Small-time expansions of the distributions, densities, and option prices of stochastic volatility models with L\'evy jumps

We consider a stochastic volatility model with L\'evy jumps for a log-return process $Z=(Z_{t})_{t\geq 0}$ of the form $Z=U+X$, where $U=(U_{t})_{t\geq 0}$ is a classical stochastic volatility process and $X=(X_{t})_{t\geq 0}$ 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 $\bbp(Z_{t}\geq z)$, $z>0$, and for the call-option prices $\bbe(e^{z+Z_{t}}-1)_{+}$, $z\neq 0$, assuming smoothness conditions on the {\PaleGrey density of $\nu$} 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 $\phi(x){\bf 1}_{x\geq{}z}$ 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 also {\Green obtained} under mild conditions.