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: stochastic dynamic programming and equivalent martingale approach in discrete and continuous time. Copula and dependence concepts.

1. Probability for Finance – E. Kopp, J. Malczak, T. Zastawniak – Cambridge University Press 2014
2. Introduction to Probability – D.P. Bertsekas, J.N. Tsitsiklis – Athena Scientific, 2nd edition, 2008
3. Essential Mathematics for Market Risk Management – S. Hubbert – Wiley 2012
4. Statistics and Finance: An Introduction – D. Ruppert – Springer 2004
5. Basic Stochastic Processes – Z. Brzeziank, T. Zastawniak – Springer 2000
6. Statistics of Financial Markets – J. Franke, W.K. Hardle, C.M. Hafner – Springer 2015
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. Probability for Risk Management – M.J. Hassett, D.G. Stewart – Actex Pubblications 2006
11. Mathematical Techniques in Finance – A. Cerny – Princeton University Press 2009
12. Stochastic Control in Discrete and Continuous Time – A. Seierstad – Springer 2009