Demystifying the Bergomi–Guyon expansion

Battle Count: 135

The Bergomi-Guyon expansion is a well-known tool for approximating implied volatility smiles under stochastic volatility, directly relevant to options pricing and volatility surface calibration. The paper's main contribution is algorithmic efficiency and theoretical clarity (eliminating mysterious cancellations), which simplifies implementation. However, it is a theoretical/mathematical note rather than a trading strategy paper. The practical impact is on the speed and correctness of smile expansion computations used in derivatives pricing desks.

The recursion algorithm (3.9) is straightforward to implement once the tree combinatorics and cumulant series are understood. The authors report generating the expansion through order six in 0.086s and order ten in 19.3s. However, understanding the underlying forest expansion, diamond tree algebra, and the heat equation reformulation requires significant mathematical background in stochastic calculus, combinatorics, and PDE theory. The Python implementation is provided and functional.

The paper provides a complete Python 3.12 implementation of the recursion algorithm (3.9) on GitHub. Coefficients through order six are provided in a text file. A tutorial is included. The authors verified output matches the moment computation of AGR20 through order six. The mathematical derivations are fully self-contained with proofs for all propositions and theorems.