Fast calculation of cheapest-to-deliver curves

Alexander Kemarsky; Wouter Van Der Helm; Vladimir Piterbarg

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Multi-currency collateral agreements introduce optionality that requires discounting with a “cheapest-to-deliver” curve, which can be computed via Monte Carlo but benefits from faster analytic approximations.
Following the post-2008 focus on accurate collateralised pricing, this work revisits this long-standing problem and proposes an approximation that improves speed, accuracy, and scalability to any number of currencies.
The method leverages established tools such as the Clark algorithm for maxima of normal variables and Gauss–Hermite quadrature, alongside optimized discretization and factor-reduction choices.
The resulting approach is highly accurate, easy to implement, computationally efficient, and validated through extensive testing across realistic market scenarios.

Abstract

Trades subject to a collateral agreement that allows cash in multiple currencies need to be discounted with the so-called cheapest-to-deliver curve, which embeds the optionality inherent in the choice of collateral currencies. These curves can be computed essentially exactly via a Monte Carlo simulation, but analytical approximations allow for faster calculation. We revisit this long-standing problem and propose an approximation that improves on the existing methods in terms of speed, accuracy and extendibility to any number of currencies.

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