Pricing and Valuation Concepts
What you'll learn
- LOS 28.bDescribe the carry arbitrage model without underlying cashflows and with underlying cashflows.
The party to the forward contract that agrees to buy the financial or physical asset has a long forward position and is called the long. The party to the forward contract that agrees to sell/deliver the asset has a short forward position and is called the short.
We will illustrate the basic forward contract mechanics through an example based on the purchase and sale of a Treasury bill. Note that while forward contracts on T bills are usually quoted in terms of a discount percentage from face value, we use dollar prices here to make the example easy to follow.
Consider a contract under which Party A agrees to buy a $1,000 face value 90 day Treasury bill from Party B 30 days from now at a price of $990. Party A is the long and Party B is the short. Both parties have removed uncertainty about the price they will pay or receive for the T bill at the future date.
If 30 days from now T bills are trading at $992, the short must deliver the T bill to the long in exchange for a $990 payment. If T bills are trading at $988 on the future date, the long must purchase the T bill from the short for $990, the contract price.
Typically, no money changes hands at the inception of the contract, unlike futures contracts in which each party posts an initial deposit called the margin as a guarantee of performance.
At any point in time, including the settlement date, the party to the forward contract with the negative value will owe money to the other side. The other side of the contract will have a positive value of equal amount. Following this example, if the T bill price is $992 at the (future) settlement date, and the short does not deliver the T bill for $990 as promised, the short has defaulted.
Forward Contract Price Determination
The No Arbitrage Principle
The price of a forward contract is not the price to purchase the contract; in fact, the parties to a forward contract typically pay nothing to enter into the contract at its inception. Here, price refers to the forward price of the underlying. This price may be a U.S. dollar or euro price, but it is often expressed as an interest rate or currency exchange rate.
For T bills, the price will be expressed as an annualized percentage discount from face value; for coupon bonds, it will usually be expressed as a yield to maturity; for the implicit loan in a forward rate agreement (FRA), it will be expressed as an annualized MRR; and for a currency forward, it is expressed as an exchange rate between the two currencies involved. However it is expressed, this rate, yield, discount, or dollar amount is the forward price in the contract.
The price that we wish to determine is the forward price that makes the values of both the long and the short positions zero at contract initiation. We will use the no arbitrage principle: there should be no riskless profit to be gained by a combination of a forward contract position with positions in other assets. This principle assumes that (1) transactions costs are zero, (2) there are no restrictions on short sales or on the use of short sale proceeds, and (3) both borrowing and lending can be done in unlimited amounts at the risk free rate of interest.
This concept is so important that we’ll express it in a formula:
A Simple Version of the Cost of Carry Model
In order to explain the no arbitrage condition as it applies to the determination of forward prices, we will first consider a forward contract on an asset that costs nothing to store and makes no payments to its owner over the life of the forward contract. A zero coupon (pure discount) bond meets these criteria. Unlike gold or wheat, it has no storage costs; unlike stocks, there are no dividend payments to consider; and unlike coupon bonds, it makes no periodic interest payments.
The general form for the calculation of the forward contract price can be stated as follows:
| Symbol | Meaning |
|---|---|
| Forward price | |
| Spot price at inception of the contract, | |
| Annual risk free rate | |
| Forward contract term in years |
Consider a 3 month forward contract on a zero coupon bond with a face value of $1,000 that is currently quoted at $500, and suppose that the annual risk free rate is 6%. Determine the price of the forward contract under the no arbitrage principle.
Solution
Now, let’s explore in more detail why $507.34 is the no arbitrage price of the forward contract.
Cash and Carry Arbitrage When the Forward Contract Is Overpriced
Suppose the forward contract is actually trading at $510 rather than the no arbitrage price of $507.34. A short position in the forward contract requires the delivery of this bond three months from now. The arbitrage that we examine in this case amounts to borrowing $500 at the risk free rate of 6%, buying the bond for $500, and simultaneously taking the short position in the forward contract on the zero coupon bond so that we are obligated to deliver the bond at the expiration of the contract for the forward price of $510.
At the settlement date, we can satisfy our obligation under the terms of the forward contract by delivering the zero coupon bond for a payment of $510, regardless of its market value at that time. We will use the $510 payment we receive at settlement from the forward contract (the forward contract price) to repay the $500 loan. The total amount to repay the loan, since the term of the loan is three months, is:
The payment of $510 we receive when we deliver the bond at the forward price is greater than our loan payoff of $507.34, and we will have earned an arbitrage profit of $510 − $507.34 = $2.66. Notice that this is equal to the difference between the actual forward price and the no arbitrage forward price. The transactions are illustrated in Figure 28.1.
Figure 28.1: Cash and Carry Arbitrage When Forward Is Overpriced
| Today | Cash flow | Three months from today | Cash flow |
|---|---|---|---|
| Spot price of bond | $500 | Forward price | $510 |
| Short forward | $0 | Settle short position by delivering bond | +$510.00 |
| Buy bond | −$500 | ||
| Borrow at 6% | +$500 | Repay loan | −$507.34 |
| Total cash flow | $0 | Total cash flow, arbitrage profit | +$2.66 |
Forward overpriced: borrow money ⇒ buy (go long) the spot asset ⇒ go short the asset in the forward market
So far, the underlying instrument was a zero coupon bond, and hence had no associated cash flows during the forward contract period. If the holder of the underlying receives cash flows (as coupon or dividends), the present value of those cash flows will be subtracted from the spot price of the underlying. We will see this later in this reading.
Reverse Cash and Carry Arbitrage When the Forward Contract Is Underpriced
Suppose the forward contract is actually trading at $502 instead of the no arbitrage price of $507.34. We reverse the arbitrage trades from the previous case and generate an arbitrage profit as follows: We sell the bond short today for $500 and simultaneously take the long position in the forward contract, which obligates us to purchase the bond in 90 days at the forward price of $502. We invest the $500 proceeds from the short sale at the 6% annual rate for three months.
In this case, at the settlement date, we receive the investment proceeds of $507.34, accept delivery of the bond in return for a payment of $502, and close out our short position by delivering the bond we just purchased at the forward price.
The payment of $502 we make as the long position in the contract is less than the investment proceeds of $507.34, and we have earned an arbitrage profit of $507.34 − $502 = $5.34. The transactions are illustrated in Figure 28.2.
Figure 28.2: Reverse Cash and Carry Arbitrage When Forward Is Underpriced
| Today | Cash flow | Three months from today | Cash flow |
|---|---|---|---|
| Spot price of bond | $500 | Forward price | $502 |
| Long forward | $0 | Settle long position by buying bond | −$502.00 |
| Short sell bond | +$500 | Deliver bond to close short position | $0.00 |
| Invest short sale proceeds at 6% | −$500 | Receive investment proceeds | +$507.34 |
| Total cash flow | $0 | Total cash flow, arbitrage profit | +$5.34 |
In this case, because the forward contract is underpriced, the trades are reversed from cash and carry arbitrage. To set up the reverse cash and carry arbitrage:
From the calculations just listed, we can see that the no arbitrage forward price that yields a zero value for both the long and short positions in the forward contract at inception is the no arbitrage price of $507.34.