BFS 2002

Poster Presentation

On the relation between approximation rates of stochastic integrals and properties of its integrands

Stefan Geiss, Christel Geiss

The approximation of stochastic integrals appears in Stochastic Finance while replacing continuously adjusted portfolios by discretely adjusted ones, where often equidistant time nets are used. With respect to the quadratic risk there are situations in which a significant better asymptotic quadratic error is obtained via arbitrary deterministic time nets (for example for the Binary option).
We investigate how much the asymptotics for equidistant nets and arbitrary deterministic nets differ from each other in case of European type options obtained by deterministic pay-off functions applied to a price process modeled by a diffusion.
In particular we give a characterization, that the approximation rate for the quadratic risk for equidistant nets (with $n$ time knots) behaves like $n^{-\eta}$, $0<\eta\le 1/2$, in terms of (1) the asymptotics of certain variances of the hedging strategy, (2) the asymptotics of a certain $L_2$-convexity of the value process, (3) a decomposition of the portfolio using the K-functional from interpolation theory.