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 L\'evy density {\PaleGrey  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}_{xz} 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 obtained {\Green  under mild conditions.
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}_{xz} 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.
D
L\'evy density
A
density of \nu
R
\varphi
\phi
R
\varphi
\phi
D
obtained
A
obtained