The course aims to introduce the fundamental concepts of Probability Theory, with particular emphasis on discrete random variables and their applications to discrete-time financial models. Probability theory will be developed gradually and employed to illustrate tree models for asset price evolution and to provide the basis for the valuation of simple derivative instruments. EXPECTED LEARNING OUTCOMESStudents are expected to:
progressively assimilate the fundamental concepts of Probability Theory, with particular reference to discrete random variables and their distributions;
apply these concepts to the analysis of random phenomena and to simple market models;
formalize and solve problems (problem solving) related to the valuation of basic derivative securities;
understand some theoretical aspects and be able to present them clearly;
carry out selected basic mathematical proofs within the framework of discrete probability.
KNOWLEDGE AND UNDERSTANDING By the end of the course, students will have acquired the fundamental concepts of discrete probability and understood how they can be employed in the construction and analysis of tree models for asset price evolution and for the valuation of simple derivative securities. MAKING JUDGEMENTS By the end of the course, students will have developed the ability to formalize concrete financial problems and to use the tools of discrete probability to address them and propose appropriate solutions. COMMUNICATION SKILLS By the end of the course, students will be able to summarize and present the theoretical concepts and results learned, and to clearly and rigorously explain the reasoning underlying the solutions to problems and the financial applications considered.
Course Prerequisites
Basic knowledge of mathematics, acquired through the courses taught in the first and second year of the Bachelor’s degree program.
Teaching Methods
The course consists of 48 hours of lectures. During the classes, exercises will be assigned and their solutions will be discussed by the lecturer, with the aim of verifying the practical application of the theoretical concepts presented. In addition to lectures, cycles of seminars delivered by experts and professionals may be organized as supplementary activities. Attendance is optional but strongly recommended; the final examination is the same for attending and non-attending students.
Assessment Methods
Student assessment is based on a mandatory written exam covering the topics discussed during the course. The exam will consist of exercises and theoretical questions, with scores assigned according to the difficulty and relevance of each item. The final score will be expressed in thirtieths, with the possibility of honors (cum laude). Students who obtain at least 18/30 in the written exam may, on a voluntary basis, take an oral exam. In this case, the final grade will take both parts into account.
Texts
S. Ross: Introduction to probability models, 13/ed, Elsevier, 2023. J. C. Hull: Options, Futures and other Derivatives, 11/ed, Pearson, 2022. Lecture notes provided by the professor.
Contents
Fundamentals of probability theory and discrete random variables. Main discrete distributions, expectation and variance. Joint distributions of discrete random variables, independence, marginal and conditional distributions. Binomial trees for asset price evolution (Cox–Ross–Rubinstein model) and basic trinomial model. Applications to the valuation of derivative securities.
Course Language
Italian.
More information
Weekly office hours: Monday, Tuesday and Friday at the end of lectures (1:00 PM), or by appointment via email. Office hours can also be held in English.
