The smoothing effect of the ANOVA decomposition

We show that the lower-order terms in the ANOVA decomposition of a function f(x)@?max(@f(x),0) for [email protected]?[0,1]^d, with @f a smooth function, may be smoother than f itself. Specifically, f in general belongs only to W"d","~^1, i.e., f has one essentially bounded derivative with respect to any component of x, whereas, for each [email protected]?{1,...,d}, the ANOVA term f"u (which depends only on the variables x"j with [email protected]?u) belongs to W"d","~^1^+^@t, where @t is the number of indices [email protected]?{1,...,d}@?u for which @[email protected]/@?x"k 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 also employing 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.