69827 - Financial Mathematics A.C.

Learning outcomes

By the end of the course, students know the basic mathematical concepts and methods of financial mathematics. In particular, they know the stochastic calculus tools and the models that describe the evolution of stock prices and interest rates, and that allow one to value financial derivatives. Students are capable of - understanding traditional models for asset pricing; - performing simulations and numerical investigations based on these models.

Course contents

Basics of financial mathematics and probability theory; continuous and discrete compounded interest rate; stochastic processes; Wiener processes and martingales; stochastic differential equations; scochastic models for stock prices; Ito's stochastic calculus; derivatives and financial options; no-arbitrage pricing (under the equivalent martingale measure); vanilla European options (Black-Scholes model and binomial model); basics of exotic options and stochastic volatility models; numerical approximation of option prices, computational aspects; Monte Carlo simulation (pricing by discounted expectation), the finite difference method; pricing options with Matlab.

Readings/Bibliography

Besides the slides prepared by the teacher, the suggested textbook is:

Luca Ballestra, Matematica Finanziaria, McGraw-Hill Create, 2019, ISBN: 9781307461459

For a better understanding of the theoritical models and methods, the following textbook is suggested too:

Menoncin Francesco, Mercati finanziari e gestione del rischio, ISEDI, 2006. 

Options, Futures, and Other Derivatives - Pearson

Since they are suggested textbooks, any Edition is fine. 

On the iol course' website students may find the slides of the course as well as Matlab codes that will be used in the practical lessons

Teaching methods

The first part of the course (about 22 hours) consists of both theoretical lessons and numerical examples explained by the teacher. The second part of the course is hold with the aid of Matlab, so that students can implement on the computer the option pricing models and the numerical methods they have learnt during the theoretical lessons.

Assessment methods

The exam is an oral discussion on three macro-subjects of the course, corresponding to three different questions chosen by the teacher (one of which could be replaced by the discussion of a project, as explained below).
To answer properly, students might be asked to write down mathematical formulas and passeges, as well as short numerical
examples (which have been presented in class, anyway). Each macro-subject is explained in the slides, or it is one
of the practical problem solved (for which the Matlab codes are available on the iol website).
Students will not be asked to answer questions on subjects, examples or practical cases that have not been taught in class.
Students can freely ask to subtitute one of the three macro-subjects with a project on a case study assigned by the teacher. Each student will develop the project before the exam, working either alone or in a team of no more than four persons. Every project requires the use of the Matlab software. Every student who decides to develop a project (either alone or in team) will show the project during the exam (interacting with the teacher), so as to replace one the above three macro-subjects (in other words, the exam will consists on the discussion of the project and on a talk on only two macro-subjects).
Each of the three macro-subjects (or of the two macrosubjects and the project) is equally weighted as far as the final grade is concerned. That is, both each macro-subject and the eventual project will be assigned up to 10 points, and to a student who answers very accurately, precisely and clearly to all the questions (and, eventually, who makes a brilliant discussion of the project) will be given the 30 cum laude grade. The (eventual project) will have the same maximum score (ten points) as each of the macro-subjects.

In summary, at the exam students can decide if

1) either answering to three open questions (chosen by the teacher)

2) or answering to two open questions (chosen by the teacher) and discussing a project. The project is assigned by the teacher prior to the exam and will be dealt with by students with the aid of Matlab and, eventually, by working in a small team.

The maximum grade that a student can have (30 cum laude) does not depend on if the student decides to develop the "practical" project or not. 

The use of formularies is not allowed. Pocket calculators are not needed. 

The final grade in the overall exam (Mathematics) will be the average of the grades taken in the single modules (Mathematical Analysis and Financial Mathematics).

Teaching tools

The theoretical lessons will be mainly taught using slides followed by explanations from the teacher. Practical lessons
will also be hold, which are based on the Matlab software.

Office hours

See the website of Luca Vincenzo Ballestra