The course is teached in English. The probabilistic toolkit will be developed through Lectures to be held during the classroom, and additionally by means of real world cases. This is beacause the relevant probability theory would be applied to problems arising from risk management and financial engineering (selected topics). Sometimes, real world cases consist of numerical evaluation of financial dataset, in term of probabilistic models, using spreadsheets developed during the classroom. Excel and a glimpse of Visual Basic for Application (VBA) are the main device to manipulate spreadsheets. Furthermore, students are asked to solve exercises to test their understainding of the main probabilistic definition, theorems and formulas.
1st MODULE (3 CFU)
Topic: Review of basic probability theory.
Learning goals: Probabilistic tools from the calculus viewpoint, with some elements of measure theory.
Topic description: Discrete and general probability spaces. Conditional probability and independence. Discrete and continuous random variables and their distributions. Most frequently used probability distribution in finance (univariate). Moments and characteristic function. Location and dispersion indexes. Covariance. Quantiles. Conditional expectation. Inequalities. Hazard function and the probability integral transform.
2nd MODULE (3 CFU)
Topic: Multivariate (static and dynamic) probability models.
Learning goals: Probability distributions of random vectors and stochastic processes.
Topic description: Random vectors and joint distributions. Marginals and finite dimensional distributions of infinitely many random variables. Distribution of a stochastic process. Convergence theorems. Monte-Carlo simulation. Some commonly used stochastic processes in finance (Bernoulli, random walk, Wiener, AR(1), martingale, Markov. Poisson). A glimpse to stochastic calculus.
3rd MODULE (3 CFU)
Topic: Stochastic models in finance (selected).
Learning goals: Applying random distributions and other probabilistic tools to some typical financial models.
Topic description: Modeling market invariants (log-returns, risk and expected return of a portfolio). Coherent risk measures and Value-at-Risk. Binomial asset pricing model. Geometric Brownian motion and Black-Scholes model. Market probabilities vs risk-neutral probabilities. Fat tails in practice (time permitting). Copula and dependance concepts. Empirical distribution function and plug-in estimators (time permitting). Expectiles. Robustness (time permitting). Some stochastic orders and conditional risk measures.
Additional Textbooks (not mandatory)