  We develop a numerical method for pricing multidimensional vanilla options in
the Black-Scholes framework. In low dimensions, we improve an adaptive
integration algorithm proposed by two of the authors by introducing a new
splitting strategy based on a geometrical criterion. In higher dimensions, this
new algorithm is used as a control variate after a dimension reduction based on
principal component analysis. Numerical tests are performed on the pricing of
basket, put on minimum and digital options in dimensions up to ten.
