Time decay (theta), explained · Options 101

In this lesson

Explain theta, why decay accelerates near expiration for ATM options, and what happens to extrinsic value at expiration.

5
short parts: 12
questions: ~15
minutes

Read the whole lesson on one page

Expiration and time decay: the lesson

Time is part of the price

Take the 50 call on a $50 stock with 60 days left. It has no intrinsic value, yet it costs 2.42. You're paying for 60 days of chances.

Every day that passes takes some of those chances away. That steady leak is time decay, and its size is called theta.

Decay speeds up

Value of a 50 call on a flat $50 stock as expiration approaches, IV 30%. The line drops faster near the right edge.

Model values, not real quotes.

Step 1: From 60 to 30 days left, the call loses 0.70, about 0.023 a day.
Step 2: From 7 to 1 days left, it loses 0.52, about 0.087 a day.
Step 3: Same stock, no move at all, and the daily loss more than triples.

Why it speeds up

Think about how far the stock could wander. With 60 days left it has plenty of time. With 7 days left, much less. At the money, an option's value tracks roughly the square root of the time left.

Step 1: Start with 2.42 at 60 days (the square root of 60 is about 7.75).
Step 2: At 30 days: 2.42 × 5.48 ÷ 7.75 is about 1.7.
Step 3: At 7 days: 2.42 × 2.65 ÷ 7.75 is about 0.83.
Step 4: At 1 day: 2.42 × 1 ÷ 7.75 is about 0.31.

Not every option decays the same

Put the 50 call next to the 55 call on the same $50 stock. With 60 days left the 50 call costs 2.42 and the 55 call 0.79. With 14 days left they cost 1.17 and 0.07.

50 call (at the money)
55 call (out of the money)

A 50 call and a 55 call on a flat $50 stock, IV 30%. The at-the-money call starts higher and falls further in dollars. Both end at zero.

Model values, not real quotes.

Decay can only take away extrinsic value, and an option can't lose more than it has. The 55 call needs the stock to climb past 55, so it starts with little time value. The 50 call sits at the money, where the outcome is least certain, so it carries the most to lose.

Step 1: From 60 to 30 days left, the 50 call falls from 2.42 to 1.72. That's 0.70, or $70 per contract.
Step 2: The 55 call falls from 0.79 to 0.30. That's 0.49, or $49 per contract.
Step 3: As a share of its price, the 50 call lost about 29% and the 55 call about 62%.

Who decay helps

Buyer (long option)	Seller (short option)
Theta is negative: decay costs them	Theta is positive: decay pays them
Needs the stock to move enough to beat decay	Profits if the stock stays put
Risk: losing the premium	Risk: one big move can erase many days of decay

Worked example: A 50-strike call on a flat $50 stock (IV 30%, no rates or dividends)

Illustrative numbers, not real quotes.

Hold the stock at $50 and let the days pass. Here is the 50 call's value as time runs out:

Days left	Call value	Lost since the row above
60	2.42
30	1.72	0.70 over 30 days (about 0.023/day)
14	1.17	0.55 over 16 days (about 0.034/day)
7	0.83	0.34 over 7 days (about 0.049/day)
1	0.31	0.52 over 6 days (about 0.087/day)

Per contract, the last stretch costs about $8.70 a day vs about $2.30 a day two months out.

The loss per day more than triples, from about 0.023 to about 0.087, with the stock not moving at all.
All of it is extrinsic value. The call is at the money, so intrinsic value is 0 the whole way.

Now compare it with a far out-of-the-money 55 call on the same stock:

Days left	50 call (ATM)	55 call (OTM)
60	2.42	0.79
30	1.72	0.30
14	1.17	0.07
7	0.83	0.01

The ATM call has more to lose, so it loses more dollars a day.
The OTM call loses less in dollars, but over 30 days it still falls from 0.79 to 0.30, about 60% of its price.

Education only, not investment advice. Options involve risk. Read the OCC's Characteristics and Risks of Standardized Options. Example numbers are illustrative.