# Asymptotics of Local Volatility Models

Andrzej Palczewski^{*}

We explore a direct link between local and implied volatilities in the form of quasilinear parabolic differential equations for both time independent and time dependent local volatility models. Using these equations we establish closed-form asymptotic formulae for the implied volatility near expiry. We compute the zero and first order approximations to the implied volatility and obtain expressions which are easy for numerical computations. The main new result is the proof of the convergence of the obtained asymptotic expansion. We show that the difference between the exact implied volatility and its approximations is bounded by an appropriate power of the expansion coefficient.

Mathematics Subject Classification: 91B28 35B25

Keywords: implied volatility, local volatility,

Minisymposion: Stochastic Models, Control and Applications