In this talk, we present perturbative methods to derive (semi)analytic expressions of pricing kernels and accurate arbitrage-free analytic prices. We focus on an extension of the Hull-White model that includes volatility smile and skew and use this short-rate model to price a caplet referencing backward-looking compounded rates payments, in a manner consistent with the smile and skew levels observed in the market. We then show how the caplet pricing formulae are translated into analytic expressions of the implied volatility.