Analytic Pricing of SOFR Futures Contracts with Smile and Skew

We introduce a perturbative formalism to solve the backward-looking futures pricing problem. The formalism is based on a time-ordered exponential series which allows to derive the functional form of the integral kernel associated to the backward-Kolmogorov diffusion PDE. We present an analytic pricing formula for SOFR futures contracts under an extension of the Hull-White model which incorporates not only the intrinsic convexity adjustments captured by Mercurio [2018], but also the skew and smile observed in options markets as done in Turfus and Romero-Bermúdez [2023].