Single-Stock Futures Part 6

How Futures Contracts are Priced

Previously, we learned the basics of futures contracts, how to calculate initial margin requirements, and how marking to market works. These calculations are all based on the futures price. Now it's time to turn our attention to how futures contracts get those prices in the first place.

While other types of derivatives, such as options, involve complex pricing formulas, futures contracts are relatively easy to price. This is because futures are an agreement to take delivery at a future date. Remember, they are not the option to take delivery. Because we know today the price we've agreed to trade in the future, the value of that contract is nothing more than the current price plus the cost of carry, assuming the underlying asset provides no income (such as dividends) during the year. The current price is sometimes called the spot price since that is the price at which the asset can be purchased "on the spot."

What is the cost of carry? That just refers to the interest you could have earned on your money had you not purchased the underlying asset for delivery, which is also called a financing cost. For some commodities futures contracts, the cost of carry may be a little more difficult to determine. For example, a farmer may have to store grain for several months and then deliver it by rail. In addition, they may have to buy insurance to guard against fire or water damage. In fact, it is not uncommon for regional differences in storage costs to cause different futures prices on different exchanges.

However, for financial assets, there are virtually no storage costs (the assets are held electronically) and very low "shipping" costs (they are sent through the Fed wire system) so the cost of carry is usually considered to be only the foregone interest.

For example, assume interest rates are 5%. If the underlying asset is currently trading for $100 and we enter an agreement to trade it for $100 in one year, we could replicate that futures contract by purchasing the asset for $100 today and then holding onto it and delivering it for $100 in a year. In doing so, we would miss out on the interest we could have earned on the $100 we had to spend on the asset today. That amount would be $100 * 5% * 1 year = $5, so the futures contract should be worth $105. Remember, a futures contract is an obligation to buy or sell. If we buy the spot asset for $100 and sell a one-year futures contract at $105, we have guaranteed the sale of the underlying asset at $105, which is simply the initial cost of the underlying plus interest.

If the futures contract is not trading for $105, arbitrageurswill take advantage of the pricing discrepancy for a risk-free profit. For instance, assume the futures contract is priced at $106. We know it "should be" trading for $105 but we see it is not. The futures are overpriced in relation to the spot price. In this case, arbitrageurs would borrow $100 and owe $105 in one year. They will take the borrowed $100 and buy the underlying asset in the spot market and hold onto it for one year. At expiration, they will deliver the asset for $106 and then repay the $105 loan, leaving them with a guaranteed $1 for no money at risk -- an arbitrage profit. This is known as cash-and-carry arbitrage-- the trader buys the spot asset (also called the cash market) and carries it to expiration.

Cash-and-carry arbitrage tends to raise the spot price and lower the futures price until the arbitrage opportunity disappears. In a previous course we said that arbitrageurs played an important role in the efficient functioning of the futures markets. Now you see why. Without an arbitrageur in this example, market participants would be forced to pay too high a price at $106. The ability to generate risk-free profits gives arbitrageurs great incentive to look for -- and correct -- those opportunities.

Instead, what if the futures were trading below $105, say $104? Now the futures contracts are too cheap compared to the spot price. Arbitrageurs would buy the underpriced futures at $104 and short the spot asset for $100. There is no cost to enter the futures contract so the arbitrageur just puts the $100 credit in a risk-free interest account. ¹

Because the arbitrageur shorted the underlying asset, he now has the obligation to buy it back at some point in time, and there are no restrictions on the length of time. The arbitrageur is not at risk as a normal short-seller, since he bought the futures contract and is guaranteed a purchase price of $104 in one year. The $100 credit grows to a value of $105 in one year, and he takes that money to buy the underlying asset for $104, thus closing out the short position and leaving him an arbitrage profit of $1. Whenever the futures price is too low, arbitrageurs use reverse cash-and-carry. Reverse cash-and-carry tends to lower the spot price and raise the futures price until the arbitrage opportunity disappears.

Key Points

• If the futures contract is priced above fair value, arbitrageurs will perform cash-and-carry arbitrage (Buy the asset, sell the futures)
• If the futures contract is priced below fair value, arbitrageurs will perform reverse-cash-and-carry arbitrage (Buy the futures, sell the asset)
• These two types of arbitrage keep the price of the futures contract very close to the theoretical value

Notice how these two examples used one dollar above fair value and one dollar below fair value. We calculated fair value to be $105 and assumed futures prices of $106 and $104 respectively. As a result, the arbitrage profits were $1 in both cases. The arbitrage profit is equal to the amount the futures contract is mispriced above or below fair value.

So whether the futures price rises or falls below fair value, arbitrageurs will make a guaranteed profit. We can state fair value for assets that pay no dividends as a simple formula:

Futures fair valueprice = Spot price * (1+interest rate) ^time

where time is expressed in years (or fractions of a year) and the interest rate is expressed as a decimal.

In our example, the spot price was $100, interest rates were 5% (which is .05 expressed as a decimal) and the holding period was one year. Using the above formula, the futures price should therefore be $100 * (1.05)¹ = $105, which is what we determined earlier.

If we assume the above contract is for a fraction of a year, such as six months, we just use the corresponding fraction as the exponent (the raised number) and the futures contract should be trading for:

Futures price = Spot price * (1.05)^1/2 = $1.025

Because we only need to hold the spot asset for 6 out of 12 months (half a year) we simply raise 1.05 to the 6/12, or 1/2 power.

When the futures contract is trading for exactly the cost of carry, it is said to be trading at fair value.

Alternate Formula
There is another formula you can use that is especially nice if you do not have an exponent key on your calculator. Rather than raising (1+interest rate) to the number of years, we can simply multiply the interest rate by that same number as follows:

Fair Value = Spot price * (1+ (interest rate * time))

In our first example, we found that a one-year contract at 5% was worth $105, assuming the current price was $100. Using the alternate formula, we could come up with an answer of $100 * (1+ (.05 *1)) = $105. In this case, we get the exact answer but that will not always be.

For instance, what if we were pricing a nine-month contract? We can see that the two formulas produce slightly different results:

First formula:

$100 * (1.05)^3/4 = 103.73

Alternate formula:

$100 * (1+(.05 * 3/4)) = 103.75

Note: Make sure you multiply the numbers in the parentheses first then add the 1. In this example, you would multiply .05 * 3/4 and then add 1 to that answer. Once you get that answer, then multiply by $100. If you don't do it in that order, you will get the wrong answer!

In most cases, the differences are not worth worrying about. The small variations are due to the assumptions in compounding of interest. I only mention this if you are trying to follow along with some of the exercises and using a calculator without an exponent key. The alternate formula will get you very close to the same answers.

The following table summarizes the two strategies:

Assumptions: Spot price = $100 Fair Value = $105

If futures are $106: Futures too high, so buy spot and sell futures -- cash-and-carry arbitrage: Cash and carry arbitrage procedure: 1) Borrow $100 and owe $105 in one year. 2) Buy asset for $100 with borrowed funds and sell futures for $106. 3) Wait one year and deliver asset for $106. 4) Repay loan and profit by $106 - $105 = $1. Effect: Raises the price of the underlying stock or index and lowers the price of the futures contract.

If futures are $104: Futures too low, so buy futures and sell spot -- reverse cash and carry arbitrage: Arbitrage procedure: 1) Short spot and receive $100 credit with obligation to buy back short position at later time. 2) Invest $100 in risk-free account and buy futures at $104. 3) Wait one year and replace borrowed asset by taking delivery of futures for arbitrage profit of $105 - $104 = $1. Effect: Raises the price of the futures contract and lowers the price of the underlying stock or index.

The above argument holds true for any futures price above or below fair value. We can therefore say that the only futures price where no arbitrage can occur is at $105 -- the fair value. Because arbitrage is a guaranteed profit for no cash outlay, you can be assured that arbitrageurs will keep the actual value of the futures contract very close to its theoretical fair value.

We have been assuming equal borrowing and lending rates in order to make these arbitrage opportunities work. For retail investors, this is rarely the case but it is often possible for many of the large brokerage and futures firms, which are precisely the ones carrying out the arbitrages. So while you may not be able to participate in the arbitrage profits, it doesn't mean you shouldn't take the time to understand the process. This process is important because it keeps prices fair for all of us.

Probably the most important reason for understanding fair value is that you will often hear financial stations such as CNBC talking about it prior to market open. Many investors place trades in the pre-market based on those indications and you will likely hear all kinds of myths and half-truths about what it really means. If you don't understand fair value, you can easily be convinced to enter trades based on the indications that are exactly opposite of what you should be doing. We'll look at how to use fair value in the next course, but first let's make sure you understand the process and the formula we discussed earlier.

Examples

1) Assume that the underlying asset is currently trading for $80 and interest rates are 3%. What is the fair value of a six-month futures contract?

All we need to do is use the formula and plug in the numbers as follows:

Fair value = $80 * (1.03) ^1/2 = $81.19

Therefore, the fair value of the contract is $81.19.

If the price in the real world were higher than this, arbitrageurs would use cash-and-carry arbitrage. If it were lower, they would use reverse-cash-and-carry arbitrage.

2)Assume that the underlying asset is currently trading for $140 and interest rates are 7%. What is the fair value of a 1.5-year futures contract?

$140 * (1.07) ^1.5 = $154.95

Using the alternate formula:

$140 * (1+ (.07 * 1.5)) = $154.70

The fair value is $154.95

Assume the futures price is quoting $156. How would you perform the arbitrage?

The price is too high, so we need to buy the spot (because it's cheap) and sell the futures (because they're expensive). This is cash-and-carry arbitrage.

However, arbitrage requires that no out-of-pocket money be used, so we must borrow $140 and we will owe $140 * 1.07^1.5 = 154.95 at expiration.

We will buy the spot asset with the borrowed money and let the asset sit in our account. We will short the futures contract at $156, thus giving us the obligation to deliver the underlying asset for $156. Because the asset is sitting in the account, we know we will be able to deliver it at expiration no matter what happens to the price.

At expiration, we will receive $156 and owe $154.95 and capture an arbitrage profit of $1.05.

Assume the futures price is quoting $153. How would you perform the arbitrage?

Now the futures price is too low since the fair value is $154.95 yet it is only trading for $153. We will buy the futures and sell the spot asset, which is reverse-cash-and-carry.

When we sell the spot asset, we will receive a credit of $140 and leave that in our account to earn interest. We will also buy the futures contract for $153. This ensures that we will be able to buy the asset for no more than $153 at expiration and guarantees our profit.

At expiration, the $140 credit has grown to a value of $154.95. We now take delivery of the asset through the futures contract and pay $153. This nets a guaranteed profit of $1.95

Thought Questions: In this last example, we assumed that we took delivery of the underlying asset and paid $153 to net a profit of $1.95. If the underlying asset were actually trading for $145 at expiration, would we be better off by purchasing the underlying asset in the open market rather than taking delivery through the futures contract and paying $153?

¹The initial margin deposit is generally not considered a cost. That's because it is not a down payment but rather a good-faith deposit.

Futures Price Will Converge to Spot Price

You may remember from previous course sections that we said the futures price and spot (current) price would be equal at expiration of the contract. We offered a pragmatic argument as to why that would happen. That is, if the futures contract guarantees delivery of the asset at some point in the future and that future date is today, shouldn't the prices be the same?

Now it's time to find out why the two prices are forced to converge and cannot go off in separate directions. If the futures contract will is not trading for (or at least very close to) the spot price, then -- you guessed it -- arbitrage is possible.

For example, in the last course, we gave an example of an asset with a current price of $100 with interest rates at 5%. Under those conditions, we showed that a one-year futures contract would be priced at $105. If the futures contract were priced above or below $105, then arbitrage would be possible.

With time remaining on the contract, a price difference can exist, as in this example, between the current spot price and the futures price. However, that difference cannot be too great or too small, otherwise arbitrage is possible. The difference in prices is solely due to time. So if there is no time remaining on the contract, there should be no difference in prices. If there is any difference, we take the same steps as in the last course to perform the arbitrage. The only difference is that the steps will be performed immediately rather than having to wait until expiration.

For example, let's say the spot price is $100 and the futures contract is priced at $101 at expiration. An arbitrageur could borrow money, buy the spot asset, and sell the futures contract and thus guarantee the sale at $101 the next day. Because the money is only borrowed for one day, very little interest is due and the arbitrageur makes a profit. Once again, these actions put buying pressure on the spot and selling pressure on the futures contract; this brings the spread closer together. Now, it's true that some interest may be due and, if so, that will be reflected in the price difference between the spot and futures. The point is that the difference will be very small, and that is why we say the futures price will converge to the spot price as expiration approaches.

In the last course, we found that the pricing of futures contracts is based on the following formula:

Futures fair valueprice = Spot price * (1+interest rate) ^time

Using the same numbers above, a 5% interest rate, and zero time remaining on the contract, we can see that the two prices will be exactly the same:

Fair value = $100 * (1+ .05) ⁰ = $100

Mathematically, anything raised to the zero power equals one, so with no time remaining the two prices are exactly equal. Of course, once the contract is expired you cannot trade it, so the assumption of zero time is a little unrealistic. What if the time is one day?

Fair value = $100 * (1+ .05) ^1/365 = $100.01

Again, we see that the futures fair value prices are very, very close to the current spot price at expiration and this is what you will observe in the real world of trading.

The only way to prevent arbitrage is for the futures contract price to converge to the spot price at expiration. After all, this is the concept of a futures contract -- to determine the spot price in the future. If that future point (expiration) has arrived, obviously the spot price is the same as futures price. The following figure demonstrates how the individual futures and spot markets are allowed to fluctuate somewhat independently but must meet at expiration:

Why is this important to know? It's important because many people new to futures are afraid of the following scenario. Say someone buys a contract at $100 and the underlying is trading for $110 at expiration, so it appears he should make $10 on the trade. His fear is that the exchanges will post haphazard quotes and charge anything they feel, thus reducing their profit or even turning it into a loser. This arbitrage argument shows that's not true. Remember that we said in the beginning that arbitrageurs perform important economic functions in making prices fair. Now you've seen just how important they are! Arbitrageurs ensure that the futures price will converge to the spot price.

It's also important to know since, again, there's no mathematical difference between taking delivery of the underlying asset through the futures contract as compared to just closing it in the open market. The reason is that the two prices -- spot and futures -- must be the same at expiration.

In the next course, we're going to take a closer look at these arbitrage arguments and see how they can help you predict the direction of the stock market at the opening bell.

Key Points

• Arbitrage forces the futures price to be trading very close to the spot price at expiration
• Because the two prices are close to the same, it doesn't make a financial difference if you take delivery of the underlying asset or just close out the contract in the open marke

Thought Questions: The spot asset is $100 and the futures contract is $99 on expiration day. How would you perform the arbitrage?