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Given a Lévy process X with Lévy measure ν, conditions ensuring that lim t→0 1 t E f (X t) = f (x)ν(dx) are given. The moment functions f considered here can be unbounded as well as satisfy simpler regularity conditions than those considered in some previous works. Also, the rate of convergence is determined when f vanishes in a neighborhood of the origin… (More)

Let X = (Xt) t≥0 be a Lévy process with absolutely continuous Lévy measure ν. Small time expansions of arbitrary polynomial order in t are obtained for the tails P (Xt ≥ y), y > 0, of the process, assuming smoothness conditions on the Lévy density away from the origin. By imposing additional regularity conditions on the transition density pt of Xt, an… (More)

- José Enrique Figueroa-López, Christian Houdré, Robert P. Kertz, Marcus C. Spruill, Shijie Deng, Richard Serfozo +3 others
- 2003

ACKNOWLEDGEMENTS There have been many individuals in my life without whom the completion of this work would not have been possible. It is my intension to briefly recognize their contribution to my personal and professional life. I would like to thank my advisor, Dr. Christian Houdré, for his support and orientation through all my studies at Georgia Tech. In… (More)

for Winning the Endgame" by Jeff Augen: " The final hours of each expiration cycle are characterized by unusual market forces and price distortions caused by the breakdown of traditional pricing calculations" Motivation Empirical features of implied volatility Empirical features of implied volatilities

A Lévy model combines a Brownian motion with drift and a pure-jump homogeneous process such as a compound Poisson process. The estimation of the Lévy density, the infinite-dimensional parameter controlling the jump dynamics of the process, is studied under a discrete-sampling scheme. In that case, the jumps are latent variables whose statistical properties… (More)

Estimation methods for the Lévy density of a Lévy process are developed under mild qualitative assumptions. A classical model selection approach made up of two steps is studied. The first step consists in the selection of a good estimator, from an approximating (finite-dimensional) linear model S for the true Lévy density. The second is a data-driven… (More)

- JOSÉ E. FIGUEROA-LÓPEZ, RUOTING GONG, CHRISTIAN HOUDRÉ
- 2010

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… (More)

The implied volatility slope has received relatively little attention in the literature on short-time asymptotics for financial models with jumps, despite its importance in model selection and calibration. In this paper, we fill this gap by providing high-order asymptotic expansions for the at-the-money implied volatility slope of a rich class of stochastic… (More)

We consider a Markov process {X (x) t } t≥0 with initial condition X (x) t = x, which is the solution of a stochastic differential equation driven by a Lévy process Z and an independent Wiener process W. Under some regularity conditions, including non-degeneracy of the diffusive and jump components of the process as well as smoothness of the Lévy density of… (More)

We consider a Markov process X which is the solution of a stochastic differential equation driven by a Lévy process Z and an independent Wiener process W. Under some regularity conditions, including non-degeneracy of the diffusive and jump components of the process as well as smoothness of the Lévy density of Z outside any neighborhood of the origin, we… (More)