Efficiency of alternative multivariate ICDF transformations (Rosenblatt, Nataf, copula) for RQMC-based Fourier pricing

Investigate the efficiency of applying the Rosenblatt transformation, the Nataf transformation, and copula-based methods to handle multivariate inverse cumulative distribution function mappings required for randomized quasi-Monte Carlo domain transformations in the Fourier pricing of multi-asset options, and ascertain under which conditions these alternatives preserve sufficient regularity to achieve near-optimal RQMC convergence rates.

Background

In the proposed framework, randomized quasi-Monte Carlo (RQMC) is applied to Fourier-space integrals defined over unbounded domains by transforming them to the unit hypercube via an inverse CDF (ICDF). For dependent assets, computing a multivariate ICDF is challenging; the paper circumvents this by expressing the target distribution in a normal-mixture form and using Cholesky or eigenvalue decompositions to eliminate dependence, thereby avoiding direct multivariate ICDF evaluation.

The authors note that alternative approaches for handling multivariate ICDFs include the Rosenblatt transformation, the Nataf transformation, and copula-based constructions. While these methods could be viable, the authors explicitly defer analyzing and quantifying their efficiency, leaving a concrete methodological question open regarding their performance and suitability in maintaining the smoothness properties needed for RQMC to attain favorable convergence.