Deriving Vega-Adjusted Delta When Volatility Depends on Forward Rates
Summary
The note explains why a swaption’s delta under SABR may include a volatility contribution. If implied volatility is modeled as a function of the forward rate, the derivative of the instrument’s value with respect to that rate has two parts: the direct price sensitivity to the forward and the vega multiplied by volatility’s sensitivity to the forward. This total derivative is the adjusted delta.
The question raises a practical issue for profit-and-loss attribution: using adjusted delta alongside ordinary vega attribution may count some forward-driven volatility change twice. The answer establishes the chain-rule interpretation of adjusted delta, but does not specify a complete attribution convention or demonstrate a numerical decomposition. In practice, the remaining volatility contribution would need to be defined consistently with the volatility dynamics already included in the delta measure; the brief answer itself does not spell out that procedure.
Key ideas
- When volatility depends on the forward rate, the instrument’s total forward sensitivity includes an indirect volatility effect.
- The adjusted delta adds vega times the sensitivity of volatility to the forward.
- This measure represents the derivative of value along the model-implied volatility relationship.
- Using adjusted delta with an unchanged vega attribution can raise double-counting concerns in P&L explain.
- The note gives the derivative rationale but does not prescribe a full attribution method.
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Full text
# Pnl explain using adjusted SABR delta
# Pnl explain using adjusted SABR delta
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 by RonsenbergVI (score 1)
https://quant.stackexchange.com/a/47472
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.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.