Investments Theory


Describe the characteristics of forwards, futures, options and swaps and of the markets in such investments and of the main investment assets and of the markets in such assets.

Use the Capital Asset Pricing Model and or a multifactor model  to calculate the required return on a particular asset, given appropriate inputs, and hence calculate the value of the asset. Explain the concepts of: efficient market, complete market, no-arbitrage, hedging.

Apply the risk-neutral or state price deflator approaches to valuing derivative securities and apply them. Use the properties of various stochastic models of the term structure of interest rates. Explain the limitations of the models described above and describe attempts to address them.

Use mean-variance portfolio theory to calculate an optimum portfolio and describe the limitations of this approach. Use mean-variance portfolio theory to calculate the expected return and risk of a portfolio of many risky assets, given appropriate inputs. 

General characterization





Responsible teacher

Marta Cristina Vieira Faias Mateus


Weekly - 3

Total - 42

Teaching language



Knowledge of Probability Theory [Measurable sets, negligible sets; Measurable functions, random variables; probability law of a random variable; Lebesgue integral; convergence theorems: Lebesgue and Fatou; Banach spaces of integrable function classes].

Knowledge of Stochastic Processes [Distribution of a stochastic process; construction of stochastic processes; examples of stochastic processes: Wiener process and martingales in discrete time and continuous time].


  1. Berk, J., DeMarzo, P., Corporate Finance, 4th ed. Pearson, 2014.
  2. Bjoerk T., Arbitrage Theory in Continuous Time, 3rd ed. Oxford Finance, 2009.
  3. Bodie, Z., Kane, A. and Marcus, A., Essentials of Investments, McGraw-Hill, 2008.
  4. Elton E.J., Gruber M.J, Martin J.G., Brown S.J., Goetzmann W.N. (2014) Modern Portfolio Theory and Investment Analysis, 9th ed. Wiley, 2014.
  5. Hull J.C.  Options, Futures and other Derivatives, 9th ed. Prentice Hall, Upper Saddle River, NJ, 2014.
  6. Oksendal B. Stochastic Differential Equations: an Introduction with Applications, 6th ed. Springer, 2010.

  7. Pires, C.P., Mercados e Investimentos Financeiros, Escolar Editora, 3ª edição, 2011.

  8. Sharpe, W., Alexander, G. and Bailey, J., Investments, Prentice Hall, 1999.

Teaching method

The classical methodology used in Mathematics at the university level. The contents are presented and discussed trying to stress the most important ideas and practical procedures. There are study materials: textbook, classroom notes with problems, some with solutions, and a list of questions indicating exactly what the student has to know.

Evaluation method


Obtaining frequency depends on the successful completion of 2 projects (rating ≥ 7,5 on a 0 to 20 scale), in each one.



Continuous assessment consists of conducting, during the semester, 2 projects (mandatory to obtain frequency) and 2 tests, each of which is rated from 0 to 20 points.

Let P1, P2,  T1 and T2 be the classifications obtained in the 1st and 2nd projects, 1st and 2nd tests, respectively. The student will be approved if

0,2 × P1 + 0,3×T1 +  0,15 × P2 + 0,35×T2  ≥  9,5 .

 In this case the final classification will be given by this average rounded to the units.



 All students enrolled in the Course (with frequency) can apply to the exam.

The final grade is computed according to the given formula: 

 FG= 0,2 × P1 + 0,3×EP1 +  0,15 × P2 + 0,35×EP2  , where P1 and P2 are the project grades and EP1 and EP2 the exam 1st and 2nd part classifications (0 to 20 points scale).

If the classification FG is higher or equal to 9.5 the student is approved with this classification, rounded to the nearest integer. 



 Any student wishing to perform a grade improvement must register for this purpose at CLIP (information at the Academic Office).

The improvement may regard one or more of the evaluated modules.

The improvement final classification is obtained according to 2. If this result is higher than  the one already obtained, it will be taken as a final grade. Otherwise, there is no grade improvement.

Subject matter

1. Objectives of individual and institutional investors.

2. Financial markets and types of financial investment: fixed and variable income assets.

3. Models for determining the optimal investment:
     - Factor Model
     - Capital Asset Pricing Model (CAPM)
     - Arbitrage Model (APT)

4. Pricing of financial assets: bonds, stocks, options

5. Markets free of arbitrage and complete markets; examples: binomial model and Black-Scholes model.

6. Other stochastic models (e.g. diffusions) for the pricing of derivative financial products.

7. Management of investment risks. Risk Measures: V@R, TV@R, EV@
Coherent risk measures

8. Performance evaluation of the management of financial asset portfolios.


Programs where the course is taught: