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

Abstract

Summary. In this short article, we describe how the correlation of typical diffusion processes arising e.g. in financial modelling can be exploited -by means of asymptotic analysis of principal components -to make Feynman-Kac PDEs of high dimension computationally tractable. We explore the links to dimension adaptive sparse grids [GG03], anchored ANOVA decompositions and dimension-wise integration [GH10], and the embedding in infinite-dimensional weighted spaces [SW98]. The approach is shown to give sufficient accuracy for the valuation of index options in practice. These numerical findings are backed up by a complexity analysis that explains the independence of the computational effort of the dimension in relevant parameter regimes.

Error analysis of truncated expansion solutions to high-dimensional parabolic PDEs

“…Remark 12. Computing u ξ according to (26) requires the computation of one r-dimensional PDE, N −r of (r+1)-dimensional PDEs and (N −r)(N −r−1)/2 of (r+2)-dimensional PDEs.…”

“…The relation between PDE expansions and anchored ANOVA is pointed out in [26] and [30]; a higher order extension is also sketched in [13]. A practically useful feature of decomposition methods is their inherent parallelism, which is exploited in [29] and more recently in [31].…”

Numerical valuation of derivatives in high-dimensional settings via partial differential equation expansions

“…3 Generalisations and relation to anchored ANOVA Anchored ANOVA-decompositions are used in [9] to obtain dimension-adaptive approximations to option values expressed as integrals over high-dimensional spaces; [23] point out a relation to the expansions from [22] by utilising the integral representation of the solution to the heat equation; [21] discusses these ideas jointly in the PDE context. We formulate the problem in slightly more general terms here as befits the applications later on.…”

“…, a N ). This has been successfully applied to quadrature problems from finance in [9], and its relation to the PDE expansions in [22], which form the basis for the present work, is highlighted in [21] and [23]. Key to the efficiency of this approximation as a numerical method is that the relative importance of u v decays rapidly with increasing |v|, as |v| is the dimension of the coordinate space of u v .…”

Implicit Entropic Market Risk-Premium from Interest Rate Derivatives

“…Gerstner and Griebel (2003) andReisinger (2013) are examples of multidimensional grid approaches with complexity reduction for integral valuations in multidimensional environments.…”