### Small-time asymptotics of call prices and implied volatilities for exponential Levy models

- Series
- Dissertation Defense
- Time
- Tuesday, January 6, 2015 - 15:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Allen Hoffmeyer – School of Mathematics, Georgia Tech

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.