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Delta
Delta estimates the option-price change for a small change in the underlying, with other inputs held constant.
Lesson · Options Trading
Delta and Gamma
Measure first-order directional sensitivity with delta, observe how gamma changes delta, and compare linear estimates with full theoretical repricing.
Synthetic educational examples
Core concepts
Delta is a slope; gamma is how fast the slope changes.
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Position delta
Multiplying option delta by the contract multiplier and quantity estimates share-equivalent directional exposure.
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Gamma
Gamma estimates how much delta changes for a $1 move in the underlying and is often greatest near the strike close to expiration.
Worked example
The 100 call, 30 DTE, IV 30%, one contract, with the stock moving from $100 to $105 — the lab’s default state, with the working shown.
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Delta now 0.534: first-order estimate 0.534 × $5 × 100 = +$267.Delta is the slope at $100. The estimate assumes the slope holds for the whole $5.
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Gamma 0.0462: delta rises about 0.046 per dollar, so add ½ × 0.0462 × 5² × 100 = +$58, giving +$325.The gamma term is the correction for the slope changing along the way.
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Full repricing: $3.61 → $6.83, +$322. Delta after the move is 0.743.The linear estimate used 0.534 the whole way and undershot by $55; the gamma estimate overshoots slightly because gamma itself changes.
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Position delta: 0.534 × 100 = 53 share-equivalents for one contract; three contracts are 160.Multiply by the multiplier and by the quantity. It is what the position gains per $1 of stock, right now, and it will be different after the move.
Interactive lesson
Set a move and compare the delta-only estimate, the delta-plus-gamma estimate and the full repricing.
Horizontal axis: underlying price. Vertical axis: theoretical option delta. Delta and gamma are estimates, not guarantees.
Use delta and gamma
Treat Greeks as changing sensitivities rather than fixed contract facts.
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Read current delta
Use sign and magnitude to understand the immediate directional exposure.
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Convert to position delta
Multiply by 100 and contract quantity for standard equity options.
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Inspect gamma
Higher gamma means delta can change more quickly as the stock moves.
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Recalculate after material changes
Stock, time, and IV changes can make the original Greeks stale.
Delta and gamma
Score 0 / 0Choose the best answer.
Lesson rule
Change the inputs, calculate the result, and explain the risk before selecting a contract.
Common mistakes
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Treating delta as constantDelta changes with stock price, time, IV, and other inputs.
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Ignoring the multiplierA 0.50 delta standard contract represents roughly 50 share deltas, not half a share.
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Using probability language as certaintyDelta-based probability interpretations are model-dependent approximations.
Completion standard
✓
Finish when you can turn a delta into a first-order dollar estimate, convert it to share-equivalent exposure with the multiplier and quantity, and explain why the estimate undershoots on a large move.
The drill asks for all three; the worked example above does each once.
Model scope: Theoretical values use a simplified European Black–Scholes model with a fixed 4.5% annual rate and no dividend input, so a number here can be reproduced in any calculator that takes those inputs. Listed products, early exercise, dividends, spreads, fees, and market conditions can produce different prices and outcomes.