PROBABILITY FOR FINANCE

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; random variables and their distribution (discrete case and continuous case); most commonly used models of probability distribution in finance; location and dispersion indexes; quantile functions; conditional expectation (integration with respect to a probability measure).

2nd MODULE (3 CFU)

Topic: Multivariate (static and dynamic) models under uncertainty

Learning Goals: Extending probability distributions to random vectors and stochastic processes, as models of financial phenomena.

Topic Description: Random vectors and joint distributions; marginal distributions; stochastic processes and finite dimensional distributions (FIDIS); some theorems about convergence of random variables; Monte Carlo simulation and the inverse transform method; characteristic function and moment generating function; some commonly used stochastic processes in finance: Bernoulli, random walk, Wiener, AR(1), martingale, Markov Chains; elements of stochastic calculus (Itô integral and diffusive stochastic differential equations).

3rd MODULE (3 CFU)

Topic: Stochastic Models and Finance

Learning Goals: Applying random distributions (univariate and mulivariate) and stochastic processes to some typical financial models: pricing, risk and performance measurement.

Topic Description: Modeling markets invariants (log-returns); coherent and general risk measures (e.g. Value-at-Risk, Expecetd Shortfall); pricing simple derivative contracts: the binomial approach (discrete time) and the Black-Scholes approach (continuous time, no arbitrage); stochastic control through simple cases: Hamilton-Jacobi-Bellman equation (stochastic dynamic programming) and equivalent martingale approach.

1. Lecture Notes del docente
2. Probability: The Science of Uncertainty (with Applications to Investments, Insurance and Engineering) – M.A. Bean – American Mathematical Society
3. Essential Mathematics for Market Risk Management – S. Hubbert – Wiley 2012
4. Probability Models – J. Haigh – Springer 2004
5. Basic Stochastic Processes – Z. Brzeziank, T. Zastawniak – Springer 2000
6. Basic Probability Theory – R.B. Ash – Free Electronic Copy
7. Probability Essentials – J. Jacod, P. Protter – Springer 2004
8. Elementary Stochastic Calculus (with finance in view) – T. Mikosch – World Scientific 1998
9. A First Look at Rigorous Probability Theory – J.S. Rosenthal – World Scientific – 2006
10. An Elementary Introduction to Mathematical Finance: Options and other Topics – S.M. Ross – Cambridge University Press – 2011
11. Probability for Risk Management – M.J. Hassett, D.G. Stewart – Actex Pubblications 2006
12. From Measures to Ito Integrals – E. Kopp – Cambridge University Press – 2011