Payment-Delay Convexity Adjustments for Compounded RFR Rates

Summary

The document asks how to adapt a payment-delay convexity adjustment used for Libor to daily compounded risk-free rates, with SOFR as the example. It describes a normal-model expression involving a discount factor between the accrual end and payment dates, the correlation and volatilities of two forward rates, and a time interval tied to the accrual period. The payment date is assumed to fall after the period end.

The author proposes a tentative substitution: use backward-looking RFR volatilities and shift the relevant time interval to begin at the period end. The central issue is whether this intuitive replacement follows from a proper derivation for daily compounded rates. No derivation, market calibration, numerical example, or resolution is supplied, so the proposed formula should be treated as a question rather than a usable pricing result. It highlights that compounding and the observation schedule may require a more careful treatment than replacing rate labels and dates in a Libor formula.

Key ideas

• The stated Libor adjustment depends on a discount factor, rate correlation, two volatilities, and a time interval.
• The payment date is assumed to occur after the accrual period ends.
• The author suggests substituting backward-looking RFR volatilities for Libor volatilities.
• The suggested date adjustment is presented as tentative, not derived.
• Daily compounding may require a separate derivation before the expression is used for pricing.

Tags

# Payment Delay Convexity Adjustment Formula for RFR Rates


# Payment Delay Convexity Adjustment Formula for RFR Rates












For Libor we have the following Convexity adjustment formula for payment delay (under normal model)

$$CA = P(0,T_e,T_p)\rho\sigma_e^L\sigma_p^L\Delta_e^p(T_s-t_0)$$

where

- $T_s$ is the period start date

- $T_e$ is the period end date

- $T_p$ is the payment date (assuming here $T_p$ > $T_e$)

- $P(0,T_e,T_P)$ is discount factor from $T_e$ to $T_p$

- $rho$ is correlation between 2 forward rates $F(t_0,T_s,T_e)$ and $F(t_0,T_e,T_p)$

- $\sigma_e^L$ is vol of $F(t_0,T_s,T_e)$

- $\sigma_p^L$ is vol of $F(t_0,T_e,T_p)$

How would this formula change for RFR compounded (SOFR) rates? A naïve way would be to just replace the Libor vols with backward looking RFR vols and change the $T_s$ to $T_e$ something like this

$$CA = P(0,T_e,T_p)\rho\sigma_e^{RFR}\sigma_p^{RFR}\Delta_e^p(T_e-t_0)$$

wondering is somebody did a proper derivation for daily compounded rates like SOFR.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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