Theta (option time decay)
Theta is the option greek that measures how much an option's theoretical value changes as time passes with price, volatility and rates held constant, usually quoted per calendar day. Long options have negative theta, short options positive theta, and the decay of an at-the-money option accelerates as expiry approaches.
Senzoukria · Glossary · Updated September 2026
At a glance
- Definition
- Change in option value per unit of time passing (∂V/∂t)
- Usual unit
- Premium per calendar day (annual theta ÷ 365)
- Sign
- Negative for long options, positive for short options (with rare exceptions for deep in-the-money European puts)
Definition and formula
Theta is the rate at which an option's model value changes as calendar time moves forward, everything else unchanged. Under Black-Scholes without dividends, the annual theta of a call is −S·φ(d1)·σ/(2√τ) − r·K·e^(−rτ)·N(d2), and that of a put is −S·φ(d1)·σ/(2√τ) + r·K·e^(−rτ)·N(−d2), where τ is time to expiry in years, φ the normal density and N the cumulative normal. Dividing by 365 gives a per-calendar-day figure; some desks divide by 252 trading days instead, so two platforms can quote different thetas for the same contract.
The first term, driven by volatility and 1/√τ, dominates for short-dated options. The rate term is small for short maturities and changes sign between calls and puts.
Worked example: decay speeds up
Take an at-the-money call with spot and strike at 100, implied volatility 20%, zero rate and dividend. The table shows the Black-Scholes value and one-day theta at three horizons. The last column is the share of the remaining premium that one day of time removes.
| Days to expiry | Option value | Theta per day | Share of value lost in a day |
|---|---|---|---|
| 30 | 2.29 | −0.038 | 1.7% |
| 7 | 1.10 | −0.079 | 7.1% |
| 1 | 0.42 | −0.209 | 50% |
Theta is the price of gamma
For a delta-hedged option with zero rates, the Black-Scholes equation reduces to θ = −½·σ²·S²·Γ. In the 30-day example, gamma is 0.0695, so ½ × 0.20² × 100² × 0.0695 = 13.9 per year, which is exactly the annual theta of −13.9. A long option holder pays theta to own convexity; a short option holder collects it and carries the gamma risk. This is why a dealer book described as short gamma in a GEX model is also, by construction, a book that collects time decay.
- Theta is a model output: it assumes implied volatility stays constant while a day passes.
- Whether weekends count is a convention. Calendar-day theta charges them, trading-day theta does not.
- Out-of-the-money options lose value in absolute terms more slowly, but a larger share of their price.
In Senzoukria
The GEX overview shows a Theta Decay tile in dollars per day. It sums open interest × theta × 100 over every call and put in the loaded chain, from the holder's side, without the dealer sign used for gamma, so it reads as the daily decay of the open-interest book. When the provider publishes no theta, or no leg carries both theta and open interest, the tile says the field is not in the chain instead of showing zero. In Option Flow, hovering the delta cell of a print shows that contract's gamma and theta from the latest chain snapshot. On the Databento path, where the app computes greeks itself, theta is expressed per calendar day.
Related
In the same section
- Extrinsic value
- Gamma scalping
- Thin book
- Testable hypothesis
- Tick bars
- Technical indicator
- Tick data
- Tape speed
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Frequently asked questions
- Why does theta accelerate near expiry?
- Because the volatility term of theta scales with 1/√τ. For an at-the-money option the remaining time value shrinks like √τ, so each day removes a growing share of it. In the example above, one day removes 1.7% of the value at 30 days and half of it on the last day.
- Is a positive theta position safe?
- No. Positive theta comes with negative gamma: the position earns time decay while price is quiet and loses when price moves more than implied volatility priced. Theta and gamma are two sides of the same trade, not a free income stream.
- Does theta matter to a futures trader who does not trade options?
- Indirectly. Aggregate theta and gamma shape how option books are hedged, and 0DTE decay concentrates gamma near the money during the last hours of a session. Those are context variables for reading the futures tape, not signals.