  This paper investigates the use of multiple directions of stratification as a
variance reduction technique for Monte Carlo simulations of path-dependent
options driven by Gaussian vectors. The precision of the method depends on the
choice of the directions of stratification and the allocation rule within each
strata. Several choices have been proposed but, even if they provide variance
reduction, their implementation is computationally intensive and not applicable
to realistic payoffs, in particular not to Asian options with barrier.
Moreover, all these previously published methods employ orthogonal directions
for multiple stratification. In this work we investigate the use of algorithms
producing convenient directions, generally non-orthogonal, combining a lower
computational cost with a comparable variance reduction. In addition, we study
the accuracy of optimal allocation in terms of variance reduction compared to
the Latin Hypercube Sampling. We consider the directions obtained by the Linear
Transformation and the Principal Component Analysis. We introduce a new
procedure based on the Linear Approximation of the explained variance of the
payoff using the law of total variance. In addition, we exhibit a novel
algorithm that permits to correctly generate normal vectors stratified along
non-orthogonal directions. Finally, we illustrate the efficiency of these
algorithms in the computation of the price of different path-dependent options
with and without barriers in the Black-Scholes and in the Cox-Ingersoll-Ross
markets.
