Demystifying the Bergomi-Guyon expansion

Florian Bourgey and Jim Gatheral

Papers from arXiv.org

Abstract: Al\`os, Gatheral and Radoi\v{c}i\'c derived the Bergomi-Guyon expansion of the implied variance smile from the forest expansion of the cumulant generating function. Its coefficients are sums of products of diamond trees, with prefactors that are polynomials in the log-strike $k$. Matching moments order by order produces, at order $\epsilon^\ell$, terms of degree greater than $\ell$ in $k$ that mysteriously cancel. We show that, in suitable variables, the matching condition can be formulated as a nonlinear heat equation. The resulting recursion computes the prefactor of each product of trees from those of products of fewer trees, without generating the higher-degree terms; its only model-independent input is a cumulant series, which we compute in closed form. Consequently, at order $\epsilon^\ell$ every prefactor has degree exactly $\ell$ in $k$. The prefactors are universal and need only be computed once. Code and coefficients are provided.

Date: 2026-09
References: Add references at CitEc
Citations:

Downloads: (external link)
https://arxiv.org/pdf/2609.17869 Latest version (application/pdf)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:arx:papers:2609.17869

Access Statistics for this paper

More papers in Papers from arXiv.org
Bibliographic data for series maintained by arXiv administrators ().