# What is “swimming delta” as a risk attribute in pnl explain?

What is swimming delta as in risk attribution?

• "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. – Dimitri Vulis Apr 10 at 13:28

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.

• You shoukd probably add that that is not the total derivative, just a more complete version than the partial without the shadow delta. – will Apr 10 at 15:40
• Yes, I am not happy with this answer and I will make change(s) soon. – noob2 Apr 10 at 16:16

"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.