Georgia Institute of Technology College of Sciences

We derive at-the-money call-price and implied volatility asymptotic expansions in time to maturity for a selection of exponential Levy models, restricting our attention to asset-price models whose log returns structure is a Levy process. We consider two main problems. First, we consider very general Levy models that are in the domain of attraction of a stable random variable. Under some relatively minor assumptions, we give first-order at-the-money call-price and implied volatility asymptotics. In the case where our Levy process has Brownian component, we discover new orders of convergence by showing that the rate of convergence can be of the form t^{1/\alpha} \ell( t ) where \ell is a slowly varying function and \alpha \in (1,2). We also give an example of a Levy model which exhibits this new type of behavior where \ell is not asymptotically constant. In the case of a Levy process with Brownian component, we find that the order of convergence of the call price is \sqrt{t}. Second, we investigate the CGMY process whose call-price asymptotics are known to third order. Previously, measure transformation and technical estimation methods were the only tools available for proving the order of convergence. We give a new method that relies on the Lipton-Lewis formula, guaranteeing that we can estimate the call-price asymptotics using only the characteristic function of the Levy process. While this method does not provide a less technical approach, it is novel and is promising for obtaining second-order call-price asymptotics for at-the-money options for a more general class of Levy processes.