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

For educational purposes only; not investment advice. Investing may result in loss.

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. Under one fixed valuation convention, 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, rates, dividends, and volatility surface change, so a long holding period should be divided into shorter intervals.

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. Keep broker cash statements and model marks as separate evidence.

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, signed 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 the data, convention, and cash ledger, not automatically “model error.”

One-call daily attribution

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

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.

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

• “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.

Related topics

• Delta and Gamma
• Theta
• Vega
• Option model risk

Authoritative sources

• Characteristics and Risks of Standardized Options — OCC
• Options Basics — Cboe Options Institute
• The Pricing of Options and Corporate Liabilities — Black and Scholes, Journal of Political Economy
• Theory of Rational Option Pricing — Merton, The Bell Journal of Economics and Management Science