Theta

Definition

Theta is the rate of change of an option’s value with respect to the passage of time — time decay. Quoted per day, it is what an option position gains or loses overnight with everything else unchanged. Long options generally have negative theta (they lose time value); short options collect it. Theta is the price of gamma: the same position cannot be long movement and long time.

How it works / structure

• Formula (Black-Scholes-Merton): theta combines a volatility term (−S·φ(d1)·σ / (2√T), always negative for long positions) and rate/dividend carry terms; per-day theta divides the annual figure by the day-count convention.
• Shape over time: at-the-money time value decays faster as expiry approaches (the decay curve steepens); in- and out-of-the-money decay profiles are flatter and can differ in direction for deep-in-the-money puts/calls with rates and dividends.
• Extrinsic value accounting: theta consumes extrinsic (time) value only; intrinsic value does not decay. The platform’s dividend_vs_extrinsic concept compares remaining extrinsic to a pending dividend for early-assignment risk (ms-expiration-exercise-assignment).
• Position aggregation: net theta sums across legs; income structures (strategy-iron-condor, strategy-covered-call) are net-positive theta by construction.
• Simulation parameters: theta emerges from repricing at each step rather than being charged separately — the engine reprices; theta is a diagnostic, not an input.

When it applies

Every options position, every day: expiry selection (short-dated positions concentrate decay), income-strategy design (net theta is the engine of short-premium strategies), and hold/exit decisions — mgmt-hold-to-expiry vs mgmt-profit-target is largely a question of how much remaining theta is worth the remaining gamma risk.

Risk profile & failure modes

• Theta is compensation, not income: collected decay is payment for bearing negative gamma and tail exposure; counting it as yield without the risk side is the canonical short-premium error (labeled as such in every income-strategy entry).
• Weekend/holiday conventions: models spread or lump non-trading -day decay differently; short-dated P&L attribution can mislead by a day.
• IV interaction: a “theta gain” can be erased by a small IV rise (greek-vega); decay is only realized if volatility and price cooperate.
• Deep-ITM sign surprises: with rates/dividends, some deep in-the-money options have positive theta for the holder — assuming “long always decays” mis-models edge cases.

Evidence & limits

Theta’s mathematics are model-defined (Black-Scholes-Merton). The empirical question — whether selling time value is systematically compensated beyond its risk — is the volatility risk premium, covered with citations in opt-implied-volatility and indicator-realized-vs-implied-vol; this entry makes no such claim.

Falsifiable-thesis examples

Illustrations only, not signals:

• “X’s at-the-money 30-day straddle will lose at least 40% of its extrinsic value in the next 15 calendar days if the stock stays within ±2% and IV within ±1 point” — falsified by the repriced chain under those conditions.
• “A net-positive-theta iron condor on Y will show positive P&L in replay over a month in which realized volatility stays below entry IV minus 3 points” — falsified by the replay result.

Cross-references

• The other side of the trade: greek-gamma
• Interactions: greek-vega, opt-implied-volatility, event-dividends-ex-dates (carry effects)
• Strategy consumers: strategy-covered-call, strategy-cash-secured-put, strategy-iron-condor, strategy-calendar-spread
• Management: mgmt-hold-to-expiry, mgmt-profit-target