For educational purposes only; not investment advice. Investing may result in loss.
Direct answer
Delta is the local sensitivity of an option value (V) to the chosen underlying-price variable:
Delta = partial V / partial S
A displayed number is incomplete without its convention. A feed may quote Delta per underlying unit or after applying the contract multiplier and quantity; preserve or suppress the put sign; use spot or forward as the risk variable; and use different rates, dividends, volatility, time, exercise, settlement, and price inputs.
For a typical U.S. equity chain, the common quote is per-share spot Delta: a long call is positive and a long put is negative. A long call at 0.40 with a 100 multiplier is about +40 share-equivalents at that instant; selling the contract reverses the position sign to -40.
Signs, scaling, and risk variables
With continuous dividend yield in the Black-Scholes model, European spot Deltas are commonly written:
Delta_call,spot = e^(-qT) N(d_1)
Delta_put,spot = e^(-qT) [N(d_1) - 1]
The raw put Delta is negative. Position direction is a separate sign:
position Delta = raw Delta × contracts × multiplier × long/short sign
If a platform shows 40 instead of 0.40, it may already have applied a 100 multiplier; a portfolio field may also include quantity. Confirm both before multiplying.
Share-equivalent Delta expresses first-order exposure in underlying units. Multiplying it by spot gives one common dollar or cash Delta notional, but vendors also use “dollar Delta” for other shock conventions. State the formula. A share-equivalent Delta of 550 at a $100 stock is $55,000 of cash-Delta notional, while a $1 spot rise implies about +$550 of first-order P&L.
Forward Delta uses the same-maturity forward price (F), rather than spot, as the risk variable. In the same model, a common call forward Delta is (N(d_1)), while spot Delta is (e^(-qT)N(d_1)); dividends and maturity make them differ. Premium-adjusted Delta also reflects how paying the premium changes the relevant notional. It is common in foreign-exchange conventions and has no single formula independent of currency and spot/forward choices; do not assume a U.S. equity chain uses it.
Worked example
Buy 10 calls with raw Delta 0.40 and sell 5 puts with raw Delta -0.30, all with a 100 multiplier:
0.40 × 10 × 100 = +400 share-equivalents
-0.30 × 5 × 100 = +150 share-equivalents
Net position Delta is approximately +550 shares. The short put is positive Delta because the short-position sign reverses the put’s negative raw Delta. If stock is $100, cash-Delta notional under the stated formula is about $55,000.
Now suppose a three-month $95 put on a $100 stock shows spot Delta -0.28 in one system and forward Delta -0.29 in another. For 20 long contracts:
- spot convention:
-0.28 × 20 × 100 = -560share-equivalents; - forward convention scaled the same way:
-0.29 × 20 × 100 = -580forward-underlying units.
The 20-unit difference does not prove either system wrong. Hedging with stock requires a spot-risk interpretation; comparing a volatility surface by constant forward Delta answers a different question. Mixing them creates a persistent apparent hedge discrepancy.
Delta comparison checklist
- Identify the risk variable: spot, forward, futures, index, or another reference.
- Confirm whether Delta is per share, per contract, quantity-weighted, a percentage, or displayed from -100 to +100.
- Preserve the raw option sign and apply the long or short position sign separately.
- Verify the actual contract multiplier and any adjusted deliverable.
- Record spot, option-price input, IV, rate, dividends, time, timestamp, and exercise model.
- Check whether the system uses Bid, Ask, Mid, Last, Mark, or a fitted volatility surface.
- For American options, expect a tree or finite-difference result that depends on dividends and early-exercise assumptions.
- Define “dollar Delta” explicitly: share-equivalent times spot, P&L for a fixed dollar move, or P&L for a percentage move.
- Across stocks, do not add raw share Deltas without dollar, beta, or factor normalization.
- Recalculate after changes in spot, IV, time, dividends, fills, exercise, assignment, or corporate actions.
- Include stock positions: one long share generally contributes +1 spot Delta.
- Use Gamma to stress how Delta changes; a hedge based on current Delta is not static.
- Compare systems on the same contract and timestamp before diagnosing a discrepancy.
- For a “25 Delta” strike, document expiry, call or put, sign, spot or forward convention, premium adjustment, IV side, and interpolation.
- Recognize that the strike matching 25 Delta moves as spot, time, and the volatility surface move.
- Use a small central revaluation check where possible:
([V(S+h) - V(S-h)] / (2h)). - Do not infer maximum loss, assignment exposure, or liquidity from Delta alone.
- Reconcile the actual hedge instrument and ratio; index, ETF, ADR, and futures references can have basis and multiplier differences.
Common misconceptions
- “Every platform’s 0.40 means the same exposure.” Scaling, risk variable, model, and inputs can differ.
- “Put Delta is negative, so selling a put is bearish.” Shorting reverses the sign and creates positive Delta.
- “Delta 0.30 means a 30% probability of profit.” Delta is sensitivity, not profit probability.
- “Call Delta equals the exact probability of expiring in the money.” In Black-Scholes, the risk-neutral in-the-money probability is associated with (N(d_2)), while Delta involves (N(d_1)) and a dividend factor.
- “Delta is fixed.” Gamma, time, and IV make it change.
- “Forty Delta means forty dollars.” It may mean 0.40 per share or about 40 share-equivalents after a 100 multiplier.
- “Dollar Delta has one definition.” Vendors use different shock and notional conventions.
- “A 25 Delta put is a permanent strike.” Constant-Delta strikes move with market inputs.
- “Spot and forward Delta are interchangeable.” They hedge or label different risk variables.
- “A Delta-neutral position is risk-free.” Gamma, Vega, Theta, jumps, basis, liquidity, and assignment remain.