Solving the FX cross-smiles problem — rate of convergence for Sinkhorn marginals, and the finite-option case

We adapt the arguments in Guyon (2020),Nutz (2022) to the problem of constructing a joint law μ∗ for two FX rates X and Y consistent with observed European option prices on all three cross-rates. The usual Gibbs-type exponential ansatz for μ∗ leads to three coupled integral equations for the Schrödinger potentials u(x), v(y), and yw(xy), and we prove O(1n) convergence for the marginals (in the total variation metric) for the running average of the Sinkhorn iterates, assuming an admissible law exists (typically using ≈50 iterations in practice). The primary application here is risk-neutral pricing of e.g. Basket, Quanto, Best-of or other variations of Rainbow options on X and Y so as to be consistent with all three smiles. We also consider the finite-option case via the usual duality for one-period markets between the (primal) entropy minimization problem over calibrated measures and the (dual) exponential utility maximization problem à la Föllmer&Schied (Föllmer and Schied, 2004).