3. Futures and Forward Contracts#
3.1. Definitions#
A derivative is a financial instrument whose value depends on, or is derived from, the value of another asset. Futures and forward contracts are derivatives that allow traders to fix the price at which an asset will trade at a given date in the future. In this sense, futures or forward contracts give the holder the obligation to buy or sell at a certain price, unlike options which give the holder the right but not the obligation to buy or sell at a certain price.
The futures or forward price is the delivery price that would be applicable to the contract if it were negotiated today so that its value is zero. The party that has agreed to buy at the futures price has a long position whereas the party that has agreed to sell at that price has a short position.
3.2. Futures Contracts#
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 3.1 (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
Note
The delivery size of the contract determines the notional value at maturity, which in turns determine the minimum margin requirement.
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.
Example 3.2 (Soybean Futures)
The following paragraph is from the CME rulebook on soybean futures:
Each futures contract shall be for 5,000 bushels of No. 2 yellow soybeans at par, No. 1 yellow soybeans at 6 cents per bushel over contract price, or No. 3 yellow soybeans at 6 cents per bushel under contract price provided that all factors equal U.S. No. 2 or better except for foreign material (refer to Rule 11104.). Every delivery of soybeans may be made up of the authorized grades for shipment from eligible regular facilities provided that no lot delivered shall contain less than 5,000 bushels of any one grade from any one shipping station.
The complete rulebook for soybean futures can be found here.
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.
Example 3.3 (SOFR Futures)
The London Inter-Bank Offered Rate in US dollars, or LIBOR USD, was for half a century the reference rate for corporate loans denominated in USD. LIBOR was also quoted in other major currencies such as the Japanese Yen (JPY) or the Swiss Franc (CHF), for example. Because of how LIBOR was computed and the risks it could pose to the global financial system, shortly after the 2008 financial crisis global regulators convene to transition away from LIBOR.
In 2014, the Federal Reserve Board and the New York Fed convened the Alternative Reference Rates Committee (ARRC), a group of private-market participants tasked with identifying robust alternatives to USD LIBOR and supporting a transition away from LIBOR. In 2017, the ARRC selected the Secured Overnight Financing Rate (SOFR) as a viable replacement from LIBOR USD.
SOFR is a secured overnight interest rate based on Treasury repurchase transactions (repos). The repo market is liquid and accurately represents the risk-free cost of funding of major banks. Unlike LIBOR, which was a term rate, SOFR is by definition an overnight rate. As such, a very active SOFR futures market has emerged. SOFR futures are cash settled and their values are computed against SOFR rates. For 1-month SOFR futures the reference index is an arithmetic average whereas for 3-month SOFR futures the reference index is a geometric average of SOFR rates.
The notional value of the 3-month SOFR futures is equal to $2,500 \(\times\) the contract grade IMM index, which is computed as 100 minus the business-day compounded SOFR per annum during the contract reference quarter. For 1-month SOFR futures, the notional value is computed similarly but against an arithmetic SOFR during the contract delivery month.
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.
Example 3.4 (E-mini S&P 500 Futures Expirations)
E-mini S&P 500 futures have quarterly contracts (Mar, Jun, Sep, Dec) listed for 9 consecutive quarters and 3 additional December contract months. As of 5/24/2022, the table below presents the available maturities of these contracts.
Contract Month |
Ticker |
First Trade |
Last Trade |
Settlement |
|---|---|---|---|---|
Jun-22 |
ESM22 |
19-Mar-2021 |
17-Jun-2022 |
17-Jun-22 |
Sep-22 |
ESU22 |
07-Jun-2021 |
16-Sep-2022 |
16-Sep-22 |
Dec-22 |
ESZ22 |
07-Jun-2021 |
16-Dec-2022 |
16-Dec-22 |
Mar-23 |
ESH23 |
07-Jun-2021 |
17-Mar-2023 |
17-Mar-23 |
Jun-23 |
ESM23 |
07-Jun-2021 |
16-Jun-2023 |
16-Jun-23 |
Sep-23 |
ESU23 |
18-Jun-2021 |
15-Sep-2023 |
15-Sep-23 |
Dec-23 |
ESZ23 |
07-Jun-2021 |
15-Dec-2023 |
15-Dec-23 |
Mar-24 |
ESH24 |
17-Dec-2021 |
15-Mar-2024 |
15-Mar-24 |
Jun-24 |
ESM24 |
18-Mar-2022 |
21-Jun-2024 |
21-Jun-24 |
Dec-24 |
ESZ24 |
07-Jun-2021 |
20-Dec-2024 |
20-Dec-24 |
Dec-25 |
ESZ25 |
07-Jun-2021 |
19-Dec-2025 |
19-Dec-25 |
Dec-26 |
ESZ26 |
17-Sep-2021 |
18-Dec-2026 |
18-Dec-26 |
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 Sep 23 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 Sep 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:
The figure below plots the evolution of the cotton spot price and of the Oct 23 futures contract on cotton.
We can see from the figure that the futures price converges over time to the spot price.
Example 3.5 (Negative Oil Spot Prices)
The COVID-19 pandemic hit countries hard in March 2020. As many economies came to a halt, oil inventories rose and refineries reached storage capacity. The buyer of an Apr 20 light sweet crude oil futures contract is required, if the contract reaches maturity, to purchase 1,000 barrels of oil. On April 20, 2020 there were many traders left with long positions that did not want to hold them until maturity the next day. In order to get out of a long futures position, you need to sell the same contract and that would net out the long position. The problem is that in order to sell an existing futures contract someone must be willing to buy it, and no trader wanted to purchase more contracts that day. Indeed, it would have meant to purchase physical oil for which there was no place to store it.
As a consequence, many sellers that day were willing to pay the buyer so they could get out of their long positions. In other words, the price of spot oil became negative. The figure below shows that the evolution of the spot price of oil from January 2019 until November 2021. We can see how on 4/20/2020 the spot price of oil reached almost negative $40 a barrel.
The cash price of oil, however, was not negative as no refinery would pay traders to buy the oil they had in storage.
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 from Jan-2018 until May-2022.
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.
3.3. 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 3.6
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.
What is your profit? In this case your profit is \(100 \times (100 - 90 - 3) = \$700.\)
What would be your profit/loss if you had bought 100 shares? You would have had a profit of \(100 \times (90 + 3 - 100) = -\$700.\)
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.
3.4. 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.
3.5. Contract Payoffs#
A long forward requires the buyer to purchase the asset at expiration for the futures price prevailing when the contract was first bought, which we denote by \(K.\) If the spot price at maturity is \(S_{T},\) then the payoff of the long position is \(S_{T} - K.\)
Indeed, if the spot price at maturity is greater than the forward price, we purchase the asset by \(K\) and immediately sell it by \(S_{T},\) generating a profit of \(S_{T} - K.\) If, on the other hand, \(S_{T} < K,\) then the payoff is negative since under the terms of the contract we are required to purchase the asset for \(K\) which we can only sell at a lower price \(S_{T}.\)
The figure below plots the payoff of a forward contract when the forward delivery price is \(K,\) as a function of the spot price \(S.\) As can be seen from the picture, the payoff of a long forward is increasing in \(S\) and cuts the x-axis at the forward delivery price \(K.\)
Example 3.7
On May 24, 2010, the treasurer of a corporation enters into a long forward contract to buy £1 million in six months at an exchange rate of $1.4422 per pound sterling. This obligates the corporation to pay $1,442,200 for £1 million on November 24, 2010. What are the possible outcomes?
We can compute the payoff of the contract for different values of the exchange rate in six months from now. This gives us an idea of the possible outcomes depending on how the exchange rate is in six months.
\(S_{T}\) |
1.2000 |
1.3000 |
1.4000 |
1.5000 |
1.6000 |
|---|---|---|---|---|---|
Payoff |
-242,200 |
-142,200 |
-42,200 |
57,800 |
157,800 |
Alternatively, we could do the same thing in a graph. The figure below plots the payoff of this long forward contract as a function of the currency exchange-rate at maturity.
The figure below shows the payoff of a short forward position which is given by \(K - S.\)
3.6. Practice Problems#
Exercise 3.1
Briefly explain what a forward contract is.
Solution to Exercise 3.1
A forward contract is a commitment to purchase or sell an asset in the future at a given fixed price for a given quantity.
Exercise 3.2
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. What will be the change in your margin account?
Solution to Exercise 3.2
The change will be \(50 \times (4,175.20 - 4,296.12) = -6,046.\)