We consider a stochastic volatility model with L\'evy jumps for a log-return
process $Z = (Z_t)_{t\ge 0}$ of the form $Z = U+X$, where $U = (U_t)_{t\ge 0}$
is a classical stochastic volatility process and $X = (X_t)_{t\ge 0}$ is an
independent L\'evy process with absolutely continuous L\'evy measure $\nu$.
Small-time expansions, of arbitrary polynomial order in time $t$, are obtained
for the tails $\bbp(Z_t \ge z)$, $z >0$, and for the call-option prices
$\bbe(e^{z+Z_t} - 1)_+$, $z\neq 0$, assuming smoothness conditions on the
L\'evy density away from the origin and a small-time large deviation principle
on $U$. Our approach allows for a unified treatment of general payoff functions
of the form $\phi(x){\bf 1}_{x\ge z}$ for smooth functions $\phi$ and $z > 0$.
As a consequence of our tail expansions, the polynomial expansions in $t$ of
the transition densities $f_t$ are obtained under rather mild conditions.