Pricing Options on Big Baskets

Christoph Reisinger

Abstract: The task of pricing options on baskets of underlying assets results in integration or PDE problems, where the dimensionality of the equation corresponds to the number of assets in the basket and can become very high, e.g. for the DAX or 500 for the S&P. Direct numerical simulation via PDE approaches on conventional grids is not feasible if the dimension exceeds three or four as the size of the problem (in terms of unknowns) increases exponentially.

Sparse grids can partially overcome this curse of dimensionality by exploiting smoothness of the solution for approximation in sparse subspaces. We introduce a combination technique that couples splitting extrapolation with the conventional combination approach to construct an approximation of high order on the sparse grid from low order approximations on small full grids. Taking advantage of the full parallelism inherent to this technique we can handle problems of dimension up to roughly ten on high perfomance clusters. Multigrid techniques guarantee an efficient and robust solution of the anisotropic discrete equations for both European and American-style options, where complementarity conditions have to be fulfilled.

For higher-dimensional cases we observe that - viewed over longer periods - stocks are typically strongly correlated, even more so, if they belong to similar sectors. Within a branch there will be again stronger couplings, etc. This nested clustering results in an exponential decay of the eigenvalues of the covariance matrix. Principal components can be identified and the equations restricted to lower dimensional manifolds, where the above techniques are applicable.

For a solution of optimal attainable accuracy with limited resources (= memory, time) we have to balance the approximation error from the dimension-reduction against the discretisation error of the resulting equation. It is evident that none of the extreme cases, i.e. neither the exact solution of a one-dimensional approximation nor a very rough numerical approximation of the full system (if at all possible) is optimal. We give a few examples, where we estimate the error by asymptotic analysis and compare with results from control theory.