Lesson 5 of 20 · 6 min

Input three: time to expiry

More time is worth more money, but the value grows with the square root of time, not with time.

Time is the input with the most counter-intuitive shape. Everybody expects more time to be worth more. Few people expect the relationship to be as flat as it is.

In the base case, the at-the-money call is worth $1.42 with 7 days left, $3.02 with 30 days, $5.43 with 90 days and $11.84 with a full year. Look at the ratios. Ninety days is nearly thirteen times as long as seven days, but the premium is only 3.8 times larger. A year is 52 times as long, for 8.3 times the premium.

The reason is that a price's expected wandering grows with the square root of elapsed time, not with elapsed time. Four times the days gives roughly twice the expected movement. Since the option's value tracks the amount of expected movement, it grows on the same square-root schedule. Check it: the square root of 13 is 3.6, close to the 3.8 observed above.

Two consequences follow directly. Buying twice the time does not buy twice the exposure to movement, so a longer contract is proportionally cheaper per day. And running the schedule backwards, a contract loses value slowly at first and quickly at the end — the flat part of a square-root curve is at the far end and the steep part is near zero. That is the shape of theta, and lesson 15 is entirely about it.

Key terms

Time to expiry (T): Time remaining, expressed as a fraction of a year. 30 days is 30/365, or 0.0822.
Square-root-of-time scaling: Expected price movement grows with the square root of elapsed time. Four times the time gives about twice the movement.
Cost per day: Premium divided by days remaining. Falls as expiry lengthens, which is the square-root relationship stated another way.

Worked example

The square root of time, measured

Base case: underlying 100, strike 100, rate 4%, volatility 25%. Only the days change.

1 day: $0.53
7 days: $1.42
30 days: $3.02
90 days: $5.43
365 days: $11.84
Predicted 7 → 90 day ratio: 3.6×

Fifty-two times the days cost only eight times the premium. The per-day cost falls by a factor of seventeen across the range.

Check yourself

An at-the-money call with 30 days left is worth $3.02 in the base case. Roughly what would you expect the 120-day contract to be worth, and why not four times as much?

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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.