  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-Berm\'udez [2023].
