  We adapt and extend Yosida's parametrix method, originally introduced for the
construction of the fundamental solution to a parabolic operator on a
Riemannian manifold, to derive Varadhan-type asymptotic estimates for the
transition density of a degenerate diffusion under the weak H\"ormander
condition. This diffusion process, widely studied by Yor in a series of papers,
finds direct application in the study of a class of path-dependent financial
derivatives known as Asian options. We obtain the Varadhan formula
\begin{equation} \frac{-2 \log p(t,x;T,y) } { \Psi(t,x;T,y) } \to 1, \qquad
\text{as } \quad T-t \to 0^+, \end{equation} where $p$ denotes the transition
density and $\Psi$ denotes the optimal cost function of a deterministic control
problem associated to the diffusion. We provide a partial proof of this
formula, and present numerical evidence to support the validity of an
intermediate inequality that is required to complete the proof. We also derive
an asymptotic expansion of the cost function $\Psi$, expressed in terms of
elementary functions, which is useful in order to design efficient
approximation formulas for the transition density.
