio, Edições CEAUL, 2005.
Bibliografia Geral:
 Textos de Apoio dos slides teórico/práticos a facultar pela equipa docente durante o trimestre;
Authors:
Reference:
Year:
Title: 4. Stuart A. Klugman, Harry H. Panjer, Gordon E. Willmot, Loss Models: From Data to Decisions, 3rd Edition. Wiley, 2008.
3. R. Kaas, M.J. Goovaerts, J. Dhaene & M. Denuit (2002). Modern Actuarial Risk Theory. Kluwer Academic Publishers, Dordrecht. 2002
2. M. I. Fraga Alves, Introdução à Teoria do Risco, Working Paper no 62, ISEG, CEMAPRE, 1997.
1. M.L. Centeno, Teoria do Risco na Actividade Seguradora, Celta editora, 2003.
Authors:
Reference:
Year:
Topics of Real Analysis
At the end of this UC, the student must be able to understand:
LG1. why the field of real numbers is the most "appropriate" to discuss the basics of Real Analysis;
LG2. the main concepts of topology;
LG3. the concepts of convergent sequences and that of Cauchy sequences; the concept of convergent series and to determine convergence of a power series;
LG4. the notions of limit and continuity of functions and the main theorems that characterize them;
LG5. the concept of differentiability of real functions of one real variable and the main Mean Value theorems; the concept of Riemann integral and its relation with differentiability;
LG6. sequence/series of real functions and properties under the natural limit operations; point and uniform convergences;
PC1. The real numbers: ordered sets; fields; the field of the real numbers.
PC2. Basic topology: cardinality of sets; metric spaces; compact sets.
PC3. Sequences and series: convergent sequences; subsequences; Cauchy sequences; series; some convergence criteria; power series.
PC4. Continuity: limits and continuity of functions defined on metric spaces; continuity and topology.
PC5. Differentiation: the derivative of a real function; fundamental theorems; Taylor's theorem; differentiation of functions of several variables; partial derivatives and derivatives of higher order.
PC6. The Riemann integral: definitions and properties; integration and differentiation.
PC7. Sequences and series of functions: uniform convergence; uniform convergence and continuity, differentiability and integrability.
The course includes a periodic evaluation process that integrates the following assessment instruments:
1) Two sets of exercises with theoreticalpractical questions, (with their average counting for 30%).
2) Final exam or group project (2 students) with an oral presentation on a mutually agreedupon topic (70%).
Observations:
To pass, the minimum grade on the Final Exam is 8.0 points.
Title: Stephen Abbott, Understanding Analysis, 2015, 1,
 S. Mendes, Tópicos de Análise Real (Notas de apoio às aulas), 2021.
 J. Campos Ferreira, Introdução à Análise Matemática, Fundação Calouste Gulbenkian, 11ª Edição, 2014.
 W. Rudin, Principles of Mathematical Analysis, McGrawHill, Third Edition, 1976.
 Curso Elementar de Equações Diferenciais, Miguel Ramos,Textos de Matemática do DMFCUL, 2000.
Authors:
Reference:
Year:
Derivatives and Risk Management
At the end of the course, students should be able:
 to characterize the main financial derivatives (excluding options);
 to understand the differences between organized and overthecounter markets and the role of intermediation;
 to compute the price of each derivative and understanding the link to the spot market associated with each derivative;
 to understand the relationship between pricing and arbitrage;
 to engage in a trading negotiation in the overthecounter markets by taking the role of the financial institution and of the client;
 to use each derivative as speculative and risk management tool;
 to identify the innovation vectors associated with each derivative and to apply them to innovative solutions in risk management problems related to financing and investment operations and other corporate operations.
1. Introduction to derivatives
2. Forward contracts
2.1. Currency and interest rate forwards (FRA)
2.1.1. Characterization: taxonomy and markets
2.1.2. Pricing and arbitrage
2.1.3. Risk management and speculation
2.1.4. Negotiating in the overthecounter market
3. Interest rate swaps (IRS)
3.1. Characterization
3.2. Pricing
3.3. Risk management
3.4. Rate management (fixed vs. floating)
3.5. Speculation
4. Futures
4.1. Characterization, participants and market organization
4.2. Stock and commodity futures
4.2.1. Characterization and pricing
4.2.2. Risk management and speculation
5. Derivatives and financial innovation
The evaluation system includes:
 Cases (30%)
 Final Exam (70%)
Students that fail or want to improve their grade in the regular grading system have one additional moment to pass: a resit exam, that is worth 100% of the final grade.
In any of the evaluation systems (regular or resit exam) it is considered that a student has course approval if he has a grade equal or above 9.5 points.
Title:  Hull, John C., 2011, Options, Futures, and Other Derivative Securities, Prentice Hall, Eighth Edition
 Ao longo do trimestre é disponibilizada documentação de apoio que serve de base a cada módulo do programa. Complementarmente são indicadas referências bibliográficas para cada tema.
Authors:
Reference:
Year:
Title: 
Authors:
Reference:
Year:
Fundamentals of Economics
By the end of the semester the student should have developed the following competences:
1. understand the relevant modelling techniques;
2. combine the conceptual, mathematical and graphical approaches to solve specific problems.
1 The consumer
Preference and utility,
Demand functions
2 The firm
Technology
Cost functions,
3 Markets
Competitive
Monopoly
Oligopoly
Evaluation of this learningteaching unit consists of a Written exam at the end of the term (100%).*,**
* the student has to have a minimum of 80% attendance in class.
** the mark of the written exam cannot be below 10 ou of 20.
Title: Varian, H. R., Intermediate Microeconomics: A Modern Approach, 8th ed, New York, W. W. Norton, 2010
Authors:
Reference:
Year:
Title: Frank, R., Microeconomics and Behavior, 10th ed, Mc GrawHill, 2008
Friedman, L., The Microeconomics of Public Policy Analysis, Princeton University Press, 2017
Gravelle, H. e R. Rees, Microeconomics, Financial Times/ Prentice Hall; 3 edition , 2004
Varian, H. R., Intermediate Microeconomic, 8th ed, New York: W. W. Norton
Authors:
Reference:
Year:
Financial Markets
At the end of this learning unit, the student must be able to:
1. Understand the structure and functioning of financial markets.
2. Understand the key concepts underlying bond analysis and valuation.
3. Price fixed rate bonds.
1. Financial Markets
1.1. Money Markets
1.2. Foreign Exchange Markets
1.3. Capital Markets (Bonds, Equities and Derivatives
2. Bond Valuation
2.1. Basic concepts
2.2. Term structure of interest rates
2.3. Pricing of fixed rate bonds
2.4. Yield measures
2.5. Term structure estimation
2.6. Rating and credit risk
Regular grading system:
 One individual exam (100%)
Students that fail or want to improve their grade in the regular grading system have one additional moment to pass: a resit exam, that is worth 100% of the final grade.
In any of the evaluation systems (regular or resit exam) it is considered that a student has course approval if he has a grade equal or above 9.5 points.
Title:  Artigos científicos a facultar pela equipa docente durante o trimestre.
 Textos de Apoio teórico/práticos a facultar pela equipa docente durante o trimestre;
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Title:  Martellini, L., P. Priaulet e S. Priaulet, Fixed Income Securities ? Valuation, Risk Management & Portfolio Strategies, Wiley Finance, 2003.
 Garbade, K. D., Fixed Income Analytics, The MIT Press, 1996.
 Fabozzi, F., Bond Markets Analysis and Strategies, Prentice Hall, 3rd Edition, 1993.
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Dissertation in Mathematical Finance
Students must be able to
1. Define a scientific problem and motivate for its relevance.
2. Define research goals and possible testing hypothesis;
3. Produce a literature review supporting the dissertation main problem
4. Collect data and apply the methodologies more suitable to test the research hypothesis;
5. Critical reflexion sustained on theoretical frameworks and empirical results.
1. Writing the introduction and abstract
2. Definition of a research problem
3. Definition of research goals
4. Literature review
5. Defining hypothesis
6. Data collection methods
7. Data analysis methods
8. Writing conclusions and defining new research paths
 Written presentation of the thesis
 Oral presentation with the synthesis of the thesis followed by a public defense with a jury.
Title: Shreve, S., 2004, Stochastic Calculus for Finance II: ContinuousTime Models, Springer
Hull, John C., Options, Futures, and Other Derivative Securities, Prentice Hall, EIGHT edition, 2011
Bem, Daryl., 2002, Writing the Empirical Journal Article, in In Darley, J.M., Zanna, M.P., & Roediger III, H.L. (Eds.), The Complete Academic: A Career Guide.
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Title:  Fisher, C. (2007). Researching and writing a dissertation: A guidebook for business students. Essex: Prentice Hall
Bryman, A. (2003). Business Research Methods. Oxford: Oxford University Press
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Econometrics of Financial Markets
The course is devoted to econometric methods related with the parameter estimation and time series modelling. We study classical and modern econometrics models, in order to be able to solve typical problems from finance.
At the end of this learning unit?s term, the student must be able to:
1. Recognize and apply the simple/multiple linear regression in specific situations.
2. Recognize and apply ARMA/ARIMA models in specific situations.
3. Recognize and apply ARCH/GARCH models in specific situations.
4. Recognize and apply VAR/VECM models in specific situations.
5. Use the main econometrics packages (Python).
All classes will be held at the computer classroom.
1. Introduction
2. Regression
2.1. Correlation and causality
2.2. Simple and Multiple Linear Regression
2.3. Estimation and diagnostic methods. Residuals assumptions.
3. Stationary and Nonstationary univariate methods
3.1. Stationarity and Unit root tests
3.2. ARMA, ARIMA, BoxJenkins methodology
4. Conditional heteroscedasticity and volatility models: ARCH (Autoregressive conditional heteroscedasticity)/ GARCH
5. Stationary and Nonstationary multivariate methods
5.1. VAR (Vector autoregression)
5.2. Granger Causality and Cointegration
5.3. VECM (vector error correction models) and Johansen methodology
6. Applications and case studies
7. Software: Python
The evaluation takes place in 2 periods: Regular and Resitting. In the Regular, students in periodic assessment, must have a min of 2/3 attendance and will be evaluated by: a) Group work (min score 10 points) 50% b) Final exam (min score 8 points) 50%.
The score must be at least 10 points to get approval on the course.
Students can enroll the Resitting exam: if they not achieve a passing score on the first season; to improve the final grade.
Title: 4. Diana Mendes, (2021), Teaching Slides, Python scripts and Notebooks (Fenix and/or Elearning).
3. Yves Hilpisch (2018), Python for Finance, 2nd Edition, O'Reilly Media, Inc.
2. Mills, T., (2019), Applied Time Series Analysis: A Practical Guide to Modeling and Forecasting, Academic Press, Elsevier Inc.
1. Brooks, C., (2019), Introductory econometrics for finance, 4nd ed., Cambridge University Press.
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Title: 3. William H. Greene, (2018), Econometric Analysis, 8th Edition, Pearson.
2. James Ma Weiming, (2019), Mastering Python for Finance: Implement advanced stateoftheart financial statistical applications using Python, 2nd Edition, Packt Publishing.
1. Juselius, K., (2006), The Cointegrated VAR Model: Methodology and Applications, Oxford University Press.
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Numerical Methods
At the end of this learning unit, the student must be able to:
1. Distinguish the various types of finite difference methods for parabolic equations, knowing their relative strengths and drawbacks and know how to program the respective algorithms.
2. Use the Monte Carlo method to simulate stochastic variables and numerically solve a differential stochastic equation by the Euler method, programming the respective algorithms.
I. Basic numerical analysis
Interpolation
Numerical Derivation and integration
Linear systems
Euler method for EDO
II. Finite differences for parabolic equations
Explicit and implicit methods (1+1D)
Stability and convergence (1+1D)
Pricing of European options using finite differences.
ADI method (1+2D)
Pricing of American Options using finite differences (1+1D)
III. Monte Carlo method
Simulation of stochastic variables
Stochastic differential equations
Regular grading system:
 One written exam with a worth of 100
Students that fail or want to improve their grade in the regular grading system have one additional moment to pass: a resit exam, which is worth 100% of the final grade.
In any of the evaluation systems (regular or resit exam) it is considered that a student has course approval if he has a grade equal or above 9.5 points.
Title: Morton, K.W. ; Mayers, D.F.  Numerical Solution of Partial Differential Equations, Cambridge, 2nd ed. (2005)
Higham, D.J.  An Introduction to Financial Option Valuation, Cambridge (2004)
Brandimarte, P.  Numerical Methods in Finance and Economics, Wiley, 2nd ed. (2006)
Atkinson, K. ; Han, W.  Elementary Numerical Analysis, Wiley, 3rd ed. (2004)
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Title: Farlow, S.J.  Partial Differential Equations for Scientists and Engineers, Dover (1993)
Boto, J.P.  Introdução ao MATLAB (apontamentos)
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Models of the Term Structure of Interest Rates
At the end of this learning unit, the student must be able to:
1. Understand and implement alternatives to the Black and Scholes model in order to cope with volatility smiles in the FOREX and equity markets.
2. Understand and implement equilibrium singlefactor and multifactor term structure models.
3. Understand, calibrate and implement noarbitrage term structure models.
1. Alternatives to the BlackScholes model: volatility smiles
a. CEV model
b. Heston (1993) model
2. Term structure of interest rates
a. Bond markets
b. Spot interest rates, forward interest rates and discount factors
c. Valuation of fixedrate bonds
d. Yieldtomaturity
e. Valuation of floatingrate bonds
f. Estimation of the spot yield curve
i. Bootstraping
ii. NelsonSiegel (1987)
g. Duration and imunization
3. Equilibrium models
a. Vasicek (1977)
b. CIR (1985)
c. Multifactor CIR model
d. General Framework of DuffieKan (1996)
e. Stochastic duration
4. Noarbitrage models
a. HJM models
b. Noarbitrage condition
c. HullWhite (1990) specification
d. Gaussian HJM model: valuation of futures and options
e. Market Models
i. Lognormal LIBOR market model: caps, floors and collars
ii. Jamshidian model: swaptions
Regular grading system:
 One individual exam (80%)
 Individual assessment cases, attendance and active participation (20%)
Students that fail or want to improve their grade in the regular grading system have one additional moment to pass: a resit exam, that is worth 100% of the final grade.
In any of the evaluation systems (regular or resit exam) it is considered that a student has course approval if he has a grade equal or above 9.5 points.
Title:  Artigos cientificos a facultar pela equipa docente durante o trimestre.
 Textos de Apoio teórico/práticos a facultar pela equipa docente durante o trimestre;
Authors:
Reference:
Year:
Title: Shreve, S., 2004, Stochastic Calculus for Finance II: ContinuousTime Models, Springer.
Rebonato, R., 1998, Interestrate Option Models, John Wiley & Sons, 2nd edition.
Musiela, M. and M. Rutkowski, 2011, Martingale Methods in Financial Modelling, 2nd edition, Springer.
Lamberton, D. and B. Lapeyre, 2007, Introduction to Stochastic Calculus Applied to Finance, 2nd edition, Chapman & Hall.
James, J, and N. Webber, 2000, Interest Rate Modelling: Financial Engineering, Wiley.
Brigo, D. and F. Mercurio, 2006, Interest Rate Models  Theory and Practice: With Smile, Inflation and Credit, 2006, 2nd edition, Springer.
Björk, T., 2009, Arbitrage Theory in Continuous Time, 3rd edition, Oxford University Press.
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Objectives
The MSc in Financial Mathematics aims at the advanced training of staff in the area of stochastic processes applied to Finance and has as its general objectives:
Develop expertise in the valuation of complex financial instruments such as financial derivatives
Develop expertise in modeling and quantifying financial risks relevant to the banking and insurance sectors
Provide students with research methodologies, procedures, and techniques that allow them to develop their research projects with a high degree of autonomy
Accreditations