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
Basics of Stochastic Analysis by Timo Seppäläinen
Learning Objectives
Knowledge acquired: stochastic differential equations derivatives
Competence acquired: in Mathematical Finance
Skills acquired:mathematical modelling in Finance
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: 150
Contact hours for: Lectures (hours): 70
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
The exam is oral and consists of:
- In a technical conversation of about an hour with the teacher on concepts and theorems related to stochastic differential equations and the equations of finance in order to bring out the knowledge acquired on the subject
- In anessay on a specific topic of mathematical finance in order to bring out the critical skills acquired.
The aim of this analytical graduation performance of the student is to reliably assess the level of achievement of the expected learning outcomes described above.
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