Small-time asymptotics, and expansions of option prices under Levy-based models

This thesis is concerned with the small-time allymptotics and expansions of call option prices, when the log-return processes of the underlying stock prices follow several Levy-balled models. To be specific, we derive the time-to-maturity allymptotic behavior for both at-the-money (ATM), out-of-the-money (OTM) and in-the-money (ITM) call-option prices under several jump-diffusion models and stochalltic volatility models with Levy jumps. In the OTM and ITM Calles, we consider a general stochalltic volatility model with independent Levy jumps, while in the ATM case, we consider the CGMY model with or without an independent Brownian component.;An accurate modeling of the option market and asset prices often requires a mixture of a continuous diffusive component and a jump component. In this thesis, we first model the log-return process of a risky allset with a jump diffusion model by combining a stochastic volatility model with an independent pure-jump Levy process. By allsuming smoothness conditions on the Levy density away from the origin and a small-time large deviation principle on the stochalltic volatility model, we derive the small-time expansions, of arbitrary polynomial order in time-t, for the tail distribution of the log-return process, and for both OTM and ITM call-option prices. Moreover, our approach allows for a unified treatment of more general payoff functions. As a consequence of our tail expansions, the polynomial expansion in t of the transition density is also obtained under mild conditions.;The allymptotic behavior of the ATM call-option price is more complicated to obtain and, in general, is given by fractional powers of t, which depends on different choices of the underlying log-return models. Here, we focus on the CGMY model, one of the most popular tempered stable models used in financial modeling. A novel second-order approximation for ATM option price under the pure-jump CGMY Levy model is derived, and then extended to a model with an additional independent Brownian component. The third-order asymptotic behavior of the ATM option price as well as the asymptotic behavior of the corresponding Black-Scholes implied volatility are also addressed.