Mathematical Foundations of Finance
Contents:
- Review of probabilistic notation
(probability spaces; random variables, product spaces; filtrations;
trees; atoms; transition probabilities)
- Definition of adapted and predictable stochastic processes
- Conditional expectations (definition and basic properties)
- Martingales, submartingales, supermartingales
(definition and basic properties)
- Stopping times and their sigma-algebras
- Bayes' formula in connection with conditional expectations
- Implications between differnt kinds of convergence
- Weak convergence
- Central limit theorem
- Portmanteau theorem
- Weak convergence and unbounded functions
- Snell envelopes
- Decomposition of supermartingales
- Optimal stopping and American options
- Brownian motion and its properties
- Brownian motion with drift
- Introduction to stochastic integration
- Itô's lemma, Girsanov-Maruyama theorem
- Stochastic differential equations
- Lévy's characterization of Brownian motion
- Application: option pricing in the Samuelson model
- Feynman-Kac formula
- Numerical approximation schemes for stochastic differential equations
Grades:
The grades for participants of the
MAS Finance program
will be based on a written examination. The examination is shifted to Monday,
February 17, 2003, from 1 until 3pm. It will take place in lecture theatre
HG E7 in ETH's main building. The examination will be closed books,
participants only need something to write (no pencil please). Participants may
prepare and use one A4-page (one side) of handwritten notes (no copies), which they
prepared themselves and which they have to show at the beginning of the exam.
In case of doubt, they may show that page earlier for approval.
For the two participants who could not agree to the new date,
the examination takes place on February 7, 2003, from 1 until 2pm in
room HG F33.3.
Literature:
- David Williams: Probability with Martingales,
Cambridge University Press, Cambridge 1991,
ISBN 0-521-40605-6.
- Stanley R. Pliska:
Introduction to Mathematical Finance,
Discrete Time Models,
Blackwell Publishers Inc.,
Malden (USA) and Oxford (UK), 1997,
ISBN 1-557-86945-6.
- Damien Lamberton
and Bernard Lapeyre:
Introduction to Stochastic Calculus
Applied to Finance,
Chapman & Hall, London, 1996,
ISBN 0-412-71800-6.
- Hans Föllmer
and Alexander Schied:
Stochastic
Finance, An Introduction in Discrete Time,
De Gruyter Studies in Mathematics 27,
De Gruyter 2002, ISBN 3-11-017119-8
Links to:
Courses and Seminars,
Finance and Insurance,
RiskLab,
MAS Finance