311504 Continuous-time finance II (V) (WiSe 2008/2009)

This course is concerned with the mathematical foundations and applications of continuous-time arbitrage pricing theory for multiple securities.

In the first part of the course, we shall prove the two fundamental theorems of asset pricing:
(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.

In the second part, we will study - as much as time permits - examples for applications: (a) a general pricing formula for European vanilla options, (b) option pricing in affine models, (c) or a simplistic analysis for default timing.

The third part of this lecture is devoted to the equilibrium foundations of continuous-time finance. In particular, we will sketch an existence proof for a security-spot market equilibrium under the assumption of dynamic spanning.

Students should have some basic acquaintance with stochastic calculus (in particular, Itô's lemma and Girsanov's theorem).

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.
B. Øksendal [2007]: Stochastic differential equations. An
introduction with applications, 6th ed., Berlin: Springer.
J.M. Steele [2001]: Stochastic calculus and financial applications, New York: Springer.

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