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

2010/09/30 by J. E. Figueroa-López, R. Gong, C. Houdré · 1 citation
Economics, Econometrics and Finance · Mathematics · #q-fin.PR #math.PR #msc:60G51 #msc:60F99

paper · pdf · doi:10.1016/j.spa.2012.01.013

Final version to appear in Stochastic Processes and their Applications

arxiv created 2012/02/21 · arxiv updated 2012/02/23

Abstract

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 \bbp(Z_t≥ z), z>0, and for the call-option prices \bbe(e^z+Z_t-1)₊, z≠ 0, assuming smoothness conditions on the \PaleGrey 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)\bf 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 \Green obtained under mild conditions.

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