19th GAMM-Seminar Leipzig on
High-dimensional problems - Numerical treatment and applications

Max-Planck-Institute for Mathematics in the Sciences
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  19th GAMM-Seminar
January, 23th-25th, 2003
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  Abstract Christoph Reisinger, Fri, 16.00-16.25 Previous Contents Next  
  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.

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