# OIS floating-floating cross-currency basis swap

I'm trying to understand whether notional resets on a floating-floating cross-currency basis swap play a role or not when the coupon payments are SOFR-based (with no spread) and they are discounted with the same SOFR rate.

From section 6.5.2.3 in Andersen-Piterbarg, we have \begin{align*} V_\text{basisswap,\\\}(0) =& L_{\\\}(0,t_i,t_{i+1})\tau_i P_{\\\}(0,t_{i+1})+P_{\\\}(0,t_n) \\ &-X(0)\left(\sum_{i=0}^{n-1}(L_{\yen}(0,t_i,t_{i+1})+e_{\yen})\tau_iP_{\yen}(0,t_{i+1})+P_{\yen}(0,t_n)\right) \\ =&1-X(0) \\ &\left(\sum_{i=0}^{n-1}\left(\frac{P^{(L)}_{\yen}(0,t_i)}{P^{(L)}_{\yen}(0,t_{i+1})}-1+e_{\yen}\tau_i\right)P_{\yen}(0,t_{i+1})+P_{\yen}(0,t_n)\right) \end{align*}

with $$e_\yen^\text{par}$$ being the market quotes making the basis swap $$V_\text{basisswap,\\\}(0)$$ price at par.

The equality above was based on the assumption that the Libor discount curve was the same as the real discount curve. This is no longer true when using Libor rates for the coupons and OIS discounting, but what if coupons are RFR-based? Also what about the notional resets? My understanding is that these cancel out.

• The curve used for the forward rate on $\yen$ should be the OIS or 1D curve, while the discounting curve for $\yen$ should be a spread curve with a multiplier to $P_{\yen}$, that is, $P_{\yen}(0, T_i) S(0, T_i)$. Here $S(0, T_i)$ can be calibrated or based on a given spread curve. Feb 4, 2022 at 15:09
• Thanks, no doubt that $\yen$ curve needs to be spread-adjusted. My question was rather, how do we prove mathematically that it makes no difference to consider the OIS cross-currency basis swap being resettable or non-resettable, since in both cases forward rates are derived from SOFR curve, which is the same one used in discounting? Feb 4, 2022 at 15:34
• It will make difference with resettable and non-resettable, as forward exchange rates will involved rather than a single exchange rate at the start. Moreover the notional exchange on a settlement date should also be considered. Feb 4, 2022 at 15:47
• Well, you can represent the USD resetting leg as a set of single period legs with constant notional equal to forward exchange rate times JPY leg notional. Each of the leg uses the same rate for discounting and coupons, so even with notional resets it must shrink to zero as the notional exchange offsets the discounted coupon payment. Perhaps there's only a small difference due to the calendar used on the USD leg. Feb 4, 2022 at 16:07
• I am not sure there is. I need to check for SOFR specifically, to make sure. However, in pre RFR times, if you set a XCCY MTM and you disable OIS discounting/DC stripping it is assuming 3m LIBOR as the interest rate on collateral and the MTM resets will therefore have no pricing different to non-MTM version, by definition of 3mL discounting and 3mL floating leg. (In Bloomberg, that would mean SWPM would strip S92 with 3mL). Feb 5, 2022 at 2:28