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

For educational purposes only; not investment advice.

Direct answer

Section titled “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.

Build a reproducible attribution ledger

Section titled “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.”

One-call daily attribution

Section titled “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

Section titled “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

Attribution risks

Section titled “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.

Common misconceptions

Section titled “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.
Section titled “Related topics”

Authoritative sources

Section titled “Authoritative sources”