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.

Upon successful completion of the course, students will be able to construct and analyze a binomial model with stochastic dynamics for the evolution of the underlying asset price, and to apply this framework to the pricing and hedging of selected classes of derivative instruments. They will demonstrate a rigorous understanding of the Black–Scholes formula, interpreted as the continuous-time limit of the binomial model, and will be able to apply it effectively to problems of derivative pricing. In addition, students will acquire foundational knowledge of continuous-time stochastic processes, with particular attention to arithmetic and geometric Brownian motion, and will develop the ability to implement Monte Carlo simulation techniques for the valuation of European-style derivatives, critically assessing the accuracy and limitations of the numerical outcomes. Finally, they will be introduced to Reinforcement Learning methodologies, considered as advanced tools for supporting data-driven decision-making in quantitative finance.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 programmazione

Lectures, theory and exercises

Attendance of the lecture class is not compulsory but is strongly recommended. Students with physical disabilities, Learning Disabilities or Special Education Needs can request specific services and tools via the Staff Sviluppo e Coordinamento Carriere e Servizi alle Studentesse e agli Studenti, consulting the University webpage: https://www.uniupo.it/en/services/servicesstudents-physical-or-learning-disabilities. Students with disabilities, learning disabilities or special education needs, once they have contacted the University Staff, can refer to the tutor in charge of the course to define the examination modalities, concerning academic aspects.

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.

By the end of the course, students will have acquired the knowledge required to construct a binomial model with stochastic dynamics of underlying asset prices and to apply this model to the pricing and hedging of standard derivative instruments. They will also have developed a rigorous understanding of the Black–Scholes formula, viewed as the continuous-time limit of the binomial model, and its application to pricing and hedging. Participants will be able to simulate the stochastic evolution of underlying asset prices for simple processes using Monte Carlo techniques and to employ the resulting outputs to price classical derivative instruments, critically interpreting the numerical results. Finally, they will have gained an introductory understanding of the main features of reinforcement learning problems.