Asymptotics of Implied Volatility in Local Volatility Models
Mathematical Finance, Forthcoming
39 Pages Posted: 27 Jan 2010 Last revised: 30 Jul 2010
Date Written: December 23, 2009
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.
Keywords: Implied volatility, local volatility, asymptotic expansion, heat kernels
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