PROBABILITY AND APPLICATIONS TO FINANCE

The aim of the course is to give a basic knowledge on probability theory in order to discuss at the end more advanced topics such as the stochastic processes useful in the main models in finance (for example the Black-Scholes model for asset pricing). Moreover all the techniques studied in the course can be applied in other scientific fields.

Knowledge and understanding: the aim of the course is to give to the students the basic knowledge of probability theory concluding with stochastic processes. Applying knowledge and understanding: students will learn advanced instruments in probability, such as stochastic processes, and they will learn how to apply them in finance. Criticism of judgment: at the end of the course students will have a more rational view of the world of probability and hence they will be able to make prevision of random events with awareness. Communication skills: students will learn a proper language for the description of models from the probabilistic point of view. Learning skills: at the end of the course students will have a complete overview on basic probability and on some stochastic processes that will allow them to face with real world models and, more in general, with many research topics in different fields.

Calculus, Linear Algebra

The first part of the course will cover basic probability theory, while in the second part we will discuss stochastic processes with particular attention to martingales in discrete time. Indeed these processes are really important in finance since they represent the basic instrument for relevant models, such as the Black-Scholes model for asset pricing. More precisely, in the first part of the course we consider combinatorics and basic probability theory, that is probability axioms, consequences of the axioms, conditional probability and independence of events. Then we will introduce random variables and we will speak about expectation and variance, indexes useful to resume the behavior of the random variable. All the knowledge on random variables will be then generalized to random vectors. The first part of the course ends with main results on limit theorems useful to describes phenomena (such as in statistics) when the number of agents involved is big. In the second part of the course we will speak about stochastic processes and conditional expectation in order to introduce then martingales and their properties. Finally we will see applications to the binomial model for asset pricing (a discrete time version of Black-Scholes model) and to betting systems.

-Ross, S. M. (2013). Calcolo delle probabilità. Italia: Apogeo. -Ethier, S. N. (2010). The Doctrine of Chances: Probabilistic Aspects of Gambling. Germania: Springer Berlin Heidelberg. -Shreve, S. E. (2004). Stochastic Calculus for Finance I: The Binomial Asset Pricing Model. Germania: Springer.

Two frontal lectures, while third lecture is devoted to exercises.

The grade is established through a written and an oral exam. For attending students the written exam can be substituted with the two midterms: the first midterm (on 19/10/2023) concerns the first part of the course (up to random vectors excluded), while the second midterm (on 30/11/2023) concerns the remaining part of the course. If a student does not pass one of the two midterms, he/she can recover it during the written exam. For non attending students the exam is composed by the written and the oral exam. In particular the oral exam contains additional arguments (with respect to the oral exam of attending students). Such arguments will be decided with the teacher during the course. For attending students: each midterm has a weight of 30% on the final grade, while the oral exam has a weight of 40%. For non attending students: the written exam has a weight of 60% on the final grade, while the oral exam has a weight of 40%.

Minimum grade 29/30 and participation during the course. Moreover a discussion with the teacher is needed to understand if the student is able to work on the topic.