This paper develops a new efficient scheme for approximations of expectations of
the solutions to stochastic differential equations (SDEs). In particular, we present a
method for connecting approximate operators based on an asymptotic expansion with
multidimensional Malliavin weights to compute a target expectation value precisely. The
mathematical validity is given based on Watanabe and Kusuoka theories in Malliavin
calculus. Moreover, numerical experiments for option pricing under local and stochastic
volatility models confirm the effectiveness of our scheme. Especially, our weak approximation
substantially improve the accuracy at deep Out-of-The-Moneys (OTMs).