Small-time Expansions of the Distributions, Densities, and option prices of stochastic volatility models with Levy jumps

Mathematical Finance/Financial Engineering Seminar
Friday, October 22, 2010 - 3:05pm
1 hour (actually 50 minutes)
Skiles 002
School of Mathematics, Georgia Tech

Hosted by Christian Houdre and Liang Peng.

We consider a stochastic volatility model with Levy jumps for a log-return process Z = (Z_t )_{t\ge 0}of the form Z = U + X , where U = (U_t)_{t\ge 0}is a classical stochastic volatility model and X = (X_t)_{t\ge 0} is an independent Levy process with absolutely continuous Levy measure \nu. Small-time expansion, of arbitrary polynomial order in time t, are obtained for the tails P(Z_t\ge z), z > 0 , and for the call-option prices E( e^{z+ Z_t| - 1), z \ne 0, assuming smoothness conditions on the Levy density away from the origin and a small-time large deviation principle on U. The asymptotic behavior of the corresponding implied volatility is also given. Our approach allows for a unified treatment of general payoff functions of the form \phi(x)1_{x\ge z} for smooth function \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.