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?

The Square-Root Pricing Rule

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.

Contango

In the chart below, the spot price is lower than the futures price which has generated an upward sloping forward curve. This market is in contango - the futures contracts are trading at a premium to the spot price. Physically delivered futures contracts may be in a contango because of fundamental factors like storage, financing (cost to carry) and insurance costs. The futures prices can change over time as market participants change their views of the future expected spot price; so the forward curve changes and may move from contango to backwardation.

Contango example

Let’s assume that the spot price of crude oil is £100, but the price of a crude oil futures contract is £110 for delivery in one month. A trader could buy this futures contract on the assumption that the price of oil will rise above £110 before the expiry date arrives.

A market that is currently in contango will experience gradual reductions in the futures price to meet the expected spot price at the delivery date of the contract. However, if the price of the futures contract remains above the spot price in contango, traders could take advantage of the discrepancy in price – this is known as arbitrage.

Backwardation

In the chart below, the spot price is higher than future prices and has generated a downward sloping forward, or inverted, curve which is in backwardation. The futures forward curve may become backwardated in physically-delivered contracts because there may be a benefit to owning the physical material, such as keeping a production process running. This is known as the convenience yield, which is an implied return on warehouse inventory. The convenience yield is inversely related to inventory levels. When warehouse stocks are high, the convenience yield is low and when stocks are low, the yield is high.

Backwardation example

Let’s assume that the spot price of natural gas is £1000, but the price of a natural gas futures contract is £900 for delivery in one month. A trader could buy this futures contract on the assumption that the price of natural gas will fall below £900 before the expiry date arrives.

A market that is currently in backwardation will experience gradual increases in the futures price to meet the expected spot price at the delivery date of the contract. However, if the price of the futures contract remains below the spot price in backwardation, there could be an arbitrage opportunity similar to a contango market.

Convergence

Over time, as the futures contract approaches maturity, the futures price will converge with the spot price, otherwise an arbitrage opportunity would exist.