Construction of a Bivariate Distribution Accounting for Correlation Skew by Alternate Projections

A method based on successive adjustments of a discrete bivariate distribution consistently with target marginal distributions is presented, with the motivation of matching the correlation skew for rates and FX markets. Each step is performing a permutation of the coordinates to achieve optimal pairing consistently with the spread or cross FX distribution, by ensuring minimal perturbation using the Hoeffding-Fréchet map, as inspired by the theory of optimal transport. A proof of convergence in the special case of evenly distributed coordinates (Monte Carlo simulations) is given using alternate projections onto discrete sets results, although a more efficient version of the algorithm is introduced and investigated with numerical examples. Numerous tests are presented for the convergence speed and calibration to CMS spread options and cross FX markets. The method is compared with the copula method (which is used as a starting point of the algorithm) and the Austing model for the FX case, to value exotic payoffs. The robustness and efficiency of the method (it requires only sorting a vector of ℝN at each iteration) makes it easy to adapt in higher dimensions (equity markets) or to use as a tool to spot arbitrage in the rates market.