Option Theta

Option theta is the estimated change in an option's price, per share, for one day's passage of time, with the stock price, implied volatility and every other input held constant. FINRA defines it as "the rate of change in an option's theoretical value for every one-day change in the time remaining until expiration, holding all else constant," and the Options Industry Council says it "represents, in theory, how much an option's premium may decay per day with all other pricing factors remaining the same." Theta measures the speed of time decay, the erosion of an option's time value, also called extrinsic value, as expiration approaches. It is a rate, not the time value itself. The word is commonly used more loosely for the amount of time value an option still has: FINRA's options page, for example, says "Theta is also known as a contract's time value," while defining theta on the same page as a rate per day.

The sign. Time works against anyone who owns an option and for anyone who has sold one. The council's page on buying calls notes that "an option typically loses time value premium with every passing day," its theta page says that as expiration approaches "it is likely Theta would become increasingly negative," and its pages on writing uncovered calls and puts state that "the passage of time will have an extremely positive impact" on those positions. So a bought call or put carries negative theta and a sold one carries positive theta. Some sources print the daily figure without a minus sign, as the council itself does in one example, so the safe reading is to ask which side of the trade the number describes. This glossary prints a bought option's theta as a negative number throughout.

Why the daily figure changes. Theta is not constant over an option's life. The council explains that "the theoretical rate of decay will tend to increase as time to expiration decreases," so for an option at or near the money the daily loss is gradual at first and larger in the final weeks. That pattern is not universal. The council adds that "at-the-money options will have the most exposure to time decay," while options "that are either deep-in-the-money or far out-of-the-money will have very little decay as they have less time premium." A far out-of-the-money contract has little time value left to lose, so it does not show the same late acceleration. Implied volatility matters too: "In general, the higher the implied volatility levels, the higher the Theta amount," because there is more time value to erode.

Weekends and models. Options are priced in calendar time, but markets trade only on business days. The council notes that "pricing models take into account weekends, so options will tend to decay seven days over the course of five trading days," and that "there is no industry-wide method for decaying options so different models show the impact of time decay differently." That is the usual reason two brokers show different thetas for the same contract.

Reading theta with the other Greeks. Theta assumes nothing else changes, and something always does. The council's theta page gives the case of a day in which an option's price does not move at all: "then one of the other variables must have changed," most likely, it says, an increase in implied volatility. A position's daily result is the combined effect of theta, delta's response to the stock and vega's response to implied volatility.

Theta is quoted per share, for one day. Multiply it by the number of shares the contracts cover to get the expected daily change for a position, keeping in mind that the figure is recalculated every day and tends to change as expiration approaches.

For example, using illustrative figures: a stock trades at $63, and a call with a $65 strike and 30 days to expiration is quoted at $1.55 with a theta of −0.04.

• If one day passes and nothing else changes, the call should be worth about $1.55 − $0.04 = $1.51, a loss of about $4 on one 100-share contract.
• Someone holding three contracts should expect to lose about 3 × $4 = $12 a day from time passing alone.
• Someone who sold one contract has a theta of +0.04 on it, so the same day adds about $4 to the writer's position.
• Suppose the stock is still at $63 with seven days left. The call might now be worth about $0.41, with a theta of about −0.065, or about $6.50 a day per contract. The daily loss has grown even though the option is worth far less, because what time value remains has fewer days left to run off.

None of this is a forecast. If implied volatility rises or the stock moves, the option's price can rise on a day theta says it should fall, and two pricing models can show different thetas for the same contract.

Pros

• Puts a daily dollar figure on the cost of holding an option while waiting.
• Shows a writer how much a position should gain each day if nothing else changes.
• Makes clear why a short-dated option near the money can lose value quickly even when the stock barely moves.

Cons

• It assumes the stock price and implied volatility stay fixed, which they rarely do for a whole day.
• There is no single industry method for spreading decay across weekends, so figures differ between platforms.
• The sign convention varies between sources, which makes it easy to misread a bought option's daily loss as a gain.
• It is a rate for today only; for options near the money it usually grows as expiration approaches.