# Different versions of sticky strike, moneyness and delta

I head a lot of versions of those three concepts: sticky strike, sticky moneyness and sticky delta, especially fot sticky delta. For example:

sticky strike: $$\dfrac{\partial \sigma_{im}(S,K,T)}{\partial K} = 0.$$

sticky moneyness: $$\dfrac{\partial \sigma_{im}(S,K,T)}{\partial \dfrac{K}{S}} = 0$$

sticky delta: $$\dfrac{\partial \sigma_{im}(S,K,T)}{\partial \ln\dfrac{K}{S}} = 0.$$

But someone identifies sticky moneyness and sticky delta. And someone call $$\dfrac{\partial \sigma_{im}(S,K,T)}{\partial \Delta} = 0$$ sticky delta.

The last version sticky delta much meets the intuition. So, which version is the correct or most commonly used one?

Let $\Sigma(S;K,T)$ denote the implied volatility of a European vanilla of strike $K$ and maturity $T$ now that the underlying spot price is worth $S$.
Sticky strike translates to $$\Sigma(S+\delta S;K,T) = \Sigma(S;K,T) \iff \color{blue}{\frac{\partial \Sigma}{\partial S}(S; K, T) = 0}$$
Sticky moneyness would require re-expressing the IV in the moneyness rather than absolute strike space by defining the function $$\hat{\Sigma}(S;m,T) = \Sigma(S;K=S m, T)$$ and then write that $$\hat{\Sigma}(S+\delta S; m, T) = \hat{\Sigma}(S; m ,T) \iff \color{blue}{\frac{\partial \hat{\Sigma}}{\partial S}(S; m, T) = 0}$$ One can show that this stickiness assumption is the one embedded in space homogeneous diffusion models since \begin{align} \frac{\partial \hat{\Sigma}}{\partial S}(S; m, T) &= \frac{\partial \Sigma}{\partial S}(S; K, T) + m \frac{\partial \Sigma}{\partial K}(S; K, T) \\ &= \frac{\partial \Sigma}{\partial S}(S; K, T) + \frac{K}{S} \frac{\partial \Sigma}{\partial K}(S; K, T) \end{align} which is zero under a space homogeneous diffusion model because the following holds (would require a separate question to show that) $$S \frac{\partial \Sigma}{\partial S}(S; K, T) = - K \frac{\partial \Sigma}{\partial K}(S; K, T)$$ The other definitions you mention are actually equivalent to the sticky moneyness, in the sense that it amounts to considering not $\Sigma(S; K, T)$ but rather a re-expression of the IV in a spatial dimension $\theta$ such that $$\hat{\Sigma}(S; \theta, T) = \Sigma(S; K = S f(\theta), T)$$
For instance in a sticky delta you would have $$\frac{\partial \hat{\Sigma}(S; \Delta, T)}{\partial S} = 0$$ Intuitively, it's equivalent to sticky moneyness because for the $\Delta$ to remain constant, everything else being equal, it's the same as for $K/S$ (or $\ln(K/S)$) to remain constant. More formally, you can re-use the same argument as the one I just hinted above.
• but what's the correct definition of sticky delta? – user6703592 Aug 7 '18 at 9:08