# Why might these options price so far from the square-root of duration?

In general, to first order, option prices rise with the square root of duration (i.e., time-to-expiration).

I was just looking at puts on U.S. ETF FXI and they grossly violate this rule. With FXI trading at 39, the current ask on strike 35 puts (i.e., ~10% OTM) is as follows:

Days to Exp     Ask     Implied Vol     Sqrt(Duration)  $$/Day 22$$0.07  29%             4.7             $$0.015 50$$0.22  27%             7.1             $$0.031 78$$0.35  25%             8.8             $$0.040 113$$0.70  24%             10.6            \$0.066


This makes no sense: Even though the implied vol on the longer duration options is lower, the price per day increases vastly faster than $$\sqrt{Duration}$$. (In fact, it is roughly $$Duration^{1.5}$$!) What am I missing?

• What are the strikes of these options?
– will
Aug 29 '19 at 19:05
• @will: 35 (presently 10% OTM) Aug 29 '19 at 19:49

$$\frac{1}{\sigma\sqrt{T - t}}\left[\ln\left(\frac{S_t}{K}\right) + \left(r + \frac{\sigma^2}{2}\right)(T - t)\right]$$