American Option Pricing on Sparse Grids
Christoph Reisinger (Uni Heidelberg)
The pricing of multivariate derivatives in the Black-Scholes context
leads to high-dimensional integration or PDE problems. By considering
stochastic models not only for the underlying assets, but also for
volatilities or other market parameters, the dimension is increased
even further. Hence efficient numerical algorithms for multi-dimensional
partial differential equations of Black-Scholes type are required.
The sparse grid combination technique provides a parallel framework
for handling medium-size problems of dimension up to roughly ten.
Additionally we consider a type of splitting extrapolation and integrate
it into the sparse grid structure for higher-order convergence. This
is essential for reliable estimation of sensitives, which are important
hedge parameters known as the "Greeks". Furthermore, they can be used
for an asymptotical analysis of markets with volatilities that show
high-frequency mean reversion. Due to the typical non-smooth shape
of the initial conditions a suitable transformation of the problem
is fundamental for this approach. Adaptivity can then easily be
introduced via so-called graded sparse grids.
Numerical results for the example of multi-dimensional basket puts
of European and American style will demonstrate the capabilities
(and limitations) of this technique.