STOCHASTIC CALCULUS FOR FINANCE

This is a core course, whose main purpose is to introduce the theoretical tools of Stochastic Calculus lying underneath the mathematical approach to Finance, and which are used to price financial products, in particular options.

At the end of the course, students should have acquired the fundamentals of Stochastic Calculus.
Note that the course is held for second year students, who have been already exposed to practical work with financial derivatives. At this stage, then, they are expected to become aware of the genesis of the formulas that they already use. 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 by means of 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; stochastic (Itô) integrals.
b.2) Ability to analyse 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 results of (b.1)(b.2).

c) Making judgements:
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 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 a formal language, to make logic deductions; 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

Students need preferably be acquainted with the contents of the courses "Derivatives and insurance" and "Stochastic models for finance".

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

Grading is based on a final written exam. The written exam contains 4-6 questions, of which:
a) 2-3 are theoretical dissertation about a given subject, intended to verify knowledge of students about the topics of the course;
b) 3-4 are exercises to be solved (similar to those discussed during lectures and practise sessions) intended to verify the ability of students to apply their knowledge of theory to problem solution.

The oral exam is optional and can be acceded with a minimum of 16/30 at the written exam.

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.

English

The materials of the course are made available for download on a cloud storage. Access to the storage need to be requested to the instructor.
Materials for review are also made available online. Note that a knowledge on the main methods of integration of functions of one variable is considered mandatory (as the majority of exercises presented in the course require to solve 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.

written and oral

This subject deals with topics related to the macro-area "Human capital, health, education" and contributes to the achievement of one or more goals of U. N. Agenda for Sustainable Development