# 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.

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.

• So vol and time value are directly proportional then? Commented May 23 at 18:58
• The value of a straddle is roughly 0.8x f x volx sqrt(t). So yes. Same for otm (diff formula). Commented May 24 at 17:37

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.

• just being a bit pedantic here but intrinsic value depends on discounting => so does depend on time to expiry Commented May 25 at 8:45
• Well yeah but let say rates are 0 🫡 Commented May 25 at 19:34
• They aren't are they? Commented May 25 at 20:37
• This was an informal explanation… if you set the rates to 0 this is correct and even without it discount factor is function of time but the rates still close to 0… we were asked for informal explanation and this detail has no importance to answer the question. Commented May 25 at 21:16