Stochastic calculus
with applications to mathematical finance 1999-2000
Course Contents
M. Loeve wrote in 1973: "Martingales, Markov dependence and stationarity are the only three
dependence concepts so far isolated which are sufficiently general
and sufficiently amenable to investigation, yet with a great number of deep properties".
Of these three concepts we will treat martingales only. A standard example
of a martingale is Brownian Motion (already used by Bachelier in 1900 to describe
stock price fluctuations), which lies at the basis of the definition and construction
of stochastic integrals.
Stochastic integrals are first defined by It\^o in the fourties
with Brownian motion as integrator. The problem here is that a pathwise
construction as Stieltjes Integrals is not possible, since the paths of Brownian Motion
have infinite variation on compact intervals. Later on the theory has been
expanded to include integration with respect to semimartingales. Actually these processes
form the most general class for which stochastic integration can be
defined in a sensible way. In this course we will concentrate on stochastic integration
with respect to locally square integrable martingales with continuous sample
paths. This yields a class of processes that is sufficiently large for a variety of
applications although there are also shortcomings.
An area of vivid research in
applied probability these days is that of Mathematical Finance where especially (but not only)
processes with continuous sample paths and stochastic integrals are
used as a vehicle to model behaviour on stock markets.
The program of the first part of the course comprises the following subjects.
Martingales, stopping times, filtrations, Doob-Meyer decomposition, Brownian Motion,
Stochastic Integral, Ito rule, representation of continuous martingales in terms of
Brownian Motion, Girsanov theorem, stochastic differential equations (existence
and uniqueness of weak and strong solutions). A good impression of this part of the
program can be obtained by having a look at chapters 1,3 and 5 of the book
Brownian Motion and Stochastic Calculus by Ioannis Karatzas & Steve Shreve (2nd edition,
Graduate Texts in Mathematics 113, Springer).
In the second part of the course (2 lectures), the ideas of the first part of the course will
be applied to term-structure models. It is clear that in the bond market the
prices of bonds cannot follow a Geometric Brownian Motion since the bonds will pay there fixed
principal values at maturity date. Of course, this also implies that,
when pricing bond options, the Black-Scholes formula for stocks as treated in the first part
of this course should be adpated.
The ideas of modelling the term structure and the consequences for option pricing in the case
of bonds will be illustrated from an empirical perspective.
No special knowledge about the theory of stochastic processes is required, but participants are
assumed to be familiar with measure theoretic probability.
Literature
Available are the
transparencies of the first part
and some additional comments
(both as PostScript files).
To Peter Spreij's homepage.
Email: spreij@science.uva.nl