The smoothing effect of the ANOVA decomposition

Abstract

We show that the lower-order terms in the ANOVA decomposition of a function f ( x ) ≔ max ( ϕ ( x ) , 0 ) for x ∈ [ 0 , 1 ] d , with ϕ 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 u ⊆ { 1 , … , d } , the ANOVA term f u (which depends only on the variables x j with j ∈ u ) belongs to W d , ∞ 1 + τ , where τ is the number of indices k ∈ { 1 , … , d } ∖ u for which ∂ ϕ / ∂ 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.