Caplet stripping in the bwd-looking RFR world with/without maturity adjustment

Question

Since the beginning of this year, LIBOR rates have ceased in some markets like GBP, CHF, and JPY and rates pricing has moved into the RFR space, using compounded overnight rates as the underlying for cap-/floor(lets). My questions now are concerned with the building of a vol surface based on broker quotes of normal par vols (absolute strikes, yearly expiries, backward looking with quarterly compounding), and, in particular, caplet stripping:

1. what is the correct approach to bootstrap implied caplet vols if one wants to value caplets based on the Lyashenko/Mercurio expiry adjustment; basically, instead of using the accrual start date $T_S$ as the expiry of the option like we did for LIBORs, one uses an "extended" maturity $T_S+(T_E-T_S)/3$ to account for the fact that we are dealing with a backward looking rate that is still stochastic up until the last fixing occurs at $T_E$ but the volatility decays linearly within the accrual period. @Daneel Olivaw gave a splendid answer & derivation based on the above paper in this thread. My goal would be something simple for caplet vols (preferably piecewise linear or even constant) but I'm struggling how to achieve this with the "new" adjusted maturity.

2. how would the stripping change if we know that the quoted implied par vols are based on the fact that interdealer brokers assume a constant vol (instead of a linearly decaying one) within the accrual period? Then, no maturity adjustment is needed - we simply use $T_E$ as the caplet expiry date. It's clear that what matters for pricing is total variance of the compound rate, and we can achieve the same variance using a tuple of either $(\sigma_{implied}^{decaying}, T_S+(T_E-T_S)/3)$ or, alternatively, the broker quote $(\sigma_{implied}^{constant}, T_E)$. How does this factor into caplet stripping?

3. any idea on surface construction for a non-quoted tenor (e.g. options on 1m or 6m compound rates)? how could we up-/down-scale the surface to price such options?

Answer 1

1, there is an assumption that the normal vol quotes are the same for the RFR caplets and LIBOR caplets, so to calculate the caplet prices for RFR vs LIBOR, we make the adjustment to the maturity Ts. 2, in reality, the quoted vol should not be the same for RFR backward-looking vs the forward-looking caplets and usually they are quoted as constant vol. Therefore, we simply take the vol quotes to calculate the caplet price. The RFR vol quotes would be the forward rate volatility through end of the accrual period Te, while the forward-looking vol is the forward rate volatility through the beginning of the accrual period Ts. If we have the vol quotes for both RFR and forward-looking caplets, we can verify the vol decaying assumption in bullet 1. 3, you need to extrapolate or interpolate the 1m or 6m caplet vols. The 3m caplet would only give you the information for the 3m tenor rate, for other tenors probably consider about SOFR swaptions or futures options.