FINANCIAL MATHEMATICS

The Stochastic Calculus course plays a fundamental role in the degree programme. It introduces the theoretical tools underlying the mathematical approach to finance, including ordinary and stochastic differential equations and Itô calculus. During the course, these tools are applied to problems in economics and finance, with particular attention to option pricing.

At the end of the course, students should have acquired the fundamentals of Stochastic Calculus.
The scope is, on the one hand, to build common grounds with theorists so to enhance communication, and on the other hand to help develop a critical thinking so to understand the advantages and the criticalities of the models in use.

More specifically:

a) Knowledge and understanding
a.1) Knowledge of definitions of basic tools of stochastic calculus, such as stochastic processes, filtrations, stochastic integral and differentials, ordinary and stochastic differential equations.
a.2) Interpretation of the above definitions employing a span of crucial financial examples.

b) Ability to apply knowledge and understanding
b.1) Ability to compute solutions of simple ordinary and stochastic differential equations, stochastic differentials, and stochastic (Itô) integrals.
b.2) Ability to analyze properties of stochastic processes on infinite probability spaces, such as mean value and variance, behaviour in the long run.
b.3) Ability to derive the Black and Scholes equation as a result of (b.1)(b.2).

c) Making judgments:
c.1) improved ability to critically understand perspectives, advantages, and criticalities of instruments in use to mathematical finance.

d) Communication
d.1) Ability to present, discuss, and prove the mathematical correctness of option pricing via the Black and Scholes model;
d.2) Ability to interact with financial model designers and theorists.

e) (Lifelong) learning skills
e.1) Improved ability to handle formal language, make logical deductions; and enhance rigorous rational thinking;
e.2) Improved ability to translate a problem into formal terms, solve it, and interpret the solution in terms of the original problem.

Prerequisites to the course are:
- the main techniques of integration for functions of one variable,
- calculus for multiple variables are given as known
- basics in probability theory

a) Ordinary Differential equations
a.1) Basic definition, economic/financial examples.
a.2) Separable equations. First order linear equations.
a.4) Existence and uniqueness for the Cauchy problem. Qualitative study of ODEs.
a.5) An application: the Solow model for economic growth.

b) Stochastic Processes
b.1) Random variables, stochastic processes. Examples of stochastic processes in Finance.
b.2) Sigma algebras and filtrations.
b.3) Conditional expectation. Martingales. Meaning of conditional expectation and of martingale property in financial examples.


c) Brownian Motion
c.1) Introducing a Gaussian disturbance in an ODE. Definition of a Wiener Process/Brownian Motion.
c.2) Construction of a Brownian Motion as limit of scaled random walks.
c.3) Properties of Brownian Motion (normal distribution of increments, quadratic variation, martingale property).

d) Ito Integral
d.1) Construction of the integral as limit of integrals of approximating simple processes.
d.2) Properties of the Ito integral.
d.3) Ito processes; Ito Doeblin formula.

e) Stochastic Differential equations
e.1) Linear equations.
e.2) Geometric Brownian Motion; solution formula, expected value.
e.3) The Vasicek Interest Rate Model.
e.4) The Cox-Ingersoll-Ross Model.

f) Black and Scholes Model for European call options
f.1) Setting of the model, assumptions.
f.2) Derivation of BS equation.
f.3) The Greeks.
f.4) The Feynman-Kač theorem. Application to Black and Scholes model, interpretation.

Steven E. Shreve (2000)," Stochastic Calculus for Finance II". Continuous Time Models, Springer, Chapters 1 – 4.

Tomas Bjork, "Arbitrage Theory in Continuous Time", Oxford University Press.

Lecture Notes

The examination consists of a written test and an oral examination. The written test specifically assesses the ability to apply the mathematical tools covered in the course to the solution of exercises and problems; the oral examination assesses the theoretical understanding of the topics, the ability to discuss the written test, and the correct use of mathematical language.

The written test lasts approximately 2.5 hours and consists of 5 questions, of which:

a) 2-3 are theoretical essays on a given topic, aimed at assessing students’ knowledge of the topics covered in the course;

b) 2-3 are exercises to be solved, similar to those discussed during lectures and practical sessions, aimed at assessing students’ ability to apply their theoretical knowledge to problem solving.

The skills acquired by students are assessed through the solution of the proposed problems; theoretical knowledge is assessed by requiring students to justify their answers in detail on the basis of the relevant theoretical results, such as definitions and theorems.

During the written test, the use of electronic devices of any kind, including calculators, is not allowed.

The oral examination takes place in the days immediately following the written test. In order to be admitted to the oral examination, students must obtain at least 16 points in the written test. The oral examination begins with a discussion of the written test and may then extend, if necessary, to the other topics covered in the course.

written and oral

The instructor is responsible for ensuring the authenticity and originality of all examinations and coursework. In cases of suspected academic misconduct, an additional on-site assessment may be required during the exams, which may differ from the standard format.

The score for the written examination ranges from 0 to 30 points and represents the grade out of thirty. Additional, more complex questions, for a maximum total of 6 points, may be added to allow for the possible award of honours.

The ordinary 30 points are distributed as follows:

* 22-24 points for basic questions;
* 6-8 points for questions of moderate difficulty;
* 6 points for more complex questions.

Answers that are not adequately justified will receive no credit. It is therefore important to explain clearly what is being done and why.

The oral examination mainly has the function of confirming the grade obtained in the written examination and, on average, may change that grade by an interval ranging from -3 to +3 points. This indication should be understood as an approximate description of what may happen, not as a formal rule for assigning the final grade.

In particular, should serious discrepancies emerge between the assessment of the written examination and that of the oral examination, the assessment of the oral examination will prevail, without any point constraints.

The topics of the course are conveyed by means of face-to-face lectures and practise sessions, and weekly office hours after class.
In particular, during the course time, office hours are held in public. Students may come and ask questions or simply sit and listen to other students’ questions and to the instructor’s answers. A further discussion is also possible on appointment.
The topics discussed in class are supported by materials made available for download on a cloud storage, and include:
a) the complete set of slides/lecture notes;
b) weekly sets of homework exercises;
c) a list of previous exams, all completely solved
d) all relevant information about the course, and real time updates.

The materials of the course are made available for download on moodle.unive.it. Note that knowledge of the main methods of integration of functions of one variable is considered mandatory (as the majority of exercises presented in the course require solving integrals). In particular, students need to be able to integrate by parts and by substitution, as well as to integrate simple rational functions.

Accessibility, Disability, and Inclusion
Accommodation and support services for students with disabilities and students with specific learning impairments

Ca' Foscari abides by Italian Law (Law 17/1999; Law 170/2010) regarding support services and accommodation available to students with disabilities. This includes students with mobility, visual, hearing and other disabilities (Law 17/1999), and specific learning impairments (Law 170/2010). If you have a disability or impairment that requires accommodations (i.e., alternate testing, readers, note takers, or interpreters) please contact the Disability and Accessibility Offices in Student Services: disabilita@unive.it.