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### Subject 1. Generic pricing and valuation of a forward contract

A forward contract price is the fixed price or rate at which the transaction scheduled to occur at expiration will take place. This price is agreed on the contract initiation date, and is commonly called the forward price or forward rate. The price here is different from value, which is what you can sell something for or what you must pay to acquire something.

• Pricing means to determine the forward price or forward rate.
• Valuation is the process of determining the value of an asset or service. It means to determine the amount of money one would need to pay or would expect to receive to engage in the transaction. For example, if one already held a position, valuation would mean to determine the amount of money one would either pay to pay or expect to receive in order to get out of the position.

There are two ways to acquire an asset for use in the future.

• First, the asset could be purchased at the spot price today and stored until it is needed.
• Second, a long position in futures could be established today and funds could be set aside in an interest-bearing account to acquire the asset in the future.
• The two strategies must have the same costs.

A simple example: an investor owns a stock that has a current value of \$130. Assume that the stock will pay a \$2 dividend at the end of the year. Assume that a forward contract that calls for delivery in one year is available for \$134.5. The investor hedges by selling or shorting one contract.

Here are some possible outcomes: A perfect hedge should return a riskless rate of return. With a perfect hedge (like the above example), the forward payoff is certain - there is no risk.

There is a predictable relationship between the current price of the underlying asset, including the costs of holding and storing it, and the forward price. This relationship can be used to develop forward pricing relationship.

Let's first consider value at expiration of a forward contract established at time 0:

VT(0, T) = ST - F(0, T)

Where:

• VT(0, T) is the value of the forward contract at expiration (time T).
• F(0,T) is the price of a forward contract initiated at time 0 and expiring at time T.
• ST is the price of the underlying asset in the spot market at expiration (time T).

Example 1

Suppose the forward price established at time 0 is \$30. Now at expiration, the spot price is \$36. The contract value must be \$6. If this equation does not hold, an arbitrage profit can be easily made.

• If the contract were \$8, then the long would be able to sell the contract to someone for \$8. Would someone be paying the long \$8 to obtain the obligation of buying a \$36 asset for \$30? Obviously not.
• The opposite is also true.

Note that the value of a forward contract can also be interpreted as its profit, the difference between what the long pays for the underlying asset, F(0, T), and what the long receives, the asset price ST.

Now the question is: how do we determine F(0, T) at time 0?

The value of a forward contract at initiation (t = 0) is:

V0(0, T) = S0 - F(0, T)/(1 + r)T

Where:

• V0(0, T) is the value of the forward contract at initiation (time 0).
• S0 is the price of the underlying asset in the spot market at initiation (time 0).
• F(0,T) is the price of a forward contract initiated at time 0 and expiring at time T.
• r is the risk-free rate.

Customarily, no money changes hands at initiation so V0(0, T) is set to zero. Thus,

F(0, T) = S0 (1 + r)T

This formula can be interpreted as saying that the forward price is the spot price compounded at the risk-free interest rate. Why?

• At time 0, we:

• buy the underlying asset at S0.
• sell a forward contract at F(0, T).

The total outlay is the spot price of the asset.
• We hold asset and lose interest on the money.
• At time T, we:

• deliver the asset.
• Receive the forward price for a payoff of F(0, T).

The transaction is risk-free, and should be equivalent to investing S0 dollars in a risk-free asset that pays F(0, T) at time T. Thus, the amount received at T must be the future value of the initial outlay invested at the risk-free rate: F(0, T) = S0 (1 + r)T.

For example, suppose the spot price is \$100, and the risk-free rate is 5%, and the contract is for one year. The forward price would be F(0, 1) = 100 (1.05)1 = 105.

In general, we can always say that at any arbitrary point (time t) during the contract's life, the forward contract value is the asset price at time t minus the present value of the exercise price:

Vt(0, T) = St - F(0, T)/(1 + r)T-t

Why? If we went long a forward contract at time 0 and we are now at time t prior to expiration, we hold a claim on the asset at expiration and are obligated to pay the forward price at expiration.

• The claim on the asset is worth its current spot price.
• The obligation to pay the forward price at expiration is worth the negative of its present value.
• The value of the forward contract, therefore, is the current spot price minus the forward price discounted from expiration back to the present.

Note two special circumstances:

• If t = 0, Vt(0, T) = V0(0, T) = S0 - F(0, T)/(1 + r)T = 0.
• If t = T, Vt(0, T) = VT(0, T) = ST - F(0, T)/(1 + r)0 = ST - F(0, T).

Example 2

A three-year forward contract was established with a price of \$75. Now, two years later (t = 2), the spot price is \$80 and the risk-free rate is 4%. The value of the forward contract is Vt(0, T) = V2(0, 3) = 80 - 75/(1.04)3-2 = 7.88.

#### Practice Question 1

John has an asset which is worth \$110. He plans to sell it in 8 months. The risk-free interest rate is 4.5%. What should the forward price be?

Correct Answer: T = 8/12 = 0.667.
S0 = 110.
R = 0.045.
F(0, T) = 110 (1.045)0.667 = 113.28.

#### Practice Question 2

John has an asset which is worth \$110. He plans to sell it in 8 months. The risk-free interest rate is 4.5%. As found in basic question 1, the forward contract should be selling at \$113.28. If the counterparty is willing to engage in such a contract at a forward price of \$120, what should John do to take advantage of the situation? What is the annualized risk-free rate of return?

Correct Answer: As \$113.28 is smaller than \$120, clearly this overpriced contract should be sold. John should hold the asset and sell the forward contract. At contract expiration date he will deliver the asset and receive \$120 for it. The rate of return will be (120/113.28) - 1 = 5.93%. The annualized risk-free rate of return is (1.0593)12/8 - 1 = 9.03%. John's position is not only perfectly hedged but also earns an arbitrage profit.

#### Practice Question 3

John has an asset which is worth \$110. He plans to sell it in 8 months. The risk-free interest rate is 4.5%. Suppose the forward contract is entered into at \$113.28 (as computed in question 1). Four months later, the price of the asset is \$105. What's the market value of the forward contract at this point in time from John's perspective?

• T - t = 8/12 - 4/12 = 4/12.
• St = 105.
• F(0, T) = 113.28.
• Vt(0, T) = V4/12(0, 8/12) = 105 - 113.28/(1.045)4/12 = -6.63.
As John is "short", the value of the contract to him in this problem is positive \$6.63.

#### Practice Question 4

John has an asset which is worth \$110. He plans to sell it in 8 months. The risk-free interest rate is 4.5%. Suppose the forward contract is entered into at \$113.28, and the price of the underlying asset is \$109 at expiration. What is the rate of return for John?

• ST = 109.
• VT(0, T) = V9/12(0, 9/12) = 109 - 113.28 = -\$4.28.
As John is "short", the value of the contract to him is positive -\$4.28. However, he incurred a loss on the asset of 110 - 109 = \$1. The net gain is \$3.28. The rate of return of for the 8-month period is (3.28/110) - 1 = 2.98%. When annualized, the rate of return equals (1.0298)12/8 - 1 = 4.5%.

Nor surprisingly, this rate is the annual risk-free rate. The transaction was executed at the no-arbitrage forward price of \$113.28, and therefore it would be impossible for John to earn a return higher or lower than the risk-free rate.

#### Practice Question 5

What must the 3-month futures price be on a stock that has a dividend yield of 2% when the current price of the stock is \$50 and the risk-free rate is 5%?

A. \$51.5.
B. \$50.
C. \$50.37.

The 3-month futures price should be \$50 (1 + 0.05 - 0.02)0.25 = 50.37.

#### Practice Question 6

The price of a stock is \$65 now. A forward contract on the stock that expires in 6 months is currently priced at \$70. The annual risk-free rate is 4.35%. Suppose the stock does not pay any dividend within the next 6 months. If you enter into such a forward contract with a dealer to sell the stock in 6 months at \$70:

A. no money changes hands.
B. you should pay the dealer \$3.53.
C. the dealer should pay you \$3.53.

S0 = \$65
F(0, T) = \$70
T = 0.5
V0(0, T) = 65 - 70/1.04350.5 = -\$3.53.

Because the contract is negative, the payment is made by the short to the long. In fact, this is an off-market forward contract.

#### Practice Question 7

A stock is selling for \$50 now. The risk-free rate is 4.5%. A dealer offers to enter into a 2-month forward contract at \$49.9. Suppose this is a "typical" forward contract so no money changes hand at initiation. Also suppose there is no dividend involved. Can you earn an arbitrage profit? If so, how?

A. The stock is priced "correctly" so there is no arbitrage opportunity.
B. Short sell the stock, invest the proceeds at 4.5%, and enter into a forward contract to buy the stock.
C. Borrow money to buy the stock, and enter into a forward contract to sell the stock.