Note on "The smoothing effect of integration in R and the ANOVA decomposition"

This is a note on Math. Comp. 82 (2013), 383-400. We first report a mistake, in that the main result Theorem 3.1, though correct, does not as claimed apply to the Asian option pricing problem. This is because assumption (3.3) in the theorem is not satisfied by the Asian option pricing problem. In this note we present a strengthened theorem, which removes that assumption. The new theorem is immediately applicable to the Asian option pricing problem with the standard and Brownian bridge constructions. Thus the option pricing conclusions of our original paper stand.