Swimming Delta and Volatility Skew in Options Risk
Summary
The document explains swimming delta as an option’s total sensitivity to the underlying when implied volatility changes with the underlying price. Standard Black–Scholes delta is a partial derivative that holds volatility constant. Under a skew-aware view, a move in the underlying can also shift the option’s implied volatility, adding a vega-weighted skew effect to the direct delta exposure.
It gives the total-derivative expression as Black–Scholes delta plus vega multiplied by the change in implied volatility per unit move in the underlying. A second explanation equates swimming delta with floating delta: under a floating-skew convention, the volatility shape moves with the forward price, while sticky skew holds volatility fixed by strike. The document notes that empirical evidence distinguishing these conventions is inconclusive and that their difference may be small compared with daily changes in volatility levels. The adjustment therefore depends on the assumed surface behavior and an empirically estimated skew slope.
Key ideas
- Black–Scholes delta measures price sensitivity while holding implied volatility constant.
- Swimming delta includes the impact of implied volatility moving with the underlying.
- The adjustment adds vega multiplied by the implied-volatility skew slope to standard delta.
- Floating skew moves the volatility surface with the forward, while sticky skew anchors it to strike.
- The cited empirical comparison does not decisively favor either convention.
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Full text
# What is "swimming delta" as a risk attribute in pnl explain?
# What is "swimming delta" as a risk attribute in pnl explain?
What is swimming delta as in risk attribution?
## Answer by nbbo2 (score 4)
https://quant.stackexchange.com/a/53185
Delta is the partial derivative of Call price C with respect to Stock price S, i.e $\frac{\partial C}{\partial S}$.
In the BSM model implied vol $\sigma$ is constant, in particular it does not depend on $S$ so there is nothing further to discuss.
When we allow for skew (but still compute IV according to the BSM model) the effect of a change in S is more complicated. When S increases, C increases by the direct effect mentioned earlier. But the increase in S also causes a drop in implied vol, which makes the call slightly less valuable.
Swimming Delta is the total derivative $\frac{dC}{dS}$ including both effects.
According to equation 5.8 in Euan Sinclair's book
$\frac{dC}{dS}=\frac{\partial C}{\partial S}+\frac{\partial C}{\partial \sigma}\frac{\partial \sigma}{\partial S}$
$=\Delta_{BSM}+\text{Vega}_{BSM}\cdot \frac{\partial \sigma}{\partial S}$
The last term is the (empirically estimated) slope of the Skew.
Therefore I think of Swimming delta as an ad-hoc correction to the Black Scholes delta.
## Answer by Dimitri Vulis (score 1)
https://quant.stackexchange.com/a/53194
"Swimming delta" is another name for "floating delta" (as opposed to sticky delta). books.google.com/books?id=LnLgAgAAQBAJ&pg=PA170 Glen Swindle's excellent book "Valuation and Risk Management in Energy Markets" happens to be in Google books and explains it well on page 170.
I'm just going to quote from his book here (see the book for formulas).
> This leads to the following question: How does the volatility surface behave as a function of the forward price? This question has implications for the effective delta of any options portfolio, even those comprised solely of vanilla options. This basic question can be refined as follows: Given a prescribed change in the underlying forward price, what inference can be made about the change in the implied volatility surface? ... The volatility look-up protocol implicitly assumes that the ATM volatility does not change because of forward price movements. This is commonly referred to as the floating-skew convention because the default estimate for the new volatility surface given the new forward price is that the volatility surface shifted in tandem with the forward price with shape unchanged. As prices move, the volatility surface appears fixed when viewed in reference to the forward price but changes for any fixed strike option... The other hypothesis most frequently discussed is sticky skew, in which the volatility surface is parameterized by absolute strike... Here the volatility surface is fixed with respect to option strike and moves when viewed from the reference of the prevailing forward price... As a consequence, delta under the floating-skew hypothesis is not the standard delta obtained from Black but is rather the total derivative... So which of these paradigms is more consistent with empirical behavior? The results are not particularly compelling in part because any difference between the two approaches appears to be small relative to daily changes in implied volatility levels.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.