Taylor Expansions of Option Prices with applications to Interest Rate Theory
Freitag, 20.4.07, 11:15-12:15, Raum 404, Eckerstr. 1
(Joint work with Josef Teichmann) \n\nWe apply results of Malliavin-Thalmaier-Watanabe for strong and weak Taylor expansions of solutions of perturbed stochastic differential equations (SDEs). In particular, we work out weight expressions for the Taylor coefficients of the expansion. The results are applied to LIBOR market models in order to deal with the typical stochastic drift and with stochastic volatility. In contrast to other accurate methods like numerical schemes for the full SDE, we obtain easily tractable expressions for accurate pricing. In particular, we present an easily tractable alternative to "freezing the drift" in LIBOR market models, which has an accuracy similar to the full numerical scheme. Numerical examples underline the results.
Pricing of financial derivatives using multivariate jump processes
Freitag, 6.7.07, 11:15-12:15, Raum 404, Eckerstr. 1
For one-dimensional Lévy processes option pricing by solving the\ncorresponding partial integro-differential equations has been studied by\nseveral authors. We extend the results to multivariate jump processes\nusing Lévy copulas. The partial integrodifferential equations are\ndiscretized by sparse tensor product Finite Element spaces. Since the\nmulti-dimensional tail integrals have singularities at the origin and on\nthe axes, variable order composite quadrature formulas are employed for\nthe computation of the integral part. Numerical examples are given.\n
This is joint work with N. Reich.
Regulatory networks in T cells: from experiments to models and back
Freitag, 13.7.07, 11:15-12:15, Raum 404, Eckerstr. 1
We have developed mathematical models for gene-regulatory modules that govern the proliferation and differentiation of T cells and tested their predictions experimentally. I will discuss three examples: (1) communication between conventional and regulatory T cells through the cytokine interleukin-2, (2) stochastic expression of cytokine genes, and (3) differentiation of Th1 memory cells. While analysing the network topology of these modules can lead to interesting results, the modeling of molecular interaction mechanisms in space and time may substantially alter - complicate or simplify - the picture obtained from a naive look at network topology alone. We uncovered bifurcation scenarios involved in cellular decision making as well as well as a highly stochastic mode of T-cell effector gene expression.