Teoria da Probabilidade e Modelos Discretos em Finanças


At the end of this course unit the student will have acquired knowledge, skills and competences that allow him / her to:

  • Understand some of the most relevant concepts underlying Kolmgorov''s modern probability theory. Understand the probabilistic fundamentals of discrete time financial market models.
  • Be able to use the fundamental techniques to build models with good properties - free from arbitrage and complete - as well as the estimation and calibration of parameters of the most important discrete-time financial market models.
  • Know some of the main practical applications of the fundamental concepts studied.

General characterization





Responsible teacher

Manuel Leote Tavares Inglês Esquível


Weekly - 2

Total - 56

Teaching language



Mathematical Analysis of univerisity level.


  • A course in financial calculus. Alison Etheridge. Cambridge: Cambridge University Press, 2002.
  • Stochastic finance. An introduction in discrete time. 2nd revised and extended ed., Hans Follmer and Alexander Schied. Berlin: de Gruyter, 2002.
  • Probability with Martingales. David Williams.  Cambridge University Press, 1991.
  • Essentials of stochastic finance. A. N. Shiryaev.  World Scientific, 1999.

Teaching method

Session with mixed theoretical and practical works.

Evaluation method

The evaluation consists of two interim tests, a practical computational work and, if necessary, a final exam. The tests and work will give rise to a continuous evaluation grade (weighted average with 70% for the tests with the second weight 2 and the first weight 1). To obtain attendance the student must have attended at least two thirds of the classes and must have performed the tests and work.

Subject matter

Formalism of Probabilities

Introduction. The probability model according to Kolmogorov ,. Laws of large numbers. A limit theorem

central. Independence; The law of 0-1de Kolmogorov. Conditional hope; The available or known information; Radon-Nikodym theorem; properties.


Local Martingales and Martingales in Discrete Time

Stop time and optional Doob theorem Martingale convergence theorems


The Binomial Model

One period Models. Portfolios and Arbitrage. Martingale measures. The fundamental theorems of asset pricing. Financial products derived in the binomial model. Neutral Risk Pricing. The multi-period binomial model. Estimation and Calibration. American options. Models for bonds and interest rates.


Discrete Models

Euler''s discretization of diffusions. Models with nonrandom coefficients. Girsanov theorem in the discrete case. Models with random coefficients.


Programs where the course is taught: