Asymptotics of Implied Volatility in Local Volatility Models
CUNY Baruch College
Elton P. Hsu
Department of Mathematics, Northwestern University
Peter M. Laurence
University of Rome I - Department of Mathematics; Courant Institute, NYU
Baruch College, CUNY
December 23, 2009
Mathematical Finance, Forthcoming
Using an expansion of the transition density function of a 1-dimensional time inhomogeneous diffusion, we obtain the ﬁrst and second order terms in the short time asymptotics of European call option prices. The method described can be generalized to any order. We then use these option prices approximations to calculate the ﬁrst order and second order deviation of the implied volatility from its leading value and obtain approximations which we numerically demonstrate to be highly accurate. The analysis is extended to degenerate diffusion's using probabilistic methods, i.e. the so called principle of not feeling the boundary.
Number of Pages in PDF File: 39
Keywords: Implied volatility, local volatility, asymptotic expansion, heat kernels
Date posted: January 27, 2010 ; Last revised: July 30, 2010
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