  We extend the short rate model of Turfus and Romero-Berm\'udez [2021] to
facilitate accurate arbitrage-free analytic pricing of SOFR, SONIA or ESTR
caplets, i.e. options on backward-looking compounded rates payments, in a
manner consistent with the smile and skew levels observed in the market. These
caplet pricing formulae and corresponding LIBOR or term-rate caplet results are
translated into effective variance (implied volatility) formulae, which are
seen to be of a particularly simple form. They show that the model is
essentially equivalent to imposing on a Hull-White model an effective variance
which is a quadratic function of the moneyness parameter (rather than a
constant) for any given maturity. Results are also illustrated graphically.
