There are two ways to acquire an asset for use in the future.
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:
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.
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:
Customarily, no money changes hands at initiation so V0(0, T) is set to zero. Thus,
This formula can be interpreted as saying that the forward price is the spot price compounded at the risk-free interest rate. Why?
The total outlay is the spot price of the asset.
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:
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.
Note two special circumstances:
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.
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.
The 3-month futures price should be $50 (1 + 0.05 - 0.02)0.25 = 50.37.
A. no money changes hands.
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.
A. The stock is priced "correctly" so there is no arbitrage opportunity.
S0 = $50
F(0, T) = $49.9
T = 2/12
V0(0, T) = 50 - 49.9/1.0452/12 = -$0.46
As the value is not zero there is an arbitrage opportunity. This should be obvious without any calculations as the forward price is lower than the spot price and there is no dividend involved.
You should short sell the stock for $50, and invest at $4.5% for two months. At the end of two months you will have $50.37. You will then execute the forward contract by paying $49.9 for the stock. The risk-free profit is $0.47.