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 closed-book and closed-notes. However, you may use a formula sheet.

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. Sigma-algebras. Mesurable spaces and functions.
2. Finite and sigma-finite 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 Radon-Nikodym.
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.

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. Discrete-time 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. Feynman-Kac 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 re-sit exam, that is worth 100% of the final grade.
In any of the evaluation systems (regular or re-sit exam) it is considered that a student has course approval if he has a grade equal or above 9.5 points.

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 Black-Scholes model.

1. Discrete-time models.
2. The Cox-Ross-Rubinstein model.
3. Optimal stopping and American options.
4. The Black-Scholes 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 re-sit exam, that is worth 100% of the final grade.
In any of the evaluation systems (regular or re-sit exam) it is considered that a student has course approval if he has a grade equal or above 9.5 points.

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: Reaction-Diffusion equations; Black-Scholes 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 Black-Sholes 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 re-sit exam, which is worth 100% of the final grade.
In any of the evaluation systems (regular or re-sit exam) it is considered that a student has course approval if he has a grade equal or above 9.5 points.

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. Knock-ins e knock-outs
6.4. Rebates
7. Lookback options
8. Asian options
9. Forward-start 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. Self-study, related with autonomous work by the student, as is contemplated in the Class Planning.

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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 re-sit exam, that is worth 100% of the final grade.

In any of the evaluation systems (regular or re-sit exam) it is considered that a student has course approval if he has a grade equal or above 9.5 points.

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.)

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 user-programmer: 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 re-sit exam, that is worth 100% of the final grade.

In any of the evaluation systems (regular or re-sit exam) it is considered that a student has course approval if he has a grade equal or above 9.5 points.

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 credit-equity 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%.

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%)

re-sit exam which is worth 100% of the final grade.

In any of the evaluation systems (regular or re-sit exam) it is considered that a student has course approval if he has a grade equal or above 9.5 points.

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 Cramér-Lundberg;
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 Theorical-Practical 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) )

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 theoretical-practical questions, (with their average counting for 30%).
2) Final exam or group project (2 students) with an oral presentation on a mutually agreed-upon topic (70%).

Observations:
To pass, the minimum grade on the Final Exam is 8.0 points.

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 over-the-counter 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 over-the-counter 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 over-the-counter 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 re-sit exam, that is worth 100% of the final grade.

In any of the evaluation systems (regular or re-sit exam) it is considered that a student has course approval if he has a grade equal or above 9.5 points.

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 learning-teaching 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.

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 re-sit exam, that is worth 100% of the final grade.

In any of the evaluation systems (regular or re-sit exam) it is considered that a student has course approval if he has a grade equal or above 9.5 points.

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.

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 Non-stationary univariate methods
3.1. Stationarity and Unit root tests
3.2. ARMA, ARIMA, Box-Jenkins methodology
4. Conditional heteroscedasticity and volatility models: ARCH (Autoregressive conditional heteroscedasticity)/ GARCH
5. Stationary and Non-stationary multivariate methods
5.1. VAR (Vector auto-regression)
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 Re-sitting. 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 Re-sitting exam: if they not achieve a passing score on the first season; to improve the final grade.

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 re-sit exam, which is worth 100% of the final grade.

In any of the evaluation systems (regular or re-sit exam) it is considered that a student has course approval if he has a grade equal or above 9.5 points.

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 single-factor and multifactor term structure models.
3. Understand, calibrate and implement no-arbitrage term structure models.

1. Alternatives to the Black-Scholes 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 fixed-rate bonds
d. Yield-to-maturity
e. Valuation of floating-rate bonds
f. Estimation of the spot yield curve
i. Bootstraping
ii. Nelson-Siegel (1987)
g. Duration and imunization
3. Equilibrium models
a. Vasicek (1977)
b. CIR (1985)
c. Multi-factor CIR model
d. General Framework of Duffie-Kan (1996)
e. Stochastic duration
4. No-arbitrage models
a. HJM models
b. No-arbitrage condition
c. Hull-White (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 re-sit exam, that is worth 100% of the final grade.

In any of the evaluation systems (regular or re-sit exam) it is considered that a student has course approval if he has a grade equal or above 9.5 points.