Ateneo - UPO

The course treats the topic of strategic decisions from the viewpoint of derivatives. Moreover, the focus will be on financial options, and, particularly, on pricing and hedging problems. European and American style options are the main reference. Binomial trees and Monte Carlo simulations are the main tools that will be applied.

Required reading:
Hull, J. C., Options, Futures and other derivatives. Italian edition by Emilio Barone. XI ed., Pearson Education.

Suggested readings:
Glasserman, P. (2004) Monte Carlo Methods in Financial Engineering. In: Stochastic Modelling and Applied Probability, Springer-Verlag, New York.

Guthrie, Graeme, 2009. "Real Options in Theory and Practice", OUP Catalogue, Oxford University Press.

Brandimarte,Paolo, From Shortest Paths to Reinforcement Learning: A MATLAB-Based Tutorial on Dynamic Programming. 2021, Springer
International Publishing

Additional materials will be published on the Moodle (DIR) page of the course by the teacher.

At the end of the course, the student must have acquired the knowledge of how to build a binomial model with stochastic evolution of the price of the underlying asset; he will also be able to use this model for pricing and hedging of some derivative instruments. The Student will also have knowledge about the Black&Scholes formula (as the limit of a binomial model) and its use for pricing purposes. They will also have acquired a basic knowledge of stochastic processes in continuous time, with particular reference to arithmetic Brownian motion and geometric Brownian motion. They will be able to price European derivative instruments through Monte Carlo simulations and critically evaluate the results obtained. They will be introduced to Reinforcement Learning methods as a data driven decision support tool.

Basic knowledge of mathematics and statistics. Minimum knowledge of Excel and Matlab

Lectures, theory and exercises. Part of the lessons will be conducted using Excel and Matlab

Attendance of the lecture class is not compulsory but is strongly recommended.

A compulsory oral exam at the end of the module, during which the student will also be asked to solve some problems on the course topics.

Introduction to derivatives. Features of contracts, long and short positions. Financing options, features of a standard contract, American and European call and put options, variables that affect the price of options. Price restrictions of options, modelling of the evolution of the underlying by means of binomial trees. Risk neutral probability. Pricing of an option via binomial model. Delta of an option. Replication portfolio (dynamic). The case of American options. The Black & Scholes model. Stochastic processes: generalities, discrete time and continuous time processes. Markov processes. Brownian motion. Models for price and returns of stocks: geometric Brownian motion and arithmetic Brownian motion. Introduction to Monte Carlo simulation methods: sampling from a uniform random variable and inverse transform method. Pricing of derivatives via Monte Carlo: algorithm convergence. Variance reduction techniques: anthitetic variates. Reinforcement learning: SARSA and Q-learning.

The knowledge of how to construct a binomial model with stochastic evolution of the prices of the underlying assets and to use such model for pricing and hedging of some classical derivative instruments will have to be acquired. Knowledge of the Black&Scholes formula (as the limit of a binomial model) and its use for pricing and hedging purposes must also be acquired. The Student must be able to simulate with Monte Carlo techniques the stochastic evolution of the price of the underlying for simple processes and use the results obtained to price some classic derivative instruments, critically interpreting the results obtained. They will also need to know the main features of reinforcement learning problems.