Attributing Option Vega P&L to Moneyness, Time, and Surface Changes

# How to quantify Vega PNL from volatility moving along a steep moneyness surface when bucket Vega shows zero?


# How to quantify Vega PNL from volatility moving along a steep moneyness surface when bucket Vega shows zero?












Based on this answer:

Question: When the implied volatility surface remains identical over two consecutive days, my current Vega P&L calculation yields zero. However, market movements cause options' effective volatility to shift along the surface—particularly impactful with steep volatility skew. How should we properly quantify this P&L component?

Current methodology:

- Discretized volatility surface at $t$: $\sigma_{ij}^t$ (moneyness $i$, term $j$)

- $\Delta \sigma_{ij} \equiv \sigma_{ij}^{t+1} - \sigma_{ij}^{t}$ (matrix subtraction)

- Bucket Vega calculation: $\small\text{BucketVega}_{ij} = \left[PV(\sigma_{ij} + 0.01) - PV(\sigma_{ij})\right] \times 100$

- Vega P&L $= \sum_i \sum_j \left( \text{BucketVega}_{ij} \times \Delta \sigma_{ij} \right)$

Conflict under sticky delta:

- Spot moves shift options to new moneyness coordinates → effective IV changes due to skew

- Identical surfaces give $\Delta \sigma_{ij} = 0$ → skew-induced P&L vanishes! (Example: ATM call becomes OTM and "slides" to lower vol point on static surface)

Required quantification: How to capture P&L from:

- Movement along static surface (skew effect)

- Time-decay (roll-down $\partial\sigma/\partial\tau$)

- True surface shape changes

## Answer by Dimitri Vulis (score 0)

https://quant.stackexchange.com/a/83661

Right, the P&L explain suggested in the answer you cited, doesn't show clearly how much P&L is attributable from the change in moneyness.

"Most people" use something along these lines, supplemented with whatever they want to see in addition. So you might add something like: Let $\sigma_{t,{T_E-t},m}$ denote the implied volatility for moneyness $m$. At time $T_0$, the moneyness was $m_0$, and the IV was $\sigma_{0,{T_E-T_0},m_0}$. At time $T_1$, the moneyness became $m_1$, and the IV changed to $\sigma_{1,{T_E-T_1},m_1}$.

(Some people prefer absolute strike levels as an axis of the vol surface as @Quantuple suggests, others prefer relative moneyness. Please see any sticky strike v sticky delta debate, e.g. Sticky delta vs sticky strike . You may want to explore empirically whether, for a large change in your underlyings prices, your IV changes less, assuming the strikes remain unchanged, or the moneyness remains unchanged.)

Attribute the IV changes to using the new vol surface, and to the change in moneyness, and to change in the time to expiration. Depending on how deeply you want to dive, you can look at the differences between the various combinations of IVs $\sigma_{0,{T_E-T_1},m_1}$, $\sigma_{1,{T_E-T_0},m_0}$, $\sigma_{0,{T_E-T_0},m_1}$, $\sigma_{1,{T_E-T_1},m_0}$... Multiply each IV change by the (parallel shift) vega at $T_0$ to estimate the P&L impact. I'm not sure whether you can easily include this analysis in the P&L disaggregation, such that all the components add up to the P&L, or need to make it separate.

Using absolute strikes to index the vol surface might hide the P&L impact of the change in moneyness, but wouldn't actually make it disappear.