揭秘 Bergomi-Guyon 展开

中文摘要 AI 辅助

Alòs、Gatheral 和 Radoičić 从累积量生成函数的森林展开中推导出了隐含方差微笑的 Bergomi-Guyon 展开。其系数是钻石树乘积之和，前置因子是对数执行价 $k$ 的多项式。逐阶匹配矩会在 $\u03b5^\ell$ 阶产生 $k$ 中次数大于 $\ell$ 的项，这些项神秘地相互抵消。我们证明，在合适的变量下，匹配条件可以表述为一个非线性热方程。由此产生的递推关系从较少树的乘积的前置因子计算出每个树乘积的前置因子，而不会生成高次项；其唯一的模型无关输入是累积量级数，我们以闭式形式计算了它。因此，在 $\epsilon^\ell$ 阶，每个前置因子在 $k$ 中的次数恰好为 $\ell$。这些前置因子是通用的，只需计算一次。我们提供了代码和系数。

英文摘要

Alòs, Gatheral and Radoičić 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 $ε^\ell$, 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 $ε^\ell$ every prefactor has degree exactly $\ell$ in $k$. The prefactors are universal and need only be computed once. Code and coefficients are provided.

发表机构

• NYU（纽约大学）
• Baruch College, CUNY（巴鲁克学院，纽约市立大学）

机构由 AI 辅助整理，请以论文原文为准。