Theta: the daily cost of time

Theta is the Greek that works while you sleep. It is the amount an option loses per day from the passage of time alone, with the stock and volatility unchanged. Buyers pay it; sellers collect it. It is the only Greek whose direction is certain: tomorrow will come.

The definition

Theta is the change in option price for one day passing, all else equal. It is quoted as a negative number for long options (the position loses) and, by convention, positive for short options.

XYZ at $50, the 30-day $50 call trades at $1.90 with theta −0.045. Tomorrow, if XYZ is still $50 and volatility is unchanged, the call should trade at about $1.855. Per contract, that is $4.50 gone.

Note that theta is quoted per calendar day, not trading day. Options lose value over weekends, though the market partially prices that in by Friday afternoon.

A theta decay table

Here is the 30-day ATM $50 call tracked to expiration with XYZ pinned at $50 and volatility constant. Values follow the square-root-of-time shape from Module 2.

Read the last column. With 30 days left, a day costs under 2% of the option's value. With five days left, a day costs 10%. With two days left, a day costs a quarter of what is left. This is the same curve as Module 2, seen from the daily side: the dollar amount of theta increases as expiration approaches, and the percentage increases faster still.

 theta (per day, dollars)
 0.18 |                                        *
 0.15 |                                     *
 0.12 |                                   *
 0.09 |                               *
 0.06 |                        *   *
 0.03 |*      *      *    *
 0.00 +----------------------------------------
      30     21     14    10   7   5   3   2   1   days left

Theta by moneyness

Theta is largest at the money, because that is where extrinsic value is largest. Deep ITM and far OTM options have little extrinsic value to lose, so their theta is small in dollars. But as a percentage of the option's price, OTM theta is brutal.

The $56 call loses only 1.3 cents a day, but that is 5% of its value every day. Hold it for two weeks with no movement and more than half of it is gone.

Theta and gamma, again

Theta is the price of gamma. The ATM option that pays $0.032 a day is the one with gamma 0.08; the far-OTM option with tiny gamma also has tiny theta. If you buy an option, you are choosing to pay theta in exchange for gamma, and the trade only works if the stock moves enough, soon enough, for the gamma gains to exceed the theta paid. A useful way to state the daily hurdle:

A long ATM option needs the stock to move roughly (theta ÷ gamma × 2)^½ per day to break even on time.

For our 30-day ATM call: (2 × 0.032 ÷ 0.08)^½ ≈ 0.89. XYZ needs to move about $0.89 a day, or 1.8%, for the gamma to pay for the theta. Fewer moves than that and the option bleeds. That figure is, not coincidentally, close to the daily move implied by the option's volatility; Module 4 makes that link explicit.

Key idea: Theta is the daily rent on extrinsic value. It grows every day toward expiration, it is largest at the money in dollars and largest out of the money as a percentage, and it is the bill for owning gamma.

Theta on a position

Like the other Greeks, theta nets. A long call and a short call at different strikes partly cancel; a short ATM option and a long OTM option against it (a credit spread, Module 6) leaves a small positive theta with a capped loss. Many traders track their book's total theta as "how much I earn or pay per day if nothing happens", which is a useful number as long as you remember that "nothing happens" is the one scenario theta describes.

Two things theta does not tell you

Theta assumes the stock does not move and volatility does not change. In practice both happen every day, and their effects can swamp theta in either direction. A short option "earning" $30 a day in theta can lose $300 on a 3% move. A long option "paying" $30 a day can gain $300 on the same move. Theta is the drift; the Greeks in the other lessons are the noise, and the noise is usually bigger.

Theta also does not help you on a holiday weekend in the way sellers hope. Market makers lower implied volatility going into long weekends to reflect the dead days, so much of the three-day decay is already in Friday's price.

Try it: Take any option you own or are considering, note its theta, and multiply by the number of days you expect to hold it. Compare that dollar figure to the profit you expect if you are right. If the theta bill is more than a third of the target profit, the trade needs either a faster move or a different structure.

Recap

• Theta is the daily loss from time alone; long options pay it, short options collect it.
• Theta accelerates into expiration: under 2% of value per day at 30 days, 10% at 5 days, 50% on the last day.
• In dollars, theta is largest at the money; as a percentage, it is worst far out of the money.
• Theta is the cost of gamma; a long option needs a move of roughly (2 × theta ÷ gamma)^½ per day to break even on time.
• Theta describes the one scenario where the stock does not move; that scenario is rarer than sellers hope and buyers fear.