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 = ( Z t ) t ≥ 0 of the form Z = U + X , where U = ( U t ) t ≥ 0 is a classical stochastic volatility process and X = ( X t ) 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 ( Z t ≥ z ) , z > 0 , and for the call-option prices E ( e z + Z t − 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 ) 1 x ≥ z for smooth functions φ and z > 0 . As a consequence of our tail expansions, the polynomial expansions in t of the transition densities f t are also obtained under mild conditions.