Calibrating SABR to Backward-Looking SOFR Cap and Floor Quotes
Summary
The discussion compares pricing approaches for forward-looking LIBOR-style caplets and backward-looking caplets based on compounded overnight rates such as SOFR. It describes using Hagan implied-volatility approximations with SABR parameters, then adjusting the volatility inputs and option timing to account for the different rate construction and payment schedule. Caps and floors are treated as collections of caplets or floorlets.
The central question is whether modified SABR parameters can be calibrated directly to market quotes for backward-looking caps and floors, using an existing LIBOR-style framework while accounting for expiry, settlement, and caplet-count differences. The accepted response reports that the paper's author confirmed this as a valid approach. The excerpt gives no calibration dataset, numerical comparison, or details of the confirmation, so it supports the feasibility of the method but not its accuracy across markets or quote conventions. Practitioners still need to align instrument definitions and market conventions when applying the calibration.
Key ideas
- Cap and floor values can be assembled from the values of their constituent caplets or floorlets.
- SABR implied-volatility approximations are used for forward-looking and backward-looking rate options.
- Backward-looking compounded-rate options require adjustments to volatility parameters and timing conventions.
- The response reports that direct calibration of modified SABR parameters to SOFR market quotes is valid.
- The excerpt provides no numerical calibration results or assessment of accuracy.
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Full text
# Pricing caps/floors on backward-looking USD SOFR with forward-looking LIBOR model
# 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?
## Answer by Hasek (score 3, accepted)
https://quant.stackexchange.com/a/70227
> 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)?
I sent an email to the author of the aforementioned paper and he confirmed that this is a valid approach.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.