Lesson 15 of 20 · 7 min

Why decay is not linear

Decay follows the square root of time in reverse, so most of it happens at the end.

If an option lost the same amount every day, a 30-day contract worth $3.02 would shed about ten cents a day. It does not behave that way at all, and the reason is already in lesson 5.

Value scales with the square root of time remaining. Run that curve backwards and the shape is obvious: square-root curves are flat where the input is large and vertical where it approaches zero. So an at-the-money option loses value slowly early and violently late.

The base-case numbers make it concrete. From 60 days to 30 days, the call falls from $4.36 to $3.02 — $1.34 over thirty days, about 4.5 cents a day. From 7 days to expiry it falls from $1.42 to zero — $1.42 over seven days, about 20 cents a day. The final week decays at more than four times the rate of that earlier month, and the last day alone accounts for $0.53, or 37 percent of what was left a week before.

There is an important exception, and it is the one people get wrong. This acceleration applies to at-the-money contracts. An out-of-the-money contract behaves differently: its theta rises for a while and then collapses toward zero, because once there is no realistic chance of reaching the strike there is no value left to lose. In the base case the 110-strike call has theta −$0.026 at 60 days, −$0.023 at 30 days, and −$0.003 at 7 days, when the contract itself is worth less than a cent.

At the money, decay accelerates into expiry. Out of the money, decay finishes early and then there is nothing left to decay.

Key terms

Non-linear decay: The property that time value is lost unevenly, slowly at first and rapidly near expiry, following square-root-of-time scaling.
Theta acceleration: The increase in the daily decay rate as expiry approaches. For at-the-money contracts it grows without limit.
Terminal collapse: The out-of-the-money case, where theta rises then falls to nearly zero because almost no value remains to be lost.

Worked example

The same contract, four rates of decay

Base case at the money, then the 110 strike for contrast. Underlying 100, rate 4%, volatility 25%.

ATM, 60 → 30 days: $4.36 → $3.02
ATM, 30 → 14 days: $3.02 → $2.03
ATM, 7 days → expiry: $1.42 → $0.00
ATM final day alone: $0.53
110 strike, theta at 60 days: −$0.026
110 strike, theta at 7 days: −$0.003

The at-the-money contract decays four times faster in its final week than a month earlier. The out-of-the-money contract had already finished decaying.

Check yourself

Why does an at-the-money option lose about 4.5 cents a day with 60 days left and about 20 cents a day in its final week?

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Educational only. Nothing in this lesson is investment, legal or tax advice, or a recommendation to enter into any transaction. Figures are illustrative and computed from the inputs stated in the lesson.