# Option P&L Attribution: Direction, Convexity, Time, and Volatility

Reconcile option profit and loss into Delta, Gamma, Theta, Vega, rates, hedge cash flows, execution costs, and a documented residual.

Canonical: https://wiki.fcontext.com/options/option-pnl-attribution/
Fact checked: 2026-07-22

> For educational purposes only; not investment advice.

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## Direct answer

Option P&L attribution explains a change in position value through directional movement, curvature, elapsed time, implied-volatility repricing, rates or carry, hedge cash flows, and execution costs. A local approximation for one option is:

`ΔV ≈ Delta × ΔS + 0.5 × Gamma × (ΔS)² + Vega × ΔIV + Theta × Δt + Rho × Δr + residual`

For a position, apply the long or short sign, contract quantity, and multiplier to every leg before summing. Attribution is a model-based reconciliation, not audited accounting truth. Greeks change as the underlying, time, and volatility surface change, so a long holding period should be divided into shorter intervals.

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## Build a reproducible attribution ledger

Use the same timestamp and valuation convention at the start and end of each interval. Record each leg's bid, ask, midpoint or model price; underlying price; IV; Greeks; quantity; multiplier; and side. Do not mix yesterday's close, today's intraday quote, and a stale option trade.

For leg `i`, define a signed exposure:

`signed contracts_i = +quantity for long, -quantity for short`

`leg contribution = Greek_i × market change × multiplier_i × signed contracts_i`

The Gamma term includes `0.5 × (ΔS)²`. Vega units require special care: many platforms quote dollar change per **one volatility point**, so an IV move from 30% to 27% is `-3` points, not `-0.03`, for that convention. Theta can be per calendar day or trading day. Verify platform units before calculating.

Actual economic P&L should include position value, trade cash flows, exercise or assignment, stock hedges, dividends, interest, commissions, and fees:

`total P&L = option mark change + hedge P&L + external cash flows - execution costs`

Then calculate:

`residual = actual P&L - sum(explained contributions)`

Residual can contain changing Greeks within the interval, higher-order interactions, skew and term-structure movement, model changes, discrete sampling, stale or wide quotes, rounding, and missing cash flows. A large residual is a prompt to investigate, not automatically “model error.”

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## One-call daily attribution

Assume one long Call, multiplier `100`, begins the interval with:

| Input | Value |
|---|---:|
| Delta | 0.50 |
| Gamma | 0.04 |
| Vega | $0.10 per volatility point |
| Theta | -$0.06 per day |

Over one day, the underlying rises `$2.00` and IV falls `3` volatility points.

`Delta = 0.50 × $2.00 × 100 = +$100`

`Gamma = 0.5 × 0.04 × $2.00² × 100 = +$8`

`Vega = $0.10 × (-3) × 100 = -$30`

`Theta = -$0.06 × 1 × 100 = -$6`

`explained P&L = $100 + $8 - $30 - $6 = +$72`

If the synchronized option marks show an actual increase of `$65`, residual is `$65 - $72 = -$7`. The directional view was right, but IV contraction and elapsed time offset `$36` of the Delta-plus-Gamma contribution. The `$7` residual should be checked against intraday Greek changes, skew movement, quote selection, rates, and costs.

### Portfolio workflow

For a multi-leg strategy, first calculate each signed leg, then aggregate. Do not apply the strategy's name as a sign shortcut. If stock was bought or sold to Delta hedge, keep its realized and mark-to-market P&L in a separate hedge column. Gamma-scalping gains cannot be evaluated without the option's Theta, IV repricing, and transaction costs.

| Ledger field | Required record |
|---|---|
| Opening and closing value | Same time and valuation convention |
| Market changes | `ΔS`, `ΔIV`, elapsed time, rates |
| Greek contributions | Delta, Gamma, Vega, Theta, Rho by leg |
| Cash flows | Trades, exercise, assignment, dividends, interest |
| Hedge P&L | Stock or other hedge, including costs |
| Residual | Actual less all documented components |

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## Attribution risks

- **Unit error:** percent versus volatility points, per-share versus per-contract, and day-count conventions can create 100-fold errors.
- **Sign error:** short legs and hedge trades must use their economic sign.
- **Timestamp mismatch:** nonsynchronous marks turn market movement into artificial residual.
- **Quote noise:** wide bid-ask spreads can move midpoint P&L without an executable change.
- **Greek drift:** starting Greeks poorly explain large moves or long intervals.
- **Surface risk:** one IV number misses skew and term-structure reshaping.
- **Jump risk:** a gap makes a low-order local approximation less reliable.
- **Cash-flow omission:** premiums, assignments, dividends, financing, and fees must be reconciled.
- **Model risk:** different rates, dividends, volatility surfaces, and exercise assumptions produce different Greeks.
- **False precision:** an exact spreadsheet total does not make approximate inputs exact.

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## Common misconceptions

- **“Delta explains all directional P&L.”** Gamma changes Delta as the underlying moves.
- **“A correct direction guarantees profit.”** IV contraction, time decay, entry price, and costs can outweigh direction.
- **“Vega 0.10 always uses ΔIV = -0.03.”** Platform units may require `-3` volatility points.
- **“Theta is a fixed daily charge.”** It is a local model sensitivity and changes with inputs.
- **“Residual is meaningless noise.”** It may reveal missing cash flows, skew changes, quote problems, or model limitations.
- **“Entry Greeks explain a month.”** Recalculate over shorter intervals and sum them.
- **“Hedge profit proves Gamma scalping worked.”** Include option decay, IV change, and all hedge costs.
- **“Midpoint P&L is realizable P&L.”** Exits occur at executable prices, not guaranteed midpoints.

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## Related topics

- [Delta and Gamma](/options/delta-gamma/)
- [Theta](/options/theta/)
- [Vega](/options/vega/)
- [Option model risk](/options/option-model-risk/)

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## Authoritative sources

- [Characteristics and Risks of Standardized Options](https://www.theocc.com/company-information/documents-and-archives/options-disclosure-document) — OCC
- [Options Basics](https://www.cboe.com/optionsinstitute/options_basics/) — Cboe Options Institute
- [The Pricing of Options and Corporate Liabilities](https://doi.org/10.1086/260062) — Black and Scholes, *Journal of Political Economy*
- [Theory of Rational Option Pricing](https://doi.org/10.2307/3003143) — Merton, *The Bell Journal of Economics and Management Science*