Georgia Institute of Technology College of Sciences

The first order small-time approximation of the marginal distribution of a L\'evy process has been known for long-time. In this talk, I present higher order expansions polynomial in time for the distributions of a L\'evy process. As a secondary objective, I illustrate the application of our expansions in the estimation of financial models with jumps as well as in the study of the small-term asymptotic behavior of the implied volatility for this class of financial models. This talk presents joint work with C. Houdr\'e and M. Forde. Associated reading (available in the web site of the speaker): (1) Small-time expansions for the transition distribution of Levy processes. J.E. Figueroa-L\'opez and C. Houdré. Stochastic Processes and their Applications 119 pp. 3862-3889, 2009. (2) Nonparametric estimation of time-changed Levy models under high-frequency data. J.E. Figueroa-L\'opez. Advances in Applied Probability vol. 41, number 4, pp. 1161-1188, 2009. (3) The small-maturity smile for exponential Levy model. J.E. Figueroa-L\'opez and M. Forde. Preprint.