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98057 - Stochastic Methods for Applications

Academic Year 2022/2023

                
                        Docente:
                        Stefano Pagliarani
                    
                        Credits:
                        6
                    
                        SSD:
                        MAT/06
                    
                        Language:
                        Italian
                    
                        Teaching Mode:
                        Traditional lectures
                        
                            Campus:
                            Bologna
                        
                            Corso:
                            First cycle degree programme (L) in
                            Mathematics (cod. 8010)

                                Also valid for
                                
                                    Second cycle degree programme (LM) in
                                    
                                        Computer Science (cod. 5898)
                                    
                            Teaching resources on Virtuale

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