  Geometry constitutes a core set of intuitions present in all humans,
regardless of their language or schooling [1]. Could brain's built in machinery
for processing geometric information take part in uncertainty representation?
For decades already traders have been citing the price of uncertainty based FX
optional contracts in terms of implied volatility, a dummy variable related to
the standard deviation, instead of pricing with units of money. This work
introduces a methodology for geometric representation of probability in terms
of implied volatility and attempts to find ways to approximate certain
probability distributions using intuitive geometric symmetry. In particular, it
is shown how any probability distribution supported on $\mathbb{R}_{+}$ and
having finite expectation may be represented with a planar curve whose
geometric characteristics can be further analyzed. Log-normal distributions are
represented with circles centered at the origin. Certain non-log-normal
distributions with bell-shaped density profiles are represented by curves that
can be closely approximated with circles whose centers are translated away from
the origin. Only three points are needed to define a circle while it represents
the candidate probability density approximating the distribution along the
entire $\mathbb{R}_{+}$. Just three numbers: scaling and translations along the
$x$ and $y$ axes map one circle to another. It is possible to introduce
equivalence classes whose member distributions can be obtained by transitive
actions of geometric transformations on any of corresponding representations.
Approximate completion of probability with non-circular shapes and cases when
probability is supported outside of $\mathbb{R}_{+}$ are considered too.
Proposed completion of implied volatility is compared to the vanna-volga
method.
