Efficient Hierarchical Approximation of High‐Dimensional Option Pricing Problems

Abstract

A major challenge in computational finance is the pricing of options that depend on a large number of risk factors. Prominent examples are basket or index options where dozens or even hundreds of stocks constitute the underlying asset and determine the dimensionality of the corresponding degenerate parabolic equation. The objective of this article is to show how an efficient discretisation can be achieved by hierarchical approximation as well as asymptotic expansions of the underlying continuous problem. The relation to a number of state-of-the-art methods is highlighted.

On the construction of sparse tensor product spaces

“…In practice, also problems on arbitrary tensor product domains Ω 1 × Ω 2 × · · · × Ω m may appear, see e.g. [2,11,16,17,21,22,18]. We believe that our results can be generalized to such m-fold tensor product domains which, however, is rather technical and not straightforward.…”

Asymptotic Expansion Around Principal Components and the Complexity of Dimension Adaptive Algorithms

“…For the test example of the DAX given above, [RW07] report an error against a Monte Carlo simulation benchmark of < 0.06% of the option value, with an absolute error of 0.000073 at the strike K = 1, which is less than 1 basis point. Results for other (and smaller) baskets are comparable, such that the approximation seems sufficient for practical applications.…”

“…The computational aspects of such an approach are discussed in [RW07]; meanwhile, this has been developed further by [HKSW10]. Here, we focus on the underlying expansion itself, and discuss its position within recent work on fundamentally similar ideas.…”

Efficientd-multigrid preconditioners for sparse-grid solution of high-dimensional partial differential equations

“…To address anisotropy, the standard way in two dimensions is either to coarsen along only one dimension (the one where error components are strongly coupled) or else to do a full coarsening but to resort to line-relaxation along the strongly coupled dimension [17]. These techniques can be extended to arbitrary higher d. A relaxation method based on hyperplane relaxation has been proposed in [18], which is analogous to line-relaxation in two dimensions. Contrary to that approach, we proposed a method based on point smoothing and partial coarsening schemes in [19].…”