Different versions of sticky strike, moneyness and delta

Question

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?

Answer 1

I feel like your notations are not accurate enough to write what you would like to write.

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.

Answer 2

Here's another non-formulaic (and intuitive) way of looking at sticky by delta and sticky by strike:

Let’s say current market conditions have spot $S_t=100$, the ATM option ($K=100$) has volatility of 0.2, and the 120 strike option has vol of 0.3. Now let's say the spot price moves from 100 to 120. Then sticky by strike implies the new ATM vol ($S_{t+1}=120=K$) is 0.3 and sticky by delta implies the new ATM vol is 0.2.