Faster, More Accurate Calculations for Cheapest-to-Deliver Curves Emerge

The financial industry is seeing advancements in the calculation of “cheapest-to-deliver” (CTD) discount curves, essential for accurately pricing derivatives subject to multi-currency collateral agreements. A new analytical approach promises to significantly improve speed, accuracy, and scalability compared to traditional methods, particularly Monte Carlo simulations.

The Challenge of Multi-Currency Collateral

When collateral agreements allow cash to be posted in multiple currencies, determining the most cost-effective currency to deliver—the cheapest-to-deliver—becomes a complex calculation. This optionality requires a specialized discount curve that accounts for the potential choices. Traditionally, these curves have been computed using computationally intensive Monte Carlo simulations.

A New Analytical Approximation

Researchers have developed an analytical approximation that leverages established mathematical tools, including the Clark algorithm for maxima of normal variables and Gauss–Hermite quadrature. Combined with optimized discretization and factor-reduction techniques, this method offers a faster and more scalable alternative to Monte Carlo simulations. According to a recent paper, the approach has been extensively tested across realistic market scenarios and demonstrates high accuracy. Source

How it Works: Identifying the Cheapest Currency

The “cheapest-to-deliver” currency is determined by identifying the currency that, when posted as collateral, results in the highest equivalent synthetic USD rate. This calculation considers overnight index swap (OIS) rates in each currency and adjusts them using cross-currency basis swaps. Source

The Importance of Accurate Collateral Pricing

The need for accurate collateral pricing intensified following the 2008 financial crisis, prompting a renewed focus on managing risk associated with multi-currency collateral. Accurate CTD curves are crucial for managing this risk effectively.

Key Takeaways

• A new analytical approximation offers a faster and more accurate method for calculating cheapest-to-deliver curves.
• The method leverages established mathematical tools like the Clark algorithm and Gauss-Hermite quadrature.
• Accurate CTD curves are essential for pricing derivatives subject to multi-currency collateral agreements and managing associated risks.

Further Understanding

The construction of a CTD discount curve involves determining the cheapest currency to borrow and post as collateral. This is sensitive to cross-currency basis swap volatility. Collateral posted in a currency is remunerated at the relevant OIS rate, which translates into a synthetic USD rate using the cross-currency basis. Source