The smoothing effect of the ANOVA decomposition

We show that the lower-order terms in the ANOVA decomposition of a function f(x) := max(φ(x), 0) for x ∈ [0, 1], with φ a smooth function, may be smoother than f itself. Specifically, f in general belongs only toW d,∞, i.e., f has one essentially bounded derivative with respect to any component of x, whereas, for each u ⊆ {1, . . . , d}, the ANOVA term fu (which depends only on the variables xj with j ∈ u) belongs to W τ d,∞ , where τ is the number of indices k ∈ {1, . . . , d} \ u for which ∂φ/∂xk is never zero. As an application, we consider the integrand arising from pricing an arithmetic Asian option on a single stock with d time intervals. After transformation of the integral to the unit cube and employing also a boundary truncation strategy, we show that for both the standard and the Brownian bridge constructions of the paths, the ANOVA terms that depend on (d+1)/2 or fewer variables all have essentially bounded mixed first derivatives; similar but slightly weaker results hold for the principal components construction. This may explain why quasi-Monte Carlo and sparse grid approximations of option pricing integrals often exhibit nearly first order convergence, in spite of lacking the smoothness required by the conventional theories.