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 $Z=(Z_t{)}_{t\ge 0}$ 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 $\text{bbp}(Z_t\ge z)$, $z>0$, and for the call-option prices $\text{bbe}(e^{z+Z_t}-1{)}_{+}$, $z\mathrm{\ne }0$, 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 $\varphi (x){\mathbf{1}}_{x\ge z}$ for smooth functions $\varphi$ 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.