Instructional goals
The aim of the course is to give a basic knowledge of the probability theory in order to treat at the end advanced topics, as the stochastic processes, that are involved in the most famous models in finance, as the model by Black and Scholes for European option pricing in absence of arbitrage. Moreover the contents of the course are versatile for applications in other fields.
Prerequisites
Calculus, Linear Algebra
Intended learning outcomes
Knowledge and understanding: The aim of the course is to give to the students a basic knowledge of the probability theory in order to study at the end the basic stochastic processes.
Applying knowledge and understanding: The students will learn advanced instruments in probability, as the stochastic processes, and they will understand how to apply them to famous models in finance.
Criticism of judgment: At the end of the course the students will have a more rational vision of the world of probability and then they will be able to make predictions in a more conscious way.
Communication skills: The students will learn an appropriate language for the description of phenomena and models from a probabilistic point of view.
Learning skills: At the end of the course the students will have knowledge in probability field and in particular about stochastic processes, that will allow the students to face more advanced topics of research in many fields.
Course Contents
The first part of the course will discuss basic arguments in probability, while in the second part we will study stochastic processe with particular attention to martingales in discrete time, that are instruments at the basis of famous models in finance, such as the model by Black and Scholes.
More precisely, as far as the first part of the course, we will start from combinatorics in order to introduce the concepts of probability and of conditional probability with the relative properties. Then we will study the random variables introducing the expectation and the variance, that are indexes that resume the behavior of the variables. This section of the course is the then generalized to random vectors. The first part of the course ends with important theorems for the estimate of probabilities, used for example in statistics when the number of samples is big.
In the second part of the course we will speak about stochastic processes and conditional expectation. Such ideas will be used to introduce martingales and relative properties. Finally we will see an application of these processes to the binomial asset pricing model, that is (in some sense) an approximation of the Black-Scholes model at discrete time.
Reference Books
-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.
Teaching Methods
Two lectures per week are devoted to frontal lectures, while the third lecture will be devoted to exercises.
Assessment Method
The grade is established through a written and an oral exam.
The written exam can also take place as two distinct midterms (the first one after half course and the second at the end of the course).
If a student fails or does not partecipate to one of the two midterms, it is possibile to recover the lacking midterm during the written exam.
Thesis assignment criteria
Minimum degree 27/30 and participation during the course.
Does the syllabus cover sustainability topics?
no
Week 1 Contenuto sessioni on line e on campus
-Permutations;
-Combinations;
-Exercises;
Week 2 Contenuto sessioni on line e on campus
-Definition of probability;
-Properties of the probability measure;
-Exercises;
Week 3 Contenuto sessioni on line e on campus
Conditional probability;
-Bayes’ formula, Independence;
-Exercises;
Week 4 Contenuto sessioni on line e on campus
-Random variables and Distribution function;
-Discrete random variables, density function of discrete random variables;
-Exercises;
Week 5 Contenuto sessioni on line e on campus
-Expectation and variance of discrete random variables;
-Indicator functions, probability as expectation;
-Exercises;
Week 6 Contenuto sessioni on line e on campus
-Continuous random variables and density function, density function of continuous random variables, expectation and variance of continuous random variables;
-Functions of random variables, universality of the uniform distribution;
-Exercises;
Week 7 Contenuto sessioni on line e on campus
-First midterm;
-Random vectors, joint laws, independence of random variables;
-Exercises;
Week 8 Contenuto sessioni on line e on campus
-Trasnformations of random vectors;
-Sum of independent random variables, covariance and correlations;
-Exercises;
Week 9 Contenuto sessioni on line e on campus
-Conditional law;
-Properties of the conditional expectation, tower property;
-Exercises;
Week 10 Contenuto sessioni on line e on campus
-Markov’s inequality, Chebyshev’s inequality, Jensen inequality;
-Empirical probability and the law of large numbers, Central limit theorem;
-Exercises;
Week 11 Contenuto sessioni on line e on campus
-Monte Carlo's method
-Stochastic processes, filtrations;
-Martingales, Submartingales, Supermartingales and relative properties;
Week 12 Contenuto sessioni on line e on campus
-Binomial asset pricing model, risk-neutral probability measure, discounted stock prices are martingales;
-Exercises;
-Second midterm;