Small-Time Asymptotics and Expansions of Option Prices Under Levy Based Models

The small-time asymptotic behavior of option prices and implied volatilities for jump-diffusion models has received much attention in recent years. In this presentation, we study the time-to ­maturity asymptotics of call option prices under a variety of models with L´evy jumps. In the out-of-the-money (OTM) and in-the-money (ITM) case, we consider a general stochastic volatility model with independent L´evy jumps for the log-return process of the underlying stock price. In this setting, small-time expansions, of arbitrary polynomial order, in time-t, are obtained for both OTM and ITM call option prices. In the at-the-money (ATM) case, a novel second-order approximation of the call option price is obtained for a large class of exponential “tempered-stable” L´evy models with or without Brownian component. As a consequence, small-time expansions of the corresponding Black-Scholes implied volatilities are also addressed in both cases. This is the joint work with J. E. Figueroa-L´opez and C. Houdr´e.

(RG) Department of Mathematics, Rutgers, The State University of New Jersey, 110 Frelinghuysen Road, Piscataway, NJ 08854-8019, U. S. A.