# 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?