Analytic RFR Option Pricing with Smile and Skew

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.