Before revision:  Our main result provides explicit expressions for the first three terms in the expansion of the implied volatility , based on three basic spectral-type statistics of the Gaussian process: the top eigenvalue of its covariance operator, the multiplicity of this eigenvalue, and the L^{2} norm of the projection of the mean function on the top eigenspace.
After revision:  Our main result provides explicit expressions for the first five terms in the expansion of the implied volatility , based on three basic spectral-type statistics of the Gaussian process: the top eigenvalue of its covariance operator, the multiplicity of this eigenvalue, and the L^{2} norm of the projection of the mean function on the top eigenspace.