# Option Delta Conventions: Signs, Scaling, Spot, and Forward Delta

Compare per-share and position Delta, sign conventions, share-equivalent and dollar scaling, spot versus forward Delta, and why 25 Delta is not a fixed strike or profit probability.

Canonical: https://wiki.fcontext.com/options/option-delta-conventions/
Fact checked: 2026-07-22

> For educational purposes only; not investment advice.

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## Direct answer

**Delta** is the local sensitivity of option value (V) to an underlying-price variable:

`Delta = partial V / partial S`

But a displayed number is incomplete without its convention. A system may report Delta per underlying unit or after applying contract multiplier and position quantity; retain or suppress the put sign; use spot or forward as the risk variable; and use different rates, dividends, volatility, time, exercise, and price inputs.

For ordinary U.S. equity options, a common chain convention shows per-share spot Delta: long calls are positive and long puts negative. A long call at 0.40 with multiplier 100 has approximately +40 share-equivalents at that instant. Selling the contract reverses the position sign to −40.

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## Signs, scaling, and risk variables

Under a continuous-dividend 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 option's raw put Delta is negative. Position direction is a separate sign:

`position Delta = raw Delta × contracts × multiplier × long/short sign`

A platform showing Delta as 40 instead of 0.40 may already scale by 100. A portfolio field may also include quantity. Confirm both before multiplying.

**Share-equivalent Delta** expresses current first-order exposure in underlying units. Multiplying it by spot produces a common **dollar or cash Delta** notional, but vendors also use “dollar Delta” for other shock conventions. Always state the formula. A share-equivalent Delta of 550 at a $100 stock has $55,000 cash-Delta notional, while its estimated P&L for a $1 spot rise is about +$550.

**Forward Delta** takes 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)). Dividend yield and maturity make them differ. **Premium-adjusted Delta** additionally reflects how premium payment changes the relevant notional; it is more common in foreign-exchange conventions and has no single formula independent of currency and spot/forward choices. Do not assume an equity chain uses it.

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## Two convention mismatches

Buy 10 calls whose raw Delta is 0.40 and sell 5 puts whose raw Delta is −0.30, all with multiplier 100:

`0.42 × 3 × 100 = +126 share-equivalents`

`-0.31 × 2 × 100 = -62 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=-560) share-equivalents;
- forward convention scaled the same way: (-0.29×20×100=-580) forward-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.

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## Delta comparison checklist

- Identify underlying risk variable: spot, forward, futures, index, or another reference.
- Confirm whether Delta is per share, per contract, already quantity-weighted, a percentage, or displayed from −100 to +100.
- Preserve raw option sign and apply long or short position sign separately.
- Verify actual contract multiplier and 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 numerical tree or finite-difference results and sensitivity to dividend 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.
- For portfolios across stocks, do not add raw share Deltas without a dollar, beta, or factor normalization.
- Recalculate after spot, IV, time, dividend, fill, exercise, assignment, or corporate-action changes.
- 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 expiration, call or put, sign, spot or forward convention, premium adjustment, IV side, and interpolation.
- Recognize that the strike matching 25 Delta changes as spot, time, and 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.

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## 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, creating positive Delta.
- “Delta 0.30 means a 30% probability of profit.” Delta is sensitivity, not profit probability.
- “Call Delta equals exact probability of expiring ITM.” In Black-Scholes, risk-neutral ITM 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.

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## Related topics

- [Delta and Gamma](/options/delta-and-gamma/)
- [Contract multiplier](/options/contract-multiplier/)
- [Delta hedging](/options/delta-hedging/)
- [Net Greeks](/options/net-greeks/)

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## Primary sources

- [Cboe: Options Calculator](https://www.cboe.com/education/tools/options-calculator/)
- [OCC: Industry Services—Delta Values](https://www.theocc.com/clearance-and-settlement/industry-services)
- [Black and Scholes (1973): The Pricing of Options and Corporate Liabilities](https://doi.org/10.1086/260062)
- [Merton (1973): Theory of Rational Option Pricing](https://doi.org/10.2307/3003143)