Pnl explain using adjusted SABR delta

Question

When looking at SABR, the starting point for a swaption's delta is the usual:

$\Delta = \partial V/\partial F $

However, since we have expressed our volatility $ \sigma $ as a function of our forward $ F $, we can compute an adjusted delta that accounts for the sensitivity of our volatility to our forward:

$\Delta{adj} = \Delta + vega * \partial \sigma / \partial F$

I'm trying to understand the intuition behind vega-adjusted SABR delta by looking at Pnl explain on a ATM swaption straddle.

Normally, first order Pnl explain would bucket underlying moves into delta Pnl and volatility moves into vega Pnl, but the adjusted delta appears to be "soaking up" some portion of the volatility change due to the infinitesimal change in the forward rate on the underlying. Thus, doing a naive Pnl explain with adjusted delta and conventional vega pnl seems to double count some portion of the volatility move.

When trying to attribute Pnl using adjusted delta, does our vega Pnl explain have to change interpretation from "outright first order volatility Pnl" to "residual volatility move after accounting for model-implied rate-volatility dynamics?"

Answer 1

Your volatility also depends on your forward level, as does the value of your derivatives; so a more accurate definition of your delta under a variable volatility is:

$$ \dfrac{\partial V(F,\sigma(F))}{\partial F } = \dfrac{\partial V(F,\sigma)}{\partial F } + \dfrac{\partial V}{\partial \sigma}\dfrac{\partial \sigma(F)}{\partial F} $$

This is because, the value of your product is not only going to change as F changes but also as $\sigma(F)$ changes. Your adjusted derivative expresses this relationship.