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

Abstract

We consider a stochastic volatility model with Lévy 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 process and $X=(X_t{)}_{t\ge 0}$ is an independent Lévy process with absolutely continuous Lévy 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 $\bbe(e^{z+Z_{t}}-1)_{+}$, $z\ne 0$, 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 $\varphi (x)1_{x\ge z}$ 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 {\Green obtained} under mild conditions.

Final version to appear in Stochastic Processes and their Applications

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• Testing for jumps in a discretely observed process
• Is Brownian motion necessary to model high-frequency data?
• Small-Time Asymptotics of Option Prices and First Absolute Moments
• Closed-form likelihood expansions for multivariate diffusions
• High-order short-time expansions for ATM option prices under the CGMY model
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Cited by in corpus (3)

• Small-time asymptotics for a general local-stochastic volatility model with a jump-to-default: curvature and the heat kernel expansion
• Small-time asymptotics of stopped Lévy bridges and simulation schemes with controlled bias
• Small-time expansions for local jump-diffusion models with infinite jump activity