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.
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)
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Theta per day (single option)
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Theta (annualized per option)
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| 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.
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.
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.
Further resources
External guidance
Expert Q&A
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