77942 - Probabilistic Methods for Finance

Learning outcomes

By the end of the course the student will be familiar with the theory of discrete-time stochastic processes and martingales. She knows how to apply these notions to the pricing and hedging of financial derivatives. 

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.

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, American derivatives.

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.

Readings/Bibliography

Pascucci, Andrea. PDE and martingale methods in option pricing. Springer Science & Business Media, 2011.

Pascucci, Andrea, and Wolfgang J. Runggaldier. Financial mathematics: theory and problems for multi-period models. Springer Science & Business Media, 2012.

Teaching methods

Lectures on the board.

Assessment methods

Oral examination with questions on the topics covered in the lectures, and brief exercises to test the ability of applying the acquired knowledge.

 

Teaching tools

Lecture notes (PDF).

Office hours

See the website of Stefano Pagliarani