  The SABR model is a benchmark stochastic volatility model in interest rate
markets, which has received much attention in the past decade. Its popularity
arose from a tractable asymptotic expansion for implied volatility, derived by
heat kernel methods. As markets moved to historically low rates, this expansion
appeared to yield inconsistent prices. Since the model is deeply embedded in
market practice, alternative pricing methods for SABR have been addressed in
numerous approaches in recent years. All standard option pricing methods make
certain regularity assumptions on the underlying model, but for SABR these are
rarely satisfied. We examine here regularity properties of the model from this
perspective with view to a number of (asymptotic and numerical) option pricing
methods. In particular, we highlight delicate degeneracies of the SABR model
(and related processes) at the origin, which deem the currently used popular
heat kernel methods and all related methods from (sub-) Riemannian geometry
ill-suited for SABR-type processes, when interest rates are near zero. We
describe a more general semigroup framework, which permits to derive a suitable
geometry for SABR-type processes (in certain parameter regimes) via symmetric
Dirichlet forms. Furthermore, we derive regularity properties (Feller-
properties and strong continuity properties) necessary for the applicability of
popular numerical schemes to SABR-semigroups, and identify suitable Banach- and
Hilbert spaces for these. Finally, we comment on the short time and large time
asymptotic behaviour of SABR-type processes beyond the heat-kernel framework.
