Why Out-of-the-Money Option Prices Do Not Follow Square-Root Time Scaling
Summary
The document examines why a set of out-of-the-money FXI puts appears to become much more expensive with added time to expiration than a simple square-root-of-time rule would predict. The quoted example lists ask prices and implied volatilities across several expirations, with longer-dated puts showing lower implied volatility but a higher price per day.
The answer explains that square-root scaling is only a rough guide near at-the-money strikes. In Black–Scholes, time interacts with the spot-to-strike relationship and with rates and volatility, so an out-of-the-money option’s price need not track the square root of time. The example motivates the question, but the document does not calculate model prices or establish that the observed quotes are mispriced. Its main caveat is that there is no simple general rule mapping option price to expiration alone.
Key ideas
- Square-root-of-time scaling is a rough approximation for options near the money.
- Out-of-the-money option prices depend on the relationship between spot and strike as well as time.
- Interest rates and implied volatility also affect how option value changes with expiration.
- The FXI quotes illustrate a deviation from the simple heuristic but do not by themselves show mispricing.
Tags
Full text
# Why might these options price so far from the square-root of duration?
# 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?
## Answer by feetwet (score 1, accepted)
https://quant.stackexchange.com/a/49020
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!Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.