Implied Volatility from Local Volatility: A Path Integral Approach
Large Deviations and Asymptotic Methods in Finance, Peter K. Friz, Jim Gatheral, Archil Gulisashvili, Antoine Jacquier, Josef Teichmann eds., pp. 247-271, Springer, 2015
25 Pages Posted: 23 Jun 2014 Last revised: 10 Jul 2015
Date Written: June 22, 2014
Assuming local volatility, we derive an exact Brownian bridge representation for the transition density; an exact expression for the transition density in terms of a path integral then follows. By Taylor-expanding around a certain path, we obtain a generalization of the heat kernel expansion of the density which coincides with the classical one in the time-homogeneous case, but is more accurate and natural in the time inhomogeneous case. As a further application of our path integral representation, we obtain an improved most-likely-path approximation for implied volatility in terms of local volatility.
Keywords: Small time asymptotic expansion, heat kernels expansion, implied volatility, local volatility model, most likely path, path integral
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