  We provide a surprising new application of classical approximation theory to
a fundamental asset-pricing model of mathematical finance. Specifically, we
calculate an analytic value for the correlation coefficient between exponential
Brownian motion and its time average, and we find the use of divided
differences greatly elucidates formulae, providing a path to several new
results. As applications, we find that this correlation coefficient is always
at least $1/\sqrt{2}$ and, via the Hermite--Genocchi integral relation,
demonstrate that all moments of the time average are certain divided
differences of the exponential function. We also prove that these moments agree
with the somewhat more complex formulae obtained by Oshanin and Yor.
