dc.contributor.advisor Houdre, Christian dc.contributor.author Hoffmeyer, Allen Kyle dc.date.accessioned 2015-06-08T18:21:03Z dc.date.available 2015-06-08T18:21:03Z dc.date.created 2015-05 dc.date.issued 2015-01-08 dc.date.submitted May 2015 dc.identifier.uri http://hdl.handle.net/1853/53506 dc.description.abstract We derive at-the-money call-price and implied volatility asymptotic expansions in time to maturity for a selection of exponential Lévy models, restricting our attention to asset-price models whose log returns structure is a Lévy process. We consider two main problems. First, we consider very general Lévy 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 Lévy process has Brownian component, we discover new orders of convergence by showing that the rate of convergence can be of the form t¹/ᵃℓ(t) where ℓ is a slowly varying function and $\alpha \in (1,2)$. We also give an example of a Lévy model which exhibits this new type of behavior where ℓ is not asymptotically constant. In the case of a Lévy process with Brownian component, we find that the order of convergence of the call price is √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 Lévy 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 Lévy processes. dc.format.mimetype application/pdf dc.language.iso en_US dc.publisher Georgia Institute of Technology dc.subject CGMY process dc.subject Levy process dc.subject Small-time asymptotics dc.subject Asymptotic expansions dc.subject Regular variation dc.subject Options pricing dc.subject Finance dc.title Small-time asymptotics of call prices and implied volatilities for exponential Lévy models dc.type Dissertation dc.description.degree Ph.D. dc.contributor.department Mathematics thesis.degree.level Doctoral dc.contributor.committeeMember Bakhtin, Yuri dc.contributor.committeeMember Koltchinskii, Vladimir dc.contributor.committeeMember Peng, Liang dc.contributor.committeeMember Figueroa-Lopez, Jose E. dc.date.updated 2015-06-08T18:21:03Z
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