Sequential nongaussian simulations using the FGM Copula

Copulas are statistical tools for modelling the multivariate structure between variables in a distribution free way. They are beginning to be used in financial literature as an alternative to the multivariate normal. After reviewing the main properties of bivariate and multivariate copulas, three multivariate families are presented: the classical multivariate normal, the Fairlie-Gumbel-Morganstern and a broad class called Archimedean copulas which can have interesting asymmetric tail dependence. A comparison between the Clayton copula and the bivariate gaussian illustrates this point. The rest of the paper focusses on the FGM copula. A sequential method for simulating realisations of random functions with this copula is given. These are generated with uniform marginal distributions but can be rescaled to have any other marginal distribution. One specific class of FGM random functions (those with null bivariate terms but nonzero trivariate terms) is then studied in detail. Because of the choice of parameter values, their 1 and 2 point statistics are indistinguishable from those of independent uniform variables. However their 3 point statistics are markedly different. We show that four of the eight “corners” of their 3-copulas are “heavier” than for independent data, and that the other four are correspondingly “lighter”. This means that even if their marginal distributions are transformed to normality, these random functions retain a distinctly nongaussian flavour. One practical implication of having heavier or lighter corners is that the fluid flow characteristics of these FGM random functions are modified. The paper concludes by drawing attention to the need to check the higher order statistics in addition to 1 and 2 point statistics when using simulation algorithms.