4. Futures and Forward Pricing#

We now turn our attention to understanding what should determine the futures price in equilibrium. And this is the first time where we will use one of the most fundamental ideas in modern finance which is the absence of arbitrage opportunities.

An arbitrage is a transaction that would cost nothing or even provide with income today and would potentially provide with even more income in the future. Getting money for free today even if we do not get anything later is a pretty good deal. In fact, the deal is so good that it would not last for long. This would be indeed inconsistent with an economic equilibrium, but the idea is actually more general. The principle of no-arbitrage does not depend on agents’ preferences, and as such it must hold in any economic system.

4.1. Non-Dividend Paying Assets#

We start by computing the no-arbitrage futures price of non-dividend paying asset.

Property 4.1 (Futures Price of a Non-Dividend Paying Asset)

If the spot price of a non-dividend paying asset is \(S,\) then the futures price for a contract deliverable in \(T\) years is given by:

(4.1)#\[\begin{equation} F = S e^{r T} \label{futures_price_no_dividends} \end{equation}\]

where \(r\) is the continuously compounded risk-free rate corresponding to the futures contract expiration.

We will see shortly that if the futures price is different form the one given in \(\eqref{futures_price_no_dividends},\) then there is an arbitrage opportunity that specialized traders would have no problem in exploiting at a massive scale. But first, let us see how to use this expression.

Example 4.1

Suppose that the spot price of a non-dividend-paying stock is $40, the 3-month futures price is $43 and the 3-month US$ interest rate is 5% per year with continuous compounding. The no-arbitrage futures price is:

\[\begin{equation*} F = 40 e^{0.05 \times 0.25} = 40.50 \end{equation*}\]

The next example shows how what would happen if the futures price of the contract described in Example 4.1 was higher than $40.50. When an asset is trading for more than its fair price, it makes sense to sell it.

Example 4.2

Suppose that the spot price of a non-dividend-paying stock is $40, the 3-month futures price is $43 and the 3-month US$ interest rate is 5% per year with continuous compounding. Is there an arbitrage opportunity?

The table below shows the cash flows that an investor would get today and in three months if she shorts one futures contract, borrows the present value of $43 to be paid in three months, and buys the stock. Therefore, borrowing means a positive cash flow today and a negative cash flow in the future, whereas buying a stock means a negative cash flow today and a positive cash in the future when we sell the stock.

\(T = 0 \vphantom{3/12}\)

\(T = 3/12\)

Short futures

0.00

\(43 - S_{T}\)

Borrow

42.47

\(-\)43

Long stock

\(-\)40.00

\(S_{T}\)

Total

2.47

0

The table shows that the cost of this strategy is negative $2.47, i.e. you would make money today by engaging in these transactions. Furthermore, this positive cash flow today has zero risk since we can see that the cash flow in three months is zero. Since by selling the futures for $43 we are able to make an arbitrage profit we conclude that the futures price in this case is too high.

We now turn our attention to the case when the observed futures price is less than its fair price. If this was the case, it makes sense to buy the futures and short-sell the stock.

Example 4.3

Suppose that the spot price of non-dividend paying stock is $40, the 3-month futures price is US$39 and the 3-month US$ interest rate is 5% per year with continuous compounding. Is there an arbitrage opportunity?

The table below shows the cash flows that an investor would get today and in three months if she buys one futures contract, invest the present value of $43 to get that amount in three months, and sells the stock. Notice that she need not have to own the stock in order to sell it short.

\(T = 0 \vphantom{3/12}\)

\(T = 3/12\)

Long futures

0.00

\(S_{T} - 39\)

Invest

\(-\)38.52

39

Short stock

40.00

\(-S_{T}\)

Total

1.48

0

Again, the cost of this strategy is negative as it generates money today. As before, the trader is fully hedged in three months. We can conclude that there is an arbitrage opportunity due to the fact that the futures price is too low.

As we mentioned before, the no-arbitrage futures price defined in \(\eqref{futures_price_no_dividends}\) works for any non-dividend paying asset, even a precious metal like gold, as long as we understand that \(S\) represents the spot price and not necessarily the cash price of the commodity.

Example 4.4

Suppose that gold spot is currently $1,870.60, and consider a futures contract on gold expiring in one year. Assume that the cost of storing gold is negligible and there are no additional benefits accruing from owning gold. The risk-free rate is 5% per year with continuous compounding. Then, the no-arbitrage futures price of gold is:

\[\begin{equation*} F = 1870.60 e^{0.05} = 1966.51 \end{equation*}\]

4.2. Assets Paying Cash Dividends#

There are many assets that will pay a dividend or an income during the life of the contract. Note that for some commodities there might be some non-negligible storage costs, which implies that the total income you derive from owning the asset is net of any costs of having the commodity in storage. If the storage costs outweigh the benefits of owning the commodity then the net dividend might become negative. In this case it is better to hold a long futures contract than the commodity itself.

We will denote by \(D\) the present value of the dividends or net income accruing to the owner of the asset or physical commodity, but not to the buyer of a futures contract, during the life of the contract.

For example, think about a stock. The income in this case are the quarterly dividends that you receive if you own the stock. If you were instead holding a long position on a futures contract written on the stock, you would not be entitled to any dividend payments during the life of the contract. The futures price needs to be adjusted accordingly.

Property 4.2 (Futures Price of a Dividend-Paying Asset)

The futures price of a dividend-paying asset is given by:

(4.2)#\[\begin{equation} F = (S - D) e^{r T} \label{futures_price_dividends} \end{equation}\]

where \(S\) is the spot price of the asset, \(T\) is the maturity of the futures, \(D\) denotes the present value of the dividends or net income earned during life of the contract, and \(r\) denotes the continuously compounded interest rate.

We note that \(D = 0\) when \(T = 0\) since no-dividends are paid if the contract expires immediately. This guarantees that the futures price is indeed equal to the spot price when \(T \rightarrow 0.\) Certainly, it would have been more accurate to denote \(D\) by \(D(T)\) but this would have made the notation cumbersome.

The next example shows how to apply Property 4.2.

Example 4.5

Consider a stock that currently trades at $50. The stock is expected to pay dividends of $1.15 and $1.20 in two and five months, respectively. If the risk-free rate is 5% per year with continuous compounding, let us compute the futures price six months from now.

For this we need to compute the present value of the dividends. Even though in practice dividend payments are uncertain, they can be considered certain for short-term maturities and hence we can discount them using the risk-free rate. We have that:

\[\begin{equation*} D = 1.15 e^{-0.05 \times 2/12} + 1.20 e^{-0.05 \times 5/12} = 2.32 \end{equation*}\]

Therefore,

\[\begin{equation*} F = (50 - 2.32) e^{0.05 \times 6/12} = 48.89 \end{equation*}\]

The futures price we computed in Example 4.5 is the price that prevents any arbitrage opportunities. If the futures price was different, an arbitrageur could engage in the following strategy and make an arbitrary large profit.

Example 4.6

Consider the same stock described in Example 4.5. What would an arbitrageur do if the six-months futures price was $50.20?

We know that the no-arbitrage futures price is $48.89. If the actual futures price is $50.20, it means that the six-month futures is too expensive and hence we should sell it. We hedge the transaction by buying one share of the stock at $50. The time-line below shows that an arbitrageur selling the six-months stock futures and buying the stock at time 0 is entitled to dividends in two and five months, plus the stock itself that can be sold at the six-month futures price prevailing today.

Figure made with TikZ

All this suggests that the arbitrageur could take the following loans to be repaid in full using the dividend payments and the sale of the stock in six months:

  • Loan 1: Borrowing \(1.15 e^{-0.05 \times 2/12} = \$1.14\) today and repaying $1.15 in two months.

  • Loan 2: Borrowing \(1.20 e^{-0.05 \times 5/12} = \$1.18\) today and repaying $1.20 in five months.

  • Loan 3: Borrowing \(50.20 e^{-0.05 \times 6/12} = \$48.96\) today and repaying $50.20 in six months.

The table below describes the cash flows received by the arbitrageur when engaging in this transaction. A positive cash flow means that the arbitrageur receives money whereas a negative cash flow means the opposite.

\(T = 0 \vphantom{2/12}\)

\(T = 2/12\)

\(T = 5/12\)

\(T = 6/12\)

Short futures

0.00

\(50.20 - S_{T}\)

Loan 1

1.14

\(-\)1.15

Loan 2

1.18

\(-\)1.20

Loan 3

48.96

\(-\)50.20

Long stock

\(-\)50.00

1.15

1.20

\(S_{T}\)

Total

1.28

0

0

0

This strategy generates a certain profit of $1.28 per share of the stock and has no risk. An arbitrageur could sell 100 million futures and hedge accordingly to generate an instantaneous risk-free profit of $128 million.

4.3. Assets Paying a Dividend Yield#

There are many assets that pay dividends continuously, like a foreign currency. Some other assets can be modelled as if they pay a continuous dividend like a stock index such as the S&P 500. In these cases it is convenient to think of dividends as a percentage yield paid over time. Given the convenience of this approach, some practitioners also use it to model individual stocks, even though in these cases the dividends are paid quarterly. In what follows we will denote the continuously-compounded dividend yield by \(q\).

The asset \(S\) then pays every instant \(t\) over a time-period \(\Delta t\) a dividend of \(q \Delta t\) units of the asset. Therefore, the dividend yield can be seen as the units of the asset growing over time at the rate \(q.\) Thus, if we start with 1 share at time 0 after \(T\) years we will have \(e^{q T}\) shares.

Property 4.3 (Futures Price of an Asset Paying a Dividend Yield)

The futures price of a dividend-yield paying asset is given by:

\[\begin{equation*} F = S e^{(r - q) T} \end{equation*}\]

where \(S\) is the spot price of the asset, \(T\) is the maturity of the futures, \(q\) is the continuous dividend or convenience yield, and \(r\) denotes the continuously compounded interest rate.

4.3.1. Index Futures#

Futures contracts written on stock market indices are called index futures. There are index futures written on the S&P 500, Nasdaq, Dax 30, CAC 40, Stoxx 50, among many others. In general, index futures are very liquid and provide good trading opportunities for both speculators and hedgers.

Because the tradable underlying asset is the basket of stocks that define the index, index futures are cash settled. Otherwise, it would be very cumbersome and most likely impossible to deliver a basket containing all the stocks in the index in the right proportions. Cash settlement means that if the contract reaches maturity, the contract is just marked-to-market against the value of the index. As a matter of fact, it is a pretty convenient way to settle to the contract.

Example 4.7

The E-mini S&P 500 futures contract is one of the most liquid and actively traded futures in the world. The contract value is defined as $50 \(\times\) the value of the S&P 500 Index. The way the margin works on this contract is as follows.

Say you deposit $12,000 in your margin account and buy one S&P 500 E-mini futures at $4,645.00. The next day the futures price increases to $4,656.75, which is a gain of $11.75 with respect to the previous settlement. That day, your account is then credited \(50 \times 1.75 = \$587,\) which increases your margin to $12,587.00. If the day after the futures price decreases to $4,652.25, then your account will lose \(50 \times 4.5 = \$225.00,\) bringing your margin down to $12,362.00.

The table below describes these transactions.

Day

Futures Price

Gain/Loss

Margin Account

0

4,645.00

12,000.00

1

4,656.75

587.50

12,587.50

2

4,652.25

-225.00

12,362.50

3

4,658.50

312.50

12,675.00

Note that futures exchanges require the margin account to be at all times above a certain minimum. If the margin account goes below the minimum margin requirement the trader will receive a margin call.

Example 4.8

Consider an index tracking a portfolio of stocks that pays a dividend yield of 3% per year with continuous compounding. The index is currently at 4,300. The risk-free rate for all maturities is 1% per year continuously-compounded. What should be the 6-month futures price of the index?

If we denote by \(F\) the futures price, then we have that:

\[\begin{equation*} F = 4300 e^{(0.01-0.03) \times 6/12} = 4257.21 \end{equation*}\]

Note that because the dividend yield is higher than the risk-free rate, the futures price is less than the current spot price.

In Example 4.8 we computed the no-arbitrage futures price of the index. When the observed futures price deviates from this relationship, arbitrageurs can then try to exploit this difference in their advantage in what is called index-arbitrage. Generally speaking, if \(F\) denotes the observed futures price, we have that:

  • If \(F < S e^{(r - q) T}\) then you should buy the futures, sell \(e^{-q T}\) units of the index and invest \(F e^{-r T}\) dollars in a risk-free money-market account.

  • If \(F > S e^{(r - q) T}\) then you should sell the futures, buy \(e^{-q T}\) units of the index and borrow \(F e^{-r T}\) dollars.

Note that in both scenarios the arbitrageur will make risk-free money today while being completely hedged when the futures reaches maturity.

Example 4.9

Consider the index discussed in Example 4.8. If the 6-month futures is trading at 4,300, an arbitrageur could engage in the following transactions:

\(T = 0 \vphantom{6/12}\)

\(T = 6/12\)

Short Futures

0

\(4300 - S_{T}\)

Buy \(S e^{-q T}\) of the Index

-4235.98

\(S_{T}\)

Borrow \(F e^{-r T}\)

4278.55

-4300

Total

42.57

0

The strategy generates a risk-free cash flow today of $42.57 per futures sold with no initial investment required. An arbitrageur could easily sell, just to give an arbitrary number, one million index futures, hedge accordingly and pocket $42.57 million.

This shows that in equilibrium the futures price cannot deviate much from its no-arbitrage price.

4.3.2. Currency Forwards and Futures#

Currency forwards and futures are contracts written on a foreign currency. Currency forwards trade over-the-counter (OTC) whereas currency futures trade in derivatives exchanges. Both are widely traded derivatives and represent a large fraction of the financial system, the FX forward market being by far the largest. As of 2019, the Bank for International Settlements (BIS) reports that the daily turnover of currency forwards is approximately $1 trillion.

When pricing currency forwards and futures, the dividend yield then represents the interest rate that you would earn if you had a certain amount of the foreign currency in a deposit account. For us, the spot price of the asset is the price of the foreign currency in US dollars (USD).

Typically, the exchange rate between two currencies is the number of domestic currency units per unit of foreign currency. We need to be careful, though, since the street market convention for the EUR/USD exchange rate implies that the quote currency is the US dollar (USD) and the base currency is the Euro (EUR). For example, the direct quotation of the EUR/USD could be $1.08/€, and represents the price in USD of 1 EUR. Note that you could always define it the other way around (indirect-quotes), and this is done for many currency pairs as well.

The market convention of calling this exchange rate EUR/USD might be misleading. It is written EUR/USD, EUR-USD or EURUSD but it really represents the number of USD per EUR, i.e. \(\$1.08 \Leftrightarrow \text{€}1.\) Be careful, though, as in some textbooks you might find it the other way around.

Example 4.10

If the EUR/USD exchange rate is 1.08, for a US investor, 1 Euro is worth $1.08, but in Europe, how many Euros is worth $1?

\[\begin{equation*} \$1 = \frac{1}{1.08} = \text{€} 0.93/\$. \end{equation*}\]

Some currency pairs such as EUR/USD or GBP/USD use the USD as the quote currency. However, most currency pairs are expressed using the dollar as the base currency, i.e., USD/JPY, USD/CNY, USD/CLP, etc.

In order to compute the currency forward price, we need to know the current spot currency price, and the interest rates of the foreign and domestic currency. Currency forward prices are usually expressed as forward-points, that is 10,000 times the difference between the forward and the spot price.

Example 4.11

The current GBP/USD exchange rate is 1.30. The interest rates in USD and GBP are 1% and 3% per year with continuous compounding, respectively. The 9-month GBP/USD forward price is then

\[\begin{equation*} F = 1.30 e^{(0.01 - 0.03) \times 9/12} = 1.2806, \end{equation*}\]

or

\[\begin{equation*} 10{,}000 \times (1.2806 - 1.3000) = -193.5 \text{ forward-points.} \end{equation*}\]

4.3.3. Commodity Futures#

There are many futures written on commodities such as crude oil, copper, gold, soybean, among others. It is important to note that for commodities, the dividend yield corresponds to the net benefit accruing to the owner of the physical commodity but not to the buyer of a futures contract and is called the convenience yield.

The convenience yield should take into consideration the gross benefits of owning the physical commodity, such as the ability to profit from temporary shortages, but also storage costs. The difference between the risk-free rate and the convenience yield is usually called the cost-of-carry.

Example 4.12

Suppose that the spot price of oil is $95 per barrel, the 1-year US$ interest rate is 5% per year with continuous compounding and the convenience yield is 2% per year.

The 1-year oil futures price is

\[\begin{equation*} F = 95 e^{0.05 - 0.02} = \$97.89. \end{equation*}\]

4.4. Valuing a Forward Contract#

A forward contract is worth zero when it is first negotiated. Later it may have a positive or negative value since the underlying asset might increase or decrease in value, and the time-to-maturity of the contract decreases.

Suppose that you bought the forward some time ago for \(K\), and you would like to know how much that contract is worth today. If the current forward price is \(K,\) you could sell the forward today and completely hedge your future exposure. Indeed, in the past you committed to purchase the asset for \(K\) at maturity, but you just committed to sell it for \(F\) at the same date. This means that you have just locked-in a certain cash flow of \(F - K\) at time \(T,\) which in present value terms is worth today \((F - K) e^{-r T}.\)

To value an existing short forward position entered some time ago at a forward price \(K,\) you could buy a forward contract at \(F\) today, locking-in a certain cash flow of \(K - F\) at maturity. The value of the short forward contract is then \((K - F) e^{-r T}.\)

Example 4.13

You entered into a short forward contract sometime ago on an asset that pays a dividend yield of 7% per year. The forward price at that time was $200. Today the contract has six months until maturity and the current forward price is $190. Also, the current risk-free rate is 5% per year with continuous compounding.

To compute the current value of the short forward position, we could imagine what would happen if we buy a forward today. That would lock-in a certain cash flow in six months of \(200 - 190 = \$10,\) whose present value today is \(10 e^{-0.05 \times 6/12} = \$9.75,\) which is the value of the short forward contract.

4.5. Futures Prices vs. Expected Future Spot Prices#

Many practitioners and academics have suggested that we could use futures or forward prices to forecast future spot prices. Intuitively, since futures prices determine the price at which an asset can be bought or sold in the future suggests that buyers and sellers should somehow use their forecasts when trading the derivative.

Whether or not futures prices are the best unbiased predictor of expected future prices depends on whether investors command a risk-premium to hold the asset. In stock markets the risk-premium is usually positive since stocks usually comove positively with the market portfolio. For currencies and commodities, though, the risk-premium could be positive, negative or zero depending on which side puts the hedging pressure.

For example, consider a commodity where producers want to hedge their production. If there are no consumers that want to hedge their consumption, then long futures position will have to be taken by speculators. If hedgers are risk-averse, they will then be willing to sell for less than the expected future price in order to unwind their risk-exposure.

Specifically, suppose that \(\mu\) is the expected return required by investors and denote by \(F\) the futures price expiring at time \(T.\)

Imagine that you want to compute the value of 1 unit of the asset paid at time \(T.\) You could do this in two different ways. Either you discount the expected value of the asset at \(\mu,\) or you hedge your exposure by selling a futures, which provides a certain cash flow \(F\) that can then be discounted at the risk-free rate. Therefore,

\[\begin{equation*} F e^{-r T} = \ev(S_{T}) e^{-\mu T}, \end{equation*}\]

or

\[\begin{equation*} F = \ev(S_{T}) e^{(r - \mu) T}. \end{equation*}\]

The previous expression shows that futures prices are unbiased estimators of future prices only when there is no systematic risk priced into the asset. More generally, futures prices will under- or over-estimate expected future prices depending on the sign of the systematic risk, as shown in the table below.

Type of Risk

Expected Return

Futures vs. Expected Price

No Systematic Risk

\(\mu = r\)

\(F = \ev(S_{T})\)

Positive Systematic Risk

\(\mu > r\)

\(F < \ev(S_{T})\)

Negative Systematic Risk

\(\mu < r\)

\(F > \ev(S_{T})\)

4.6. Practice Problems#

Exercise 4.1

Suppose that you enter into a 6-month forward contract on a non-dividend-paying stock when the stock price is $30 and the risk-free interest rate is 12% per year with continuous compounding. What is the forward price?

Exercise 4.2

A stock index currently stands at 350. The risk-free interest rate is 8% per year with continuous compounding and the dividend yield on the index is 4% per year. What should be the futures price for a 4-month contract?

Exercise 4.3

A 1-year long forward contract on a non-dividend-paying stock is entered into when the stock price is $40 and the risk-free rate of interest is 10% per annum with continuous compounding.

  1. What are the forward price and the initial value of the forward contract?

  2. Six months later, the price of the stock is $45 and the risk-free interest rate is still 10%. What are the forward price and the value of the forward contract?

Exercise 4.4

Assume that the risk-free interest rate is 9% per annum with continuous compounding and that the dividend yield on a stock index varies throughout the year. In February, May, August, and November, dividends are paid at a rate of 5% per annum. In other months, dividends are paid at a rate of 2% per annum. Suppose that the value of the index on July 31 is 1,300. What is the futures price for a contract deliverable in December 31 of the same year?