Why CTD Discount Curves Depend on Currency Basis Volatility

Summary

The note explains why a cheapest-to-deliver discount curve for a collateralized USD swap with currency choice may need to account for cross-currency basis volatility. A common construction integrates the maximum forward basis across eligible collateral currencies and applies it to the USD OIS discount curve. When future basis is approximated by today’s forward basis, this captures the current best basis path but omits uncertainty over which currency will be cheapest in the future.

The key point is that taking a maximum creates option-like value: the holder can select among currency bases, much as an option payoff selects a favorable outcome. The stated curve formula reflects the intrinsic value of that choice, while volatility contributes additional time value. The answer is a conceptual explanation rather than a derivation or calibration method, and it does not quantify the volatility adjustment or specify how to model correlations among currency bases.

Key ideas

• Taking the maximum across eligible currency bases creates an option-like feature in the discounting problem.
• A curve based on current forward bases captures the intrinsic value of collateral currency choice.
• Basis volatility adds time value because future basis movements can change which currency is cheapest to deliver.
• The explanation gives no quantitative model for estimating the volatility adjustment.

Tags

# Answer by dm63 (score 3, accepted)


# Why do we theoretically have to take cross currency basis volatility into account when constructing Cheapest To Deliver (CTD) discount curves?












Let's take a collateralized USD IRS where there is optionality in collateral currency. My understanding is that it is standard practice to compute forward XXX/USD OIS basis curves for all currencies taken into consideration. Then, as explained by Antoine Conze in Cheapest-to-deliver (CTD) discount curve, the CTD discount curve is constructed as

$D^{CTD}_{USD}(T)=D_{OIS_{USD}}(T)\text{exp}\left(−\int_{0}^{T}\smash{\displaystyle\max_{\text{XYZ}}} \{ \text{basis}_\text{XYZUSD}(t) \}dt\right)$,

assuming that you "disregard basis volatility" and that in that case "future basis is today's forward basis". I am wondering why this is not always the case. For example, isn't it true that when we value the floating leg of a vanilla IRS, we use forward rates as well and we assume that future floating rates are today's forward rates? And that we don't take any sort of interest rate volatility into consideration?

Why do we have to take this volatility into account for constructing CTD discount curves?

## Answer by dm63 (score 3, accepted)

https://quant.stackexchange.com/a/65751

Because the formula contains the expression max{currency bases}. Whenever there is a max, there’s an option. Eg a regular call option payout max{0, S-K}. The formula expresses only the intrinsic value of the basket option on the currency bases, not the time value.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.