Modern SABR Analytics by Alexandre Antonov & Michael Konikov & Michael Spector

Author:Alexandre Antonov & Michael Konikov & Michael Spector
Language: eng
Format: epub
ISBN: 9783030106560
Publisher: Springer International Publishing

(3.67)

and allows obtaining all ‘odd’ heat kernels from and all ‘even’ ones from .4

This tells little, though, how to find the important McKean’s heat kernel . We have found a new, quite elementary derivation; no special functions, no contour integrals, only some algebra and differentiation. It follows scheme where (see below) is Gaussian distribution, is obtained from by simple differentiation, as discussed above; and new steps, kernel , that solves a heat equation arising in connection with log-normal SABR model, and modifies multiplying it by function ; finally, follows as a marginal distribution from by integrating out one of two ‘horizontal’ coordinates.

We start now with Dimension one. Element of length is , distance (for it is better to take distance with sign). Laplacian (3.62) looks like . The heat equation becomes the standard parabolic one

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