  Quantization techniques have been applied in many challenging finance
applications, including pricing claims with path dependence and early exercise
features, stochastic optimal control, filtering problems and efficient
calibration of large derivative books. Recursive Marginal Quantization of the
Euler scheme has recently been proposed as an efficient numerical method for
evaluating functionals of solutions of stochastic differential equations. This
method involves recursively quantizing the conditional marginals of the
discrete-time Euler approximation of the underlying process. By generalizing
this approach, we show that it is possible to perform recursive marginal
quantization for two higher-order schemes: the Milstein scheme and a simplified
weak order 2.0 scheme. As part of this generalization a simple matrix
formulation is presented, allowing efficient implementation. We further extend
the applicability of recursive marginal quantization by showing how absorption
and reflection at the zero boundary may be incorporated, when this is
necessary. To illustrate the improved accuracy of the higher order schemes,
various computations are performed using geometric Brownian motion and its
generalization, the constant elasticity of variance model. For both processes,
we show numerical evidence of improved weak order convergence and we compare
the marginal distributions implied by the three schemes to the known analytical
distributions. By pricing European, Bermudan and Barrier options, further
evidence of improved accuracy of the higher order schemes is demonstrated.
