Small-time expansions of the distributions, densities, and option prices of stochastic volatility models with Levy jumps

We consider a stochastic volatility model with Levy jumps for a log-return process $Z=(Z_{t})_{tgeq 0}$ of the form $Z=U+X$, where $U=(U_{t})_{tgeq 0}$ is a classical stochastic volatility process and $X=(X_{t})_{tgeq 0}$ is an independent Levy process with absolutely continuous Levy measure $ u$. Small-time expansions, of arbitrary polynomial order, in time-$t$, are obtained for the tails $bbp(Z_{t}geq z)$, $z>0$, and for the call-option prices $bbe(e^{z+Z_{t}}-1)_{+}$, $z eq 0$, assuming smoothness conditions on the {PaleGrey density of $ u$} 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 $phi(x){bf 1}_{xgeq{}z}$ 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.