We consider a stochastic volatility model with L\'evy jumps for a log-return process Z=(Z _t) _{t %DIFDELCMD < \ge %%%  0} of the form Z=U+X, where U=(U _t) _{t %DIFDELCMD < \ge %%%  0} is a classical stochastic volatility process and X=(X _t) _{t %DIFDELCMD < \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%DIFDELCMD < \ge %%% \geq } 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 %DIFDELCMD < \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.
We consider a stochastic volatility model with L\'evy jumps for a log-return process Z=(Z _{t _{t %DIFDELCMD < \ge %%% \geq 0} of the form Z=U+X, where U=(U _{t _{t %DIFDELCMD < \ge %%% \geq 0} is a classical stochastic volatility process and X=(X _{t _{t %DIFDELCMD < \ge %%% \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 %DIFDELCMD < \ge %%% _{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 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 \varphi (x){\bf 1}_{x %DIFDELCMD < \ge %%% \geq{ 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 mild conditions.
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