Small-time asymptotics and expansions of option prices under Levy-based models
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This thesis is concerned with the small-time asymptotics and expansions of call option prices, when the log-return processes of the underlying stock prices follow several Levy-based models. To be speciﬁc, we derive the time-to-maturity asymptotic 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 stochastic volatility models with Levy jumps. In the OTM and ITM cases, we consider a general stochastic volatility model with independent Levy jumps, while in the ATM case, we consider the pure-jump CGMY model with or without an independent Brownian component. An accurate modeling of the option market and asset prices requires a mixture of a continuous diﬀusive component and a jump component. In this thesis, we ﬁrst model the log-return process of a risk asset with a jump diﬀusion model by combining a stochastic volatility model with an independent pure-jump Levy process. By assuming smoothness conditions on the Levy density away from the origin and a small-time large deviation principle on the stochastic 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 the call-option price which is not at-the-money. 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 asymptotic behavior of the ATM call-option prices 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 prices 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 prices as well as the asymptotic behavior of the corresponding Black-Scholes implied volatilities are also addressed.