Forward Contracts
Definitions
A long forward requires the buyer to purchase the asset at expiration for the forward price prevailing when the contract was first bought, which we denote by \(K.\) If the price of the asset at maturity is \(S,\) then the payoff of the long position in the forward contract is \(S - K.\)
Indeed, if the asset price at maturity is greater than the forward price, we purchase the asset by \(K\) and immediately sell it by \(S,\) generating a profit of \(S - K.\) If, on the other hand, \(S < K,\) 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 \(S.\)
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 1 (Payoff from buying currency forward) 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.
| Exchange Rate at Maturity | 1.2000 | 1.3000 | 1.4000 | 1.5000 | 1.6000 |
|---|---|---|---|---|---|
| Payoff | -242,200 | -142,200 | -42,200 | 57,800 | 157,800 |
The figure in the margin 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 \(K - S.\)
The short forward payoff is then the mirror image with respect to the x-axis of the long forward position. By fixing the price at \(K,\) the seller is happy when prices go down but is unhappy when prices increase.
Forward Prices
We now turn our attention to understanding what should determine the forward price in equilibrium. The 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 forward price has a long position whereas the party that has agreed to sell at that price has a short position.
We compute the forward price by assuming that in real financial markets is hard to make easy money without taking any risk. In finance, we call this notion 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. We just need to assume that people prefer more to less.
Non-Dividend Paying Assets
We start by computing the no-arbitrage forward price of non-dividend paying asset.
We will see shortly that if the forward price is different form the one given in \(\eqref{forward_price_no_dividends},\) 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 2 Suppose that the current price of a non-dividend-paying stock is $40, the 3-month forward price is $43 and the 3-month US$ interest rate is 5% per year with continuous compounding. The no-arbitrage forward price is: \[ F = 40 e^{0.05 \times 0.25} = 40.50 \]
The next example shows how what would happen if the forward price of the contract described in Example 2 was higher than $40.50. When an asset is trading for more than its fair price, it makes sense to sell it.
Example 3 Suppose that the current price of a non-dividend-paying stock is $40, the 3-month forward 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 selss one forward 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.
| \(T = 0\) | \(T = 3/12\) | |
|---|---|---|
| Short forward | \(0.00\) | \(43 - S_{T}\) |
| 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 forward for $43 we are able to make an arbitrage profit we conclude that the forward price in this case is too high.
We now turn our attention to the case when the observed forward price is less than its fair price. If this was the case, it makes sense to buy the forward and short-sell the stock.
Example 4 Suppose that the current price of non-dividend paying stock is $40, the 3-month forward 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 forward 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.
| \(T = 0\) | \(T = 3/12\) | |
|---|---|---|
| Long forward | \(0.00\) | \(S_{T} - 39\) |
| 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 forward price is too low.
As we mentioned before, the no-arbitrage forward price defined in \(\eqref{forward_price_no_dividends}\) 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 5 Suppose that the gold price is currently $1,870.60 per ounce, and consider a forward contract written 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 forward price of gold is \[ F = 1870.60 e^{0.05} = 1966.51. \] Note that we do not need to know what gold prices will do in the future to fix the price today at which we can buy gold in one year.
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 might be better to hold a long forward 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 forward 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 forward contract written on the stock, you would not be entitled to any dividend payments during the life of the contract. The forward price needs to be adjusted accordingly.
We note that \(D = 0\) when \(T = 0\) since no-dividends are paid if the contract expires immediately. This guarantees that the forward price is indeed equal to the asset 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.
Example 6 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 forward 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 known for short-term maturities and hence we can discount them using the risk-free rate. We have that: \[ D = 1.15 e^{-0.05 \times 2/12} + 1.20 e^{-0.05 \times 5/12} = 2.32 \] Therefore, \[ F = (50 - 2.32) e^{0.05 \times 6/12} = 48.89 \]
The forward price we computed in Example 6 is the price that prevents any arbitrage opportunities. If the forward price was different, an arbitrageur could engage in the following strategy and make an arbitrary large profit.
Example 7 Consider the same stock described in Example 6. What would an arbitrageur do if the six-months forward price was $50.20?
We know that the no-arbitrage forward price is $48.89. If the actual forward price is $50.20, it means that the six-month forward 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 forward 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 forward price prevailing today.
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.
| \(T = 0\) | \(T = 2/12\) | \(T = 5/12\) | \(T = 6/12\) | |
|---|---|---|---|---|
| Short forward | \(0.00\) | \(50.20 - S_{T}\) | ||
| 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 forwards and hedge accordingly to generate an instantaneous risk-free profit of $128 million.
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.
In the next section we analyze the market for foreign currencies that is a typical example of an asset paying a dividend yield.
Currency Forwards
Foreign Currencies
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 8 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? \[ \$1 = \frac{1}{1.08} = \text{€} 0.93/\$. \] Therefore, to buy $1 you need to pay €0.93.
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.
Forward Price of a Foreign Currency
Currency forwards are contracts written on a foreign currency and trade over-the-counter (OTC). The FX forward market is one of the largest in the world. 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, 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).
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 9 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 \[ F = 1.30 e^{(0.01 - 0.03) \times 9/12} = 1.2806, \] or \(10{,}000 \times (1.2806 - 1.3000) = -193.5\) forward points.
Valuing an Existing Forward Position
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 10 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.
Practice Problems
Exercise 1 A forward contract is defined as:
- The right but not the obligation to purchase or sell an asset in the future at a given fixed price for a given quantity.
- A commitment to purchase or sell an asset in the future at a given fixed price for a given quantity.
- An asset that pays a fixed amount at maturity.
- A physical position in the underlying asset.
Solution
A forward contract is a commitment to purchase or sell an asset in the future at a given fixed price for a given quantity. The answer is B.Exercise 2 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?
Solution
\[ F = 30 e^{0.12 \times 6/12} = 31.86. \]Exercise 3 Imagine that you agree to purchase 1 million euros in 9 months from now at an exchange rate of $1.12 per euro. If the exchange rate in 9 months is $1.02 per euro, what would be your payoff?
Solution
The is computed as \(1{,}000{,}000 \times (1.02 - 1.12) = -\$100{,}000.\)Exercise 4 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.
- What are the forward price and the initial value of the forward contract?
- 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?
Solution
- \(F = 40 e^{0.10 \times 1} = 44.21\) and \(V = 0\).
- \(F = 45 e^{0.10 \times 0.5} = 47.31\) and \(V = (47.31 - 44.21) e^{-0.10 \times 0.5} = 2.95\).