Theta Calculator (Options)
This calculator estimates Theta, the time decay of an option's value, using the Black-Scholes analytic formula for European options and an approximate method for American options. Outputs include annualized Theta and Theta per calendar day, displayed per option and scaled per contract.
Use this tool for quick, principled estimates. For settlement, margining, regulatory reporting, or trading decisions that require high precision for American options, perform a lattice (binomial/trinomial) pricing or use vendor-validated systems.
Governance
Record beaae8f1ff2b • Reviewed by Fidamen Standards Committee
Standard Black-Scholes analytic Theta for European call and put options. Use for non-dividend or continuous-dividend underlying where early exercise is not allowed.
Inputs
Results
Theta per day (per contract)
—
Theta per day (single option)
—
Theta (annualized per option)
—
| Output | Value | Unit |
|---|---|---|
| Theta per day (per contract) | — | currency/day |
| Theta per day (single option) | — | currency/day |
| Theta (annualized per option) | — | currency/year |
Visualization
Methodology
European Theta is computed from the Black-Scholes model assuming continuous dividend yield. The formula uses standard normal cumulative and density functions (N and N').
American options may have an early-exercise premium; this tool provides a baseline approximation by reusing the European result. For precise American results, a discrete lattice model with convergence testing is recommended.
Key takeaways
Black-Scholes provides a closed-form Theta for European options; use it as the authoritative analytic baseline.
American-option results returned here are approximations; use lattice pricing for firm-grade accuracy.
Worked examples
Example 1: S=100, K=100, 30 days to expiry, vol=20%, r=1%, q=0% → Theta per day (per contract) estimates the expected daily time decay in currency units.
Example 2: For an American put on a high-dividend stock, expect European-based Theta to understate early-exercise effects. Use a binomial model for a production-ready number.
F.A.Q.
Is Theta here per trading day or calendar day?
Theta per day is calculated on a calendar-day basis (annual Theta divided by 365). Adjust manually if you require trading-day convention.
Does the calculator handle American early exercise exactly?
No. American-style options can have early-exercise value that this calculator does not model exactly. The 'American' method provides an approximation; use a binomial or trinomial model for exact valuation.
What is the meaning of a negative Theta?
Negative Theta indicates the option is expected to lose value as time passes, all else equal. Positive Theta is possible for some short option positions or under special conditions.
How accurate are results and what are the model limits?
Results follow model assumptions: lognormal underlying returns, constant volatility, constant rates and yields, and continuous dividend yield. Real markets violate these assumptions; see the accuracy and calibration guidance in citations.
How should I use contract multiplier?
Multiply per-option Theta by the contract multiplier to express Theta per listed contract (standard equity options typically use 100).
Sources & citations
- Black and Scholes, The Pricing of Options and Corporate Liabilities (1973) — https://www.jstor.org/stable/1831029
- NIST (Numerical standards and guidance) — https://www.nist.gov
- ISO (Standards for quality and measurement traceability) — https://www.iso.org
- IEEE (Standards and best practices for numerical computations) — https://www.ieee.org
- OSHA (Regulatory compliance guidance for workplace processes) — 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: beaae8f1ff2bWhat changed (latest)
v1.0.0 • 2025-11-08 • 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 • 2025-11-08 • 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 8, 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 • 2025-11-08 • 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: 6c40d290fc2f