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

We consider a stochastic volatility model with Lévy jumps for a log-return process Z=(Zt)t≥0 of the form Z=U+X, where U=(Ut)t≥0 is a classical stochastic volatility process and X=(Xt)t≥0 is an independent Lévy process with absolutely continuous Lévy measure ν. Small-time expansions, of arbitrary polynomial order, in time-t, are obtained for the tails P(Zt≥z), z>0, and for the call-option prices E(ez+Zt−1)+, z≠0, assuming smoothness conditions on the density of ν 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 φ(x)1x≥z for smooth functions φ and z>0. As a consequence of our tail expansions, the polynomial expansions in t of the transition densities ft are also obtained under mild conditions.