Notional Resets in RFR-Discounted Cross-Currency Basis Swaps

Summary

The document poses a valuation question about floating-floating cross-currency basis swaps whose coupons reference overnight risk-free rates and whose discounting uses the same rate. It cites a par-value expression for a basis swap and notes that its derivation assumes the Libor projection curve and discount curve coincide. The question is whether that cancellation logic still holds when coupons are risk-free-rate based and whether notional exchanges or resets affect the valuation.

No answer or derivation is provided, so the text does not establish that the notionals cancel or specify the conditions under which they might. The excerpt is useful as a prompt to distinguish coupon projection, discounting, exchange of principal, and reset mechanics in cross-currency valuation. Any conclusion would require a precise specification of the swap’s cash flows, collateral and discounting framework, and treatment of foreign exchange; those details are not resolved here.

Key ideas

• The question concerns floating-floating cross-currency basis swaps with risk-free-rate coupons and matching-rate discounting.
• The cited par valuation relies on an assumption that the projection and discount curves coincide.
• The document asks whether notional resets cancel in the risk-free-rate setting but gives no conclusion.
• Valuation depends on the cash-flow and exchange mechanics as well as the projection and discount curves.

Tags

# OIS floating-floating cross-currency basis swap


# 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.

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