318006 Selected Topics in Mathematical Finance (V) (WiSe 2007/2008)

This course is concerned with the mathematical foundations
and applications of arbitrage pricing theory for multiple
securities (on stochastic financial markets in continuous
time). The special case of just one stochastic security on
the market has been treated in last semester's course on
Financial Economics 2 (Prof. Riedel) - which is a helpful
preparation, but no absolute prerequisite for the present
course.

Two main results are at the heart of the theory:
(a) the equivalence, modulo integrability conditions, of
(1) absence of arbitrage on a market with multiple
securities, (2) existence of an equivalent martingale
measure, (3) existence of a state-price deflator;
(b) the equivalence of (1) market completeness and (2)
full rank, almost surely, of the volatility matrix.
The present course will provide rigorous formulations and
proofs of these equivalence statements.

From the first equivalence assertion, a general pricing
formula for European call options will be derived. More
advanced applications, such as stochastic volatility
models, will be covered in the present course as well.

As mathematical excursions, complete proofs will be given
for Ito's formula, Girsanov's theorem, the martingale
representation theorem, the converse of Girsanov's
theorem, and the diffusion invariance principle.

The minimum technical prerequisite for this course is a
basic familiarity with stochastic calculus (e.g. Sections
4.1-4.5 of Björk [2004]). However, the course is intended
primarily for students who have followed the
aforementioned Financial Economics 2.

Bibliography

T. Björk [2004]: Arbitrage theory in continuous time, 2nd
ed., Oxford: Oxford University Press.
D. Duffie [2001]: Dynamic asset pricing theory, 3rd ed.,
Princeton (NJ): Princeton University Press.
A. Friedman [1975]: Stochastic differential equations and
applications. Vol. 1, New York: Academic Press. [Reprint,
two volumes bound in one, by Dover Publications, Mineola
(NY), 2006]
B. Øksendal [2007]: Stochastic differential equations. An
introduction with applications, 6th ed., Berlin: Springer.

The main reference for this course will be Duffie [2001].