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Key Takeaways
- Greeks attribution assigns modeled value changes to underlying price, curvature, elapsed time and implied volatility.
- Use a single starting set of Greeks and consistent units for the four-term calculation.
- The example attributes +$100 to delta, +$10 to gamma, −$4 to theta and −$12 to vega.
- Those contributions total +$94 against the worked +$96 observed mark change.
- Check the remaining +$2 against a full repricing before blaming any single Greek.
What is Greeks P&L attribution?
Greeks P&L attribution is a method that splits a change in an option position’s value into contributions from the underlying price, curvature, elapsed time and implied volatility, measured with delta, gamma, theta and vega. It uses consistent starting inputs, signed position quantity and the contract multiplier, and it shows why an option gained or lost value even when the stock moved the expected way.
How is Greeks P&L attribution calculated?
Delta times the price change gives the linear term. One-half gamma times the squared price change adds curvature. Daily theta times elapsed days gives the time term. Vega per volatility point times the IV-point change gives the volatility term.
Formula
Position value-change estimate = signed contracts × multiplier × [delta × ΔS + ½ × gamma × (ΔS)² + daily theta × elapsed days + vega per volatility point × IV-point change].- Measure all Greeks at the same starting inputs.
- An IV move from 25% to 24% is −1 volatility point for the stated vega convention.
- Use negative contract quantity for a short option and the actual contract multiplier.
- The four terms cover price to second order and time and IV to first order. A large joint move in price and IV adds cross and vol-curvature effects, which land in the residual.
- Residual = observed mark change minus the four attributed contributions.
Greeks P&L attribution example
One long call starts with delta 0.50, gamma 0.05, daily theta −$0.04 and vega $0.12 per volatility point, all per share. Over one day the stock rises $2 and IV falls one percentage point. The multiplier is 100, and the worked observed option-mark change is +$0.96 per share.
- Delta P&L: 0.50 × $2.00 = $1.00 per share
- Gamma P&L: 0.5 × 0.05 × ($2.00)² = $0.10 per share
- Theta P&L: −$0.04 × 1 day = −$0.04 per share
- Vega P&L: $0.12 × (−1.0 volatility point) = −$0.12 per share
- Total attributed: ($1.00 + $0.10 − $0.04 − $0.12) × 100 = $94 per contract
The four contributions total +$94 for one long contract. The observed mark change is +$96, leaving a +$2 residual. Delta and gamma carried the trade with $110; decay and the IV drop cost $16. Check the $2 against a full repricing before pinning it on a cause.
For one long contract, the example contributes +$100 delta, +$10 gamma, −$4 theta and −$12 vega. The sum is +$94.
The worked observed mark rises $96, leaving a $2 residual. Reprice the option in full and check the marks and any other changed inputs to explain it. The $96 is a mark-to-market gain until the call is sold.
Taylor expansion vs sequential repricing
The four-term formula is second order in underlying price and first order in time and implied volatility. It uses fixed starting Greeks, so adding the terms in another order does not change the result.
Sequential full repricing instead changes factors one at a time. The order reallocates interacting effects among its buckets. Completing all changes still reaches the same final model value, so the bucket sum equals that model’s total change.
How traders use Greeks P&L attribution
Use delta, gamma, theta and vega to identify the position’s main sensitivities. Keep the values and units for the selected contract and side.
Build price, time and volatility scenarios in the options profit calculator. Compare the resulting option values with the entry price, then include the other position legs and actual transaction cash flows in the complete trade result.
Limitations of Greeks P&L attribution
- Expect a bigger residual on a large joint move. The four terms skip the cross effects between price, time and volatility and the curvature of vega, so a big move in price and IV together leaves more unexplained.
- Reprice in full over big moves and long intervals. The terms use Greeks fixed at the starting inputs, and those Greeks change as the move and the days add up.
- Diagnose the residual before acting on it. Omitted terms, other changed inputs and stale or wide marks all feed it, so check each one.
- Fix the factor order in sequential repricing. The order decides how interacting effects split among the buckets, so compare two attributions only when they use the same order.
- Treat the mark change as unrealized until you close the trade, and keep vega and the IV change in the same per-volatility-point units.
Bottom line
Greeks P&L attribution explains an option’s profit or loss in dollars by price, curvature, time and volatility, which shows why a position moved the way it did. Use one set of starting Greeks with consistent units, then reconcile the residual with full repricing before drawing conclusions about any single bucket.
Clear answers
Frequently Asked Questions
Sources and Further Reading
- Fidelity: Greek symbols in trading
- CFA Institute: Understanding investment risk
- Columbia University (M. Haugh): Basic Concepts and Techniques of Risk Management
How entries are sourced, calculated and updated: editorial methodology.