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.
CITATION STYLE
Wang, T. H., & Gatheral, J. (2015). Implied volatility from local volatility: A path integral approach. In Springer Proceedings in Mathematics and Statistics (Vol. 110, pp. 247–271). Springer New York LLC. https://doi.org/10.1007/978-3-319-11605-1_9
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