Delta Calculator (Options)
This calculator computes the Black‑Scholes European option Delta and option prices from user inputs: spot price, strike, annual volatility, time to expiry (in days), continuously compounded risk‑free rate and dividend yield. Use it for European-style options where early exercise is not allowed and volatility is approximately constant over the life of the option.
The tool shows both call and put prices and a single Delta value (call or put) based on the selected option type. Results are numerical approximations and should be used for analysis and educational purposes; use production-grade pricing engines for live trading decisions.
Governance
Record e7f830c1c30a • Reviewed by Fidamen Standards Committee
Inputs
Results
d1 (standardized)
—
d2 (standardized)
—
European call price
—
European put price
—
Delta (option sensitivity to spot)
—
| Output | Value | Unit |
|---|---|---|
| d1 (standardized) | — | — |
| d2 (standardized) | — | — |
| European call price | — | USD |
| European put price | — | USD |
| Delta (option sensitivity to spot) | — | — |
Visualization
Methodology
We implement the Black‑Scholes closed form for European options. d1 and d2 are computed as standardised inputs to the normal cumulative distribution. The cumulative normal is evaluated via the error function (erf) to provide a numerically stable approximation.
Inputs must be annualized consistently: volatility as annual decimal (e.g. 20% = 0.20), rates as continuous annual decimals, and time converted to years as days/365. The calculator assumes continuous compounding and lognormal underlying price dynamics.
F.A.Q.
What does Delta mean?
Delta measures the sensitivity of the option price to a small change in the underlying price (∂option/∂S). It ranges typically between -1 and +1. Positive for calls and negative for puts.
Is this valid for American options or discrete dividends?
No. This implementation is for European options with continuous dividend yield. American options and discrete dividends require binomial, finite‑difference, or specialized models.
How accurate are the results?
Results are numerically accurate within limits of the Black‑Scholes assumptions and the error‑function approximation used for the normal CDF. Expect reduced accuracy for near-zero time to expiry, extremely high volatilities, or ill-conditioned numeric inputs. Floating point rounding follows IEEE 754 conventions and may introduce small errors.
How should I calibrate volatility and rates?
Calibrate volatility from market option prices (implied volatility) or historical returns using established statistical methods. Ensure rates reflect the same compounding convention (continuous) used here or convert accordingly.
What security, accuracy and governance standards are referenced?
Numerical computations follow IEEE floating point guidance (IEEE 754). Risk and model governance should follow ISO 31000 principles. System controls and cryptographic modules should adhere to applicable NIST recommendations. Workplace and operational safety should observe relevant OSHA guidelines.
Sources & citations
- Black–Scholes model (overview) — https://en.wikipedia.org/wiki/Black%E2%80%93Scholes_model
- IEEE 754 floating‑point standard — https://standards.ieee.org/standard/754-2019.html
- ISO 31000 Risk management — https://www.iso.org/iso-31000-risk-management.html
- NIST (standards and guidance) — https://www.nist.gov
- OSHA (occupational safety guidance) — https://www.osha.gov
- OCC — Options Clearing Corporation — https://www.theocc.com/
- CBOE — Options Education — https://www.cboe.com/education/
Further resources
Versioning & Change Control
Audit record (versions, QA runs, reviewer sign-off, and evidence).
Record ID: e7f830c1c30aWhat changed (latest)
v1.0.0 • 2025-11-25 • MINOR
Initial publication and governance baseline.
Why: Published with reviewed formulas, unit definitions, and UX controls.
Public QA status
PASS — golden 25 + edge 120
Last run: 2026-01-23 • Run: golden-edge-2026-01-23
Versioning & Change Control
Audit record (versions, QA runs, reviewer sign-off, and evidence).
What changed (latest)
v1.0.0 • 2025-11-25 • MINOR
Initial publication and governance baseline.
Why: Published with reviewed formulas, unit definitions, and UX controls.
Public QA status
PASS — golden 25 + edge 120
Last run: 2026-01-23 • Run: golden-edge-2026-01-23
Engine
v1.0.0
Data
Baseline (no external datasets)
Content
v1.0.0
UI
v1.0.0
Governance
Last updated: Nov 25, 2025
Reviewed by: Fidamen Standards Committee (Review board)
Credentials: Internal QA
Risk level: low
Reviewer profile (entity)
Fidamen Standards Committee
Review board
Internal QA
Entity ID: https://fidamen.com/reviewers/fidamen-standards-committee#person
Semantic versioning
- MAJOR: Calculation outputs can change for the same inputs (formula, rounding policy, assumptions).
- MINOR: New features or fields that do not change existing outputs for the same inputs.
- PATCH: Bug fixes, copy edits, or accessibility changes that do not change intended outputs except for previously incorrect cases.
Review protocol
- Verify formulas and unit definitions against primary standards or datasets.
- Run golden-case regression suite and edge-case suite.
- Record reviewer sign-off with credentials and scope.
- Document assumptions, limitations, and jurisdiction applicability.
Assumptions & limitations
- Uses exact unit definitions from the Fidamen conversion library.
- Internal calculations use double precision; display rounding follows the unit's configured decimal places.
- Not a substitute for calibrated instruments in regulated contexts.
- Jurisdiction-specific rules may require official guidance.
Change log
v1.0.0 • 2025-11-25 • MINOR
Initial publication and governance baseline.
Why: Published with reviewed formulas, unit definitions, and UX controls.
Areas: engine, content, ui • Reviewer: Fidamen Standards Committee • Entry ID: e963f553931a