Square-Root-of-Time Scaling in Option Time Value

Summary

The document discusses the claim that an at-the-money option’s time value scales with the square root of time and asks whether the idea extends to in-the-money and out-of-the-money options. One answer connects the rule to volatility scaling: under a diffusion model with independent increments and constant volatility, return standard deviation grows with the square root of elapsed time because variance grows linearly with time.

The other response argues that the underlying’s strike does not determine this scaling, and separates intrinsic value from time value. However, its informal derivation is not a general option-pricing result: option value is a nonlinear function of the underlying distribution, strike, rates, dividends, and maturity. Square-root scaling applies to volatility under stated assumptions, not directly to every option’s time value. The document is therefore a useful prompt for distinguishing a volatility approximation from a universal rule for option prices; it does not establish that all moneyness categories follow the same time-value relationship.

Key ideas

• Under a constant-volatility diffusion, volatility over a horizon scales with the square root of time.
• The square-root rule describes volatility scaling and does not by itself prove the same scaling for option time value.
• Option value depends on strike and other pricing inputs, so moneyness can affect how value changes with maturity.
• Intrinsic value is distinct from time value and is determined by the payoff from immediate exercise.
• The responses are informal and do not derive a general formula for time value across option types.

Tags

# time value of option proportional to sqrt(time)


# time value of option proportional to sqrt(time)












I'm reading Natenberg's Options Pricing and Volatility, and in Chapter 18, he mentions this about an example:

> We can further refine our approximation if we note that an at-the-money option is made up entirely of time value and that the time value of an option is proportional to the square root of time.

Is the square root of time thing true for all options (ATM/ITM/OTM)? Would appreciate either a formal or informal explanation.

## Answer by user68819 (score 1, accepted)

https://quant.stackexchange.com/a/79482

Vol scales with the square root of time (i.e. variance is linear in time), therefore the value of an option diminishes with it too.

## Answer by JohnGalt (score 0)

https://quant.stackexchange.com/a/79486

You suppose that the underlying price follows N(0,T) for 2T the price will follow N(0,2T).

If you want you get

$\frac{S(T)}{\sqrt{T}}= N(0,1)$

$\frac{S(2T)}{\sqrt{2T}}= N(0,1)$

Remember that $N(0,sigma)= \sqrt{sigma}N(0,1)$

=> $\sqrt{2}S(T)= S(2T)$

Done.

There is no link between the underlying price diffusion and the option strike. Hence, this is true for ATM/OTM etc, it only depends on the hypothesis of your underlying price distribution.

Now, the value of an option can be seen as $Time Value + Intrinsic Value$.

the $Intrinsic Value$ is the value of an option if it was exercised today, nothing related to time, hence, the relationship I gave you between T and 2T will only impact the time value.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.