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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

  1. $T_s$ is the period start date

  2. $T_e$ is the period end date

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

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

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

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

  7. $\sigma_p^L$ 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.

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