98057 - Stochastic Methods for Applications

Learning outcomes

By the end of the course the student will be familiar with the theory of discrete-time stochastic processes, in particular Markov chains and martingales, as well as with the method of dynamic programming for stochastic optimal control problems. She/he knows how to apply these notions to the pricing and hedging of financial derivatives. She/he has a basic knowledge of the fundamental notions underlying supervised (machine) learning problems.

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.

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). 

Office hours

See the website of Stefano Pagliarani