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?

1 Answer

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.

• I still don't really get why it would not be this case for "forecasting" future floating rates for an IRS. in both cases we would want to know what the payoff will be in the end. The only difference is that the payoff for the floating leg depends on some floating rate, while for our case it depends on some maximum of synthetic USD overnight rates. Isn't all information in the forward rates already? If our model told us that the rate in the future would be something else that the current forward rate, wouldn't that imply we should "take advantage of this"? Jun 28, 2021 at 18:41
• An interest rate swap is a linear instrument. You could take into account volatility , if you wanted, but you would get the same answer as just using the forward rate. Not true for option- like instruments.
– dm63
Jun 28, 2021 at 19:29
• I understand that for nonlinear derivatives we need to have some sort of model to value them (caps, floors, swaptions etc), but here we are not valuing anything, right? We want to construct some best estimate of the compounded daily rate, right? In that case i would say that forward rates are your best estimate of those daily rates. Where does this reasoning go wrong? Jun 28, 2021 at 19:40
• Yes you are valuing something. The poster of collateral has the option of posting either currency 1 or currency 2 for example. Therefore you assume they deliver currency 1 then you value their option to replace currency 1 by currency 2. This is an option that requires a model.
– dm63
Jun 28, 2021 at 21:56
• Thank you. The only thing that is a bit unclear to me is what we value exactly in the sense of what cash flows and how we convert this value to a discount rate. Do you have some paper you can refer me to? Jun 29, 2021 at 15:09