On the Skew Stickiness Ratio
Masaaki Fukasawa
TL;DR
The paper addresses rigorous computation of the Skew Stickiness Ratio (SSR) and its asymptotics in Bergomi-type stochastic volatility models. It derives a representation of SSR using the Itô-Wentzell and Clark-Ocone formulae, yielding a Malliavin-derivative form $R_t = X_t/Y_t$ with explicit components such as $X_t = \mathsf{E}[1_{\{S_t>S_T\}} \frac{S_T}{S_t} (1 - \frac{\mathcal{D}^1_t \log S_T}{\sqrt{V_t}}) | \mathscr{F}_t]$ and $Y_t = \mathsf{P}[S_t>S_T|\mathscr{F}_t] - \Phi(\frac{\sqrt{\Sigma^S_t}}{2})$. It further specializes to multi-factor Bergomi kernels, giving expressions for $\mathcal{D}^1_t \sqrt{V_s}$ and $\mathcal{D}^1_t \log S_T$. The paper then establishes two key asymptotics: (i) a short-maturity limit $\lim_{T\to t} R_t(T) = H + \tfrac{3}{2}$ for kernels with $g(u)=u^{1/2-H}k(u)$ and $H\in(0,1/2]$, and (ii) a small volatility-of-volatility limit $R(\epsilon)$ converging to a closed-form ratio of forward-variance functionals. These results provide a rigorous basis for market-consistent SSR values and guidance for hedging cross-gamma risk in Bergomi-type models, including rough Bergomi settings.
Abstract
The skew stickiness ratio is a statistic that captures the joint dynamics of an asset price and its volatility. We derive a representation formula for this quantity using the Itô-Wentzell and Clark-Ocone formulae, and we apply it to analyze its asymptotics under Bergomi-type stochastic volatility models.