Option theta

Theta measures how an option's theoretical value changes locally as time passes, holding the underlying, volatility, rates and other inputs fixed. The definition must identify the time variable:

or, if τ is time remaining to expiry:

Because τ falls when calendar time advances, the two derivatives have opposite signs under otherwise matching conventions. Many systems call the expected one-day change theta and show a negative number for a long vanilla option. Without vendor documentation, sign cannot be interpreted safely.

Units: per day, year or effective interval

Theta can be expressed:

• per calendar day;
• per trading day;
• on an annual basis;
• per option, contract or full position;
• in premium currency or after FX conversion.

An annualised theta of −7.30 does not automatically equal −0.02 per day: the divisor may be 365, 252 or a contract-specific convention, and time decay is nonlinear. Weekends, holidays, valuation time and day fractions can be handled differently. A report should state scale, calendar and timestamp.

At position level, quantity, multiplier and sign apply:

This aggregates an instantaneous sensitivity. It does not promise the same amount on every future day.

Sign and position profile

A long vanilla call or put often has negative calendar-time theta: with other inputs fixed, less time remains for a favourable payoff. A short position in the same option reverses the sign. “Theta is always negative” is nevertheless too broad:

• sign depends on the selected time variable;
• spreads combine long and short legs;
• rates, dividends and moneyness can generate special cases;
• exotics and conditional payoffs require their actual model;
• theoretical-value theta is not realised position P&L.

A positive-theta position can still lose much more through an underlying move, negative gamma or a volatility change. Collecting time decay means carrying other exposures, not earning a risk-free return.

Decay is not linear

For standard vanilla options, the time profile varies with moneyness and expiry. Near-at-the-money time value can decline more rapidly as expiry approaches, while deep ITM or OTM options may behave differently. Projecting theta × number of days assumes constant theta and is only a local approximation.

Gamma and vega change during the same period. A position can lose vega and acquire concentrated gamma near expiry. The passing of a scheduled event can reduce implied volatility as well: attributing the entire premium move to theta would be wrong.

Theta in P&L explanation

A local decomposition may be written:

The theta term is meaningful only when Δt uses the same unit as theta. The residual contains cross-terms, higher-order sensitivities, surface changes, model error, spread and gaps between theoretical and executable price. Over longer intervals or active markets, the portfolio should be revalued with final inputs and the result compared with the decomposition.

Systems may advance the date while rebuilding curves, forwards and dividends, or hold some inputs fixed. Two implementations can produce different numbers while remaining internally consistent with different conventions. Model and bump procedure belong to the metric.

Expiry, exercise and operational risk

Theta alone does not describe the expiry outcome. Automatic exercise, assignment, settlement, broker cut-offs and deliverables can turn an option into cash, underlying assets or futures. A low-premium position can create a much larger exposure after exercise or assignment. Time control therefore includes the contractual and operational calendar, not only the Greek.

A weekend does not necessarily credit three times Friday theta. Expected time information may already be embedded in prices and the model follows its own calendar convention. Only coherent before-and-after snapshots support an auditable attribution.

Aggregation

Portfolio theta should be aggregated only after aligning time unit, currency, multiplier, position sign and valuation timestamp. A net figure can mask large long and short gross time sensitivities across expiries. It should be retained by expiry bucket and read with gamma and vega scenarios.

Theta-neutral does not mean expiry-neutral. Different legs may exercise at different times, settle differently or respond to holiday calendars. These contractual differences remain even when a model snapshot nets to zero.

Checklist

1. Is theta defined against calendar time or time remaining?
2. Is it daily, annualised or tied to another interval?
3. Does it use calendar or trading days?
4. Are quantity, multiplier, currency and position sign included?
5. Which inputs remain fixed in the time bump?
6. How do gamma and vega change near expiry?
7. Which exercise, assignment and settlement rules become material?

Sources

• CME Group, Options Theta — The Greeks.
• Cboe Options Institute, Learning the Greeks: An Expert's Perspective.
• Cboe, Options Calculator — model- and input-specific theoretical outputs.
• The Options Clearing Corporation, Characteristics and Risks of Standardized Options.
• Fischer Black and Myron Scholes, The Pricing of Options and Corporate Liabilities, 1973.

Links

• Options and volatility
• Risk allocation and sensitivities
• Options
• Option delta
• Option gamma
• Option vega
• Risk measurement and control