  We find a simple expression for the probability density of $\int \exp (B_s -
s/2) ds$ in terms of its distribution function and the distribution function
for the time integral of $\exp (B_s + s/2)$. The relation is obtained with a
change of measure argument where expectations over events determined by the
time integral are replaced by expectations over the entire probability space.
We develop precise information concerning the lower tail probabilities for
these random variables as well as for time integrals of geometric Brownian
motion with arbitrary constant drift. In particular, $E[ \exp\big(\theta / \int
\exp (B_s)ds\big) ]$ is finite iff $\theta < 2$. We present a new formula for
the price of an Asian call option.
