Quantizations of probability measures and preservation of the convex order

Two probability measures admit a martingale transition if and only if they are ordered in the convex order (Kellerer, 1972). We show that the commonly used quantization method, L2L2-quantization, does not have the property of preserving the convex order. We introduce an alternative quantization method and demonstrate that it preserves the convex order. This result has implications concerning the choice of quantization methods for the numerical construction of martingales with specified marginals.