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This paper develops a new efficient scheme for approximations of expectationsof the solutions to stochastic differential equations (SDEs). In particular, wepresent a method for connecting approximate operators based on an asymptoticexpansion with multidimensional Malliavin weights to compute a targetexpectation value precisely. The mathematical validity is given based onWatanabe and Kusuoka theories in Malliavin calculus. Moreover, numericalexperiments for option pricing under local and stochastic volatility modelsconfirm the effectiveness of our scheme. Especially, our weak approximationsubstantially improves the accuracy at deep Out-of-The-Moneys (OTMs).

A weak approximation with asymptotic expansion and multidimensional Malliavin weights