Course teached as: B018782 - PROCESSI STOCASTICI Second Cycle Degree in MATHEMATICS Curriculum GENERALE
Teaching Language
Italian
Course Content
Definitions of Brownian motion, stochastic integrals, Ito formula and stochastic differential equations. Applications of these concepts in finance (options, pricing of derivative securities, interest rate curve)
Paul Wilmott - Sam Howison - Jeff Dewynne: The Mathematics of Financial Derivatives- Cambridge University Press
An Introduction to Stochastic Differential Equations by Lawrence C. Evans
Learning Objectives
The course aims to provide the students with fundamental knowledge and understanding in Financial Mathematics and Stochastic Differential Equations. One of the aims is to let the students develop basic technical skills, and critical thinking, needed when modelling and solving mathematical problems in different settings. Special attention will be paid to help the students to develop communication skills necessary for teamwork. The course covers topics and provides learning skills that are needed, or strongly suggested, to pursue a degree in mathematics or in any scientific subject.
Prerequisites
Courses to be used as requirements (required and/or recommended): Probability, Statistics, Calculus I and II, Ordinary Differential Equations
Courses required:Probability, Statistics, Calculus I and II
Courses recommended: Ordinary Differential Equations
Teaching Methods
CFU: 9
Total hours of the course: 220
Hours reserved to private study and other indivual formative activities: 148
Contact hours for: Lectures (hours): 72
Lectures: Presentation of the theory described in the course program, with teacher-student direct interaction, to ensure a full understanding of the subject.
Moodle learning platform: online teacher-student interaction and posting of additional notes.
Remark: The suggested reading includes supplementary material that may be useful for further personal studies in mathematics or in any scientific subject.
Further information
Office hours: Monday and Tuesday from 1.30 p.m. to 3.3O p.m.
Contact:
Dipartimento di Matematica e Informatica "Ulisse Dini"
Viale Morgagni, 67/a
50134 FIRENZE
Tel: 055 2751405
Email: vespri@math.unifi.it
Type of Assessment
Oral examination: A number of questions are posed. The oral examination is designed to evaluate the degree of understanding of the theory presented in the course. In the assessment, special attention is paid to communication skills, critical thinking and appropriate use of mathematical language.
Course program
Topics:
Financial Markets
Stocks and Bonds
European and Asian options
Brownian motion (outline)
Black and Scholes model
Black and Scholes equation
Partial differential equations (outline)
American options - problems with obstacle and free boundary (outline)
Numerical solution of the equations of Black and Scholes (European options)
Finite difference method
LSU, SOR, Crank-Nicholson
Binomial method
Numerical solution of the equations of Black and Scholes (American options)
exotic options
compound options
chooser options
barrier options
Asian options
lookback options
Russian options
Stop loss options
Options with transaction costs
Pricing of bonds
The yield curve
Stochastic interest rate
Equation for the pricing of bonds
Options on bonds
Swaps, Floors, Caps
Options on swaps, caps and floors
convertible bonds
Convertible bonds with an interest rate stochastic
Outline of the concepts of probability already faced in previous courses (Probability spaces, Concept of Independence, Random Variables, Media and Variance, inequality Chebichev, Borel-Cantelli Lemma, Martingale)
Brownian motion and white noise
Stochastic integrals and Ito formula
Stochastic differential equations (existence and uniqueness theorem )
Applications in finance