Futures Markets
Definitions
Futures trade in organized exchanges such as the Chicago Mercantile Exchange (CME) or the Chicago Board Options Exchange (CBOE). Futures contracts are written on a wide variety of asset classes such as physical commodities, equity indices, foreign currencies, interest rates and even abstract quantities such as volatility.
Traditionally, all futures used to trade in physical trading floors organized into segmented areas called pits. The idea was for traders and floor brokers to interact directly with each other in face-to-face transactions, in a system commonly known as open-outcry. With the advent of technology and later on the COVID-19 pandemic, almost all futures trading today occurs electronically.
Electronic trading is usually implemented by recording bids and offers in what is called an electronic limit-order book. Limit orders, if not executed immediately either because the bid is lower than the best offer price, or because the offer is higher than the best bid, are recorded in the book. These orders can be cancelled at no cost, otherwise they will stay active usually until the end of that day’s trading session. Market orders on the other hand will execute immediately at the best possible bid or offer available at order received time.
Unlike stocks, futures exchanges have longer trading hours. For example, the market hours for E-mini S&P 500 (ES), which is traded online, is Sunday through Friday 5 p.m. to 4 p.m. Central Time (CT).
Example 1 These are examples of some futures contracts:
- Buy 100 oz. of gold @ $1400 per oz. in December
- Sell £62,500 at $1.2620 per £ in June
- Sell 1,000 barrels of oil at $70 per barrel in July
Settlement and Maturity
The settlement of a futures contract happens if the trader keeps the position open until maturity. For many commodity contracts, the settlement procedure involves the physical delivery of the underlying asset. Therefore, the contract specifications must clearly specify the size, quality and physical delivery options to receive or deliver the commodity. 1
1 The delivery size of the contract determines the notional value at maturity, which in turns determine the minimum margin requirement.
For equity index and interest rates futures, the delivery takes place as a cash settlement. The value of the contract is computed as a dollar amount times the value of an reference index.
All futures contracts have an expiration date, and for a given underlying asset there are usually several expirations trading at any given point in time.
Futures vs. Spot Prices
Each of these contracts trade continuously during the exchange trading hours. Each day, for margin requirement purposes, the exchange determines the settlement price of the contract. The settlement price is what is usually reported as the price of the contract for that day. The figure below plots the evolution of the March 2024 Soybean settlement futures price.
The figure shows that futures prices are volatile, and clearly displays the recent spike in commodity prices. Also, note that as time moves forward, since the maturity of the Nov 23 futures stays fixed, the time-to-maturity of the contract declines.
We define the spot price of the underlying asset as as the closest-to-maturity futures price. For many commodities, the spot price is close but not the same as the cash price. Indeed, the delivery method of a futures contract might be different from the typical delivery method of the physical commodity. What makes a very short-maturity futures interesting for us is that it is relatively easily to sell a futures contract, whereas it is in general hard to short-sell a physical commodity. As a consequence, when analyzing commodity futures we will usually consider the shortest-to-delivery futures as our underlying asset and not necessarily the physical commodity.
More formally, if we denote by \(F(t, T)\) the futures price at time \(t\) of a contract expiring at time \(T,\) the spot price is defined as: \[\begin{equation*} S_{t} = F(t, t) \end{equation*}\]
The figure below plots the evolution of the cotton spot price and of the Mar 24 futures contract on cotton since January 2018.
We can see from the figure that the futures price converges over time to the spot price.
Exchanges also report the total number of contracts open which is usually called the open interest. Remember that for any long position there has to be a corresponding short position. Therefore, the open interest measures the total number of long positions, or equivalently the total number of short positions open at any given point in time.
The figure below displays the evolution of the open interest for all soybeans futures starting Jun-2006.
It is important to note that a large open interest could reflect strong demand for short positions, long positions, or both.
The Commodity Futures Trading Commission (CFTC) publishes detailed weekly information about open interest for all futures in their Commitments of Traders (COT) Reports.
Short Selling
There are cases when we can relate the cash or physical commodity price with the spot price of the underlying asset. For this to happen, we must be able to freely purchase the physical commodity in both positive and negative quantities so we can arbitrage away any price inconsistencies between cash and spot markets. In other words, we must be able to short-sell the asset. This would be the case of stocks, for example.
Short selling involves selling securities you do not own. Your broker borrows the securities from another client and sells them in the market in the usual way. At some stage, you must buy the securities back so they can be replaced in the account of the client. You must pay dividends and other benefits that the owner of the securities receives. There may be a small fee for borrowing the securities.
Example 2 (Short selling a stock) You short 100 shares when the price is $100 and close out the short position three months later when the price is $90. At the end of the three months and just before closing the position, a dividend of $3 per share is paid. In this case your profit is \[ 100 \times (100 - 90 - 3) = \$700. \]
If on the other hand you would have bought 100 shares, you would have had a profit of \[ 100 \times (90 + 3 - 100) = -\$700. \] Note that when you buy the stock you receive the dividends whereas when you short sell the stock you must pay those dividends back.
However, there are many futures contracts written on underlying assets that are difficult or impossible to short-sell. Think for example about commodities. It might be hard or impossible to short-sell oil or physical gold. In these cases we will resort to buying or selling the shortest futures contract in order to establish a relationship between the spot and the futures price.
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 3 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 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: \[ F = 4300 e^{(0.01-0.03) \times 6/12} = 4257.21 \] 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 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 5 Consider the index discussed in Example 4. If the 6-month futures is trading at 4,300, an arbitrageur could engage in the following transactions:
| \(T = 0\) | \(T = 6/12\) | |
|---|---|---|
| Short Futures | 0 | \(4300 - S_{T}\) |
| Buy \(S e^{-q T}\) units 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.
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 6 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 \[ F = 95 e^{0.05 - 0.02} = \$97.89. \]
Futures vs. Forward Contracts
Even though futures and forward contracts are very similar, in practice there are important differences between these two instruments:
- Futures contracts trade in exchanges such as the Chicago Mercantile Exchange (CME) whereas forwards trade the over-the-counter (OTC) where traders working for buy-side companies such as fund managers and corporate treasurers contact sell-side investors such as large international banks directly.
- Futures exchanges standardize the terms of the contract such as expiration dates, notional amount, delivery method, and quality, among others. Forwards can be negotiated so that to fit specific needs of a client.
- Futures exchanges require traders to keep a margin account which consists in cash or marketable securities deposited by an investor with his/her broker. The margin account balance is adjusted daily to account for daily gains or losses, and must always be above a certain minimum. Margins minimize potential losses that might occur because of a default event. Forward contracts, in general, are settled in full at expiration.
Notwithstanding these important differences, forward and futures prices with the same maturity are usually assumed to be equal. Indeed, when interest rates are deterministic (or uncorrelated with the underlying asset), futures and forward prices are the same. Given that this is the assumption we use in many chapters of this book to price options and futures, in what follows we will refer to either a futures or forward contract interchangeably unless stated otherwise.
It is important to note, however, that when interest rates are stochastic, futures and forward prices are in theory different:
- A strong positive correlation between interest rates and the asset price implies the futures price is higher than the forward price, as would be the case for Eurodollar futures (soon to be replaced by SOFR futures).
- A strong negative correlation between risk-free rates and the underlying asset implies the reverse. For interest rate futures, it is common to adjust the relevant forward rate in order to derive the futures rate. Such a modification is usually called a convexity adjustment.
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, \[ F e^{-r T} = \ev(S_{T}) e^{-\mu T}, \] or \[ F = \ev(S_{T}) e^{(r - \mu) T}. \]
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 | \(\mu < r\) | \(F > \ev(S_{T})\) |
Practice Problems
Exercise 1 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?
Solution
\[ F = 350 e^{(0.08 - 0.04) \times 4/12} = 354.70. \]Exercise 2 Consider an E-mini S&P 500 futures contract. Remember that the contract value is defined as $50 times the value of the S&P 500 Index. Yesterday’s futures price settlement was 4,296.12, whereas today, the S&P 500 futures price settled at 4,175.20. Compute the change in a trader’s margin account if she bought one futures yesterday.
Solution
The change will be \(50 \times (4{,}175.20 - 4{,}296.12) = -6,046.\)Exercise 3 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?
Solution
During the life of the contract, the index will pay dividends in August and November when the dividend yield is 5%, and in September, October and December when the dividend yield is 2%. Therefore, \[ F = 1300 e^{0.09 \times 5/12 - (0.05 \times 2/12 + 0.02 \times 3/12)} = 1331.80. \]Exercise 4 The spot price of silver is $9 per ounce. The storage costs are $0.24 per ounce per year payable quarterly in advance. Assuming that interest rates are 10% per annum for all maturities, calculate the futures price of silver for delivery in 9 months.