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

Question

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?

Answer 1

The "square-root rule" for time-to-expiration only (roughly) applies when the spot price = strike price. Even in that case there is a second-order term that is a function of the risk-free rate and implied volatility.

This can be seen in the Black-Scholes pricing formula: the time-to-expiration is included in a term that also varies with log(spot/strike), and that is then transformed by the normal distribution:

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

where S is spot, K is strike, (T - t) is time-to-expiration.

So there's no easy rule or equation for option price as a function of time-to-expiration!