The Square Root (√) Pricing Rule
If you don’t understand square roots, you’ll make one of the most common mistakes of buying short-dated options and selling long-dated ones. Traders often buy short-dated options to “lower the risk,” and they sell long-dated ones to collect more money. It makes sense until you understand there’s a square-root pricing relationship that’s working against you. What is it? And how do you use it to make better decisions?
If a one-month option is trading for $1, it seems logical that a two-month option would be worth $2. Twice the amount of time, twice the price. It sounds right, but it’s wrong, and it’ll lead to all sorts of trouble with trading. So much for using logic in your decisions. Instead, it takes four times the amount of time to double an option’s price.
Option prices are always scaled by the square root of time. The reason is that option prices are priced according to volatility, and volatility is proportional to the square root of time. Who would have guessed?
If a one-month option is $1, a 4-month option isn’t going to cost 4x as much, or $4. Instead, you must take the square root of 4, which is 2, and that becomes the multiplier. A 4-month option would trade for 2x that of the one-month option, or $2.
It seems that the 4-month option is more expensive, and that’s why traders often avoid buying longer-dated contracts. But look at the price you’re paying per day. The one-month option costs $1/30, or 3.3 cents per day. The 4-month option, on the other hand, costs $2/120, or 1.67 cents per day—exactly half as much. Longer-dated options cost more in total, but they’re much cheaper per day. It’s like buying things in bulk. A case of Coke costs more at the register, but the cost per can is cheaper. If you’re trying to make the most of options trading, you can’t afford to buy expensive options. Don’t look at them in terms of total cost. Look at them as cost per day. Longer-dated options are cheaper.
On the other hand, traders are tempted to sell longer-dated contracts, thinking they receive more money. They may, for instance, sell the 4-month call for $2 as part of a covered call. Professional traders, on the other hand, would prefer to sell the one-month option four times, which nets $4 rather than $2—exactly twice the amount.
By not understanding the square-root pricing relationship, traders end up buying the most expensive options—and selling the cheapest.
This isn’t to say that traders should never sell longer-dated contracts or buy shorter-dated ones. A good decision doesn’t just rest on the total cost, but instead, other factors such as delta, gamma, or vega, among others. Depending on the profits you’re after, or the risks you’re trying to hedge, sometimes it makes sense to buy short-dated or sell long-dated options. However, you must understand all the influences in your decision, and if you believe short-dated contracts are cheaper, the rest of your decisions will be flawed.
Here’s another example: If a 3-month option is trading for $5, what’s the value of a 27-month option? The 27-month option has 9x the amount of time, but it’s not going to cost 9x the amount, or $45. Instead, you must take the square root of 9, which is 3. Therefore, the 27-month contract will cost 3x as much, or $15. Does this hold in the real world?
It’ll be close, but skews and tilts can make them vary from the expected a little bit. However, these open up new opportunities as well, provided you know how to capitalize on them. For instance, on March 18, 2020, the Apple 30-day at-the-money call was trading for $20. The 120-day call was $39, so it’s nearly double. If these relationships get too far out of line, traders can use strategies like a diagonal spread to put a mathematical edge on their side. It’s strategies like these that ensure the square-root pricing relationships will reasonably hold in the real world.
You don’t need to know your square roots to trade options. You just need to understand that they’re priced according to the square root of time. Shorter-dated options are more expensive, and longer-dated options are cheaper, and it’s all because of the square-root pricing rule. So while you were in school wondering if square roots were ever going to be useful, you can now use them to make money.