# Pricing caps/floors on backward-looking USD SOFR with forward-looking LIBOR model

The payoff of a cap/floor is calculated as a payoff of constitutient caplets/floorlets.

The SABR volatility model has the implied volatility approximations of Hagan et al. $$\sigma^f_{IV}\approx \sigma_{Hagan}(t_0, K, F_0, \alpha, \beta, \rho, \nu)$$ which allows one to price an individual forward-looking LIBOR-like caplet/floorlet as $$V_f(0) = P(0,t_1)\cdot Black(t_0, K, F_0, \sigma^f_{IV})$$ where $$t_0$$ is an option's expiry, $$t_1$$ is a payoff settlement and all other variables seem to be pretty self-explanatory.

The paper SABR smiles for RFR caplets derives modified SABR parameters $$\hat{\alpha}, \hat{\rho}, \hat{\nu}$$ such that $$\sigma^b_{IV}\approx\sigma_{Hagan}(t_1,K,F_0,\hat{\alpha},\beta,\hat{\rho},\hat{\nu})$$ and therefore a backward-looking caplet/floorlet on a compounded overnight rate can be priced as $$V_b(0) = P(0,t_1)\cdot Black(t_1, K, F_0, \sigma^b_{IV})$$

Does that mean that one can directly calibrate vanilla SABR model for LIBOR to the market quotes of SOFR taking care of the differences in expiry and settlement, and adjusting the number of caplets in a cap (first caplet is omitted for LIBOR)?

Note that calibration is aiming to minimize the squared sum of differences between market and model volatilities, i.e. $$\min_{\alpha,\rho,\nu}\sum(\sigma_{market}-\sigma_{IV}(\alpha,\rho,\nu))^2$$ so it seems that one can directly calibrate $$\hat{\alpha},\hat{\rho},\hat{\nu}$$ to market quotes of backward-looking caps/floors with the help of already existing LIBOR model instead of calculating them from $$\alpha,\rho,\nu$$ calibrated to forward-looking caps/floors. Am I missing something important in the results of this paper?