- Docente: Stefano Pagliarani
- Credits: 6
- SSD: MAT/06
- Language: Italian
- Teaching Mode: Traditional lectures
- Campus: Bologna
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Corso:
Second cycle degree programme (LM) in
Computer Science (cod. 5898)
Also valid for First cycle degree programme (L) in Mathematics (cod. 8010)
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from Feb 24, 2023 to May 26, 2023
Learning outcomes
At the end of the course the student knows elements of some advanced probability theories with applications to computer science, such as Markov chains with discrete and continuous time. She / he is able to analyze some simple stochastic systems related to applications.
Course contents
Introduction to pricing and hedging of financial derivatives in a one-period market: options, arbitrages, Put-Call parity formula, arbitrage and risk-neutral price, incomplete markets.
Elements of martingale theory: Sigma-algebras and filtrations, conditional expectation, discrete-time stochastic processes, martingales, stopping times, Doob decomposition Th., Markov property, discrete Markov chains.
Pricing and hedging in discrete market models: self-financing and admissible strategies, equivalent martingale measure and First Fundamental Theorem of Asset Pricing, arbitrage-free markets and arbitrage price, completeness and Second Fundamental Theorem of Asset Pricing.
Binomial market model: binomial tree, absence of arbitrage and completeness, arbitrage price and hedging strategies, binomial algorithm, stability and convergence to Black-Scholes model, trinomial model and incomplete markets, examples: European and American options.
Elements of stochastic optimal control: introduction to dynamic programming method.
Elements of supervised (machine) learning: input, output and training sets; hypothesis class; expected loss and empirical risk; introduction to neural networks; deterministic and stochastic gradient descent.
Prerequisites: probability theory
Readings/Bibliography
- Pascucci, Andrea, and Wolfgang J. Runggaldier. Finanza matematica: teoria e problemi per modelli multiperiodali. Springer Science & Business Media, 2009.
- Pascucci, Andrea. Calcolo stocastico per la finanza. Springer Science & Business Media, 2008.
- Shai Shalev-Shwartz and Shai Ben-David. Understanding Machine Learning: From Theory to Algorithms. Cambridge University Press, 2014.
Teaching methods
Lectures on the board.
Assessment methods
Oral examination with questions on the topics covered in the lectures. The first topic is chosen by the student. Possibly, brief exercises to test the ability of applying the acquired knowledge.
Teaching tools
Lecture notes (PDF) covering some parts of the program available on the website virtuale.unibo.it
Office hours
See the website of Stefano Pagliarani