2
$\begingroup$

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?

$\endgroup$
2
  • $\begingroup$ What are the strikes of these options? $\endgroup$
    – will
    Aug 29 '19 at 19:05
  • $\begingroup$ @will: 35 (presently 10% OTM) $\endgroup$
    – feetwet
    Aug 29 '19 at 19:49
1
$\begingroup$

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!

$\endgroup$
2
  • $\begingroup$ Should it be forward = strike rather than spot = strike? $\endgroup$ Oct 2 '19 at 17:58
  • $\begingroup$ @CharlesFox: In this case I don't believe so because the underlying is a stock, so the only difference between the "spot" and any "forward" prices should be accounted in the risk-free rate, which is included in this formulation of Black-Scholes. $\endgroup$
    – feetwet
    Oct 2 '19 at 19:14

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.