  This paper uses Lie symmetry methods to calculate certain expectations for a
large class of It\^{o} diffusions. We show that if the problem has sufficient
symmetry, then the problem of computing functionals of the form
$E_x(e^{-\lambda X_t-\int_0^tg(X_s) ds})$ can be reduced to evaluating a single
integral of known functions. Given a drift $f$ we determine the functions $g$
for which the corresponding functional can be calculated by symmetry.
Conversely, given $g$, we can determine precisely those drifts $f$ for which
the transition density and the functional may be computed by symmetry. Many
examples are presented to illustrate the method.
