Pricing of a forward contract: The pricing of a forward contract is quite simple. Let's consider a stock trading at S0 that doesn't pay any dividend and has no benefits and costs associated with it. Now, we want to buy a forward contract on this stock that is going to expire after T years, and the annual risk-free rate is r. It seems like that to get a forward price (i.e. the price we agreed to pay at a future date for the underlying) we need to forecast the price of the stock after T years. This forecast is known as the expected spot price at expiration E(ST). It looks like that we need to discount this expected spot price at expiration with the discount rate to get the price of the forward contract. However, that's not how derivatives are priced. They are priced using arbitrage, and a risk-free position is constructed by taking a position in the forward contract and the underlying.
To earn the risk-free rate on the underlying, we should get S0*(1+r)T at the expiration, and that is the price of the forward contract. The pricing is as simple as that. So, we can say that the price of the forward contract is the spot price compounded at the risk-free rate over the life of the contract.
F0(T) = S0*(1+r)T
Now, we can include the benefits (γ) and costs (θ) of the asset. The costs will increase the forward price because the cost is incurred by the underlying holder and he would charge the future value of the costs in the forward contract. For example, if you want to sell some commodity as sugar after six months and you purchase that today and store that commodity and incur a cost for that. Obviously, you would include that cost in the forward price after six months. Similarly, the benefits will reduce the forward price.
F0(T) = S0*(1+r)T - γ*(1+r)T + θ*(1+r)T or
F0(T) = S0*(1+r)T - (γ -θ)*(1+r)T
Thus, the forward price of an asset is the spot price compounded at a risk-free rate over the life of the contract minus the future value of benefits plus the future value of costs.
The forward contract initiated at the expiration will be priced equal to the spot price at the expiration. We can get that by putting the value of T equal to zero in the above equation because at expiry; there will be no time left to expiration.
Value of a forward contract: We take a position in a forward contract at a forward price. Now, the underlying price keeps on changing and with that the value of the forward contract also changes. The value of a forward contract at any time t is simply the present value of the difference between the forward price if the contract is initiated at time t and the forward price paid by us.
Suppose if we enter into a forward contract a price of F0(T). Now, t years have passed, and the underlying is trading at a spot price of St. The total time left to expiration is (T-t) years. The forward contract initiated at time t will have a forward price of Ft(T) = St*(1+r)(T-t). Now, if we had taken a position today, we would have agreed at this price, but we had already taken the position at a price of F0(T). So, the total difference between the two prices is the profit/loss that we would have at the expiration. Profit/loss at expiration = Ft(T) - F0(T) = St*(1+r)(T-t) - F0(T). The value of the contract at time t would be simply the present value of this differential.
Vt(T) = [St*(1+r)(T-t) - F0(T)]/(1+r)(T-t) = St - F0(T)*(1+r)-(T-t)
We can also say that the value of a forward contract is the spot price of the underlying asset minus the present value of the forward price.
Since at the initiation of the contract, the differential will be zero, the value of the forward contract will be zero. That's why no parties pay any amount of money to each other at the contract initiation.
The value of the forward contract at the expiry will be simply the difference between the spot price of the underlying asset and the forward price at which we entered into the contract.
Please note that when we say the value of the forward contract, we are implying the value of the forward contract for the long party. For the short party, the value of the contract will be simply -1 times the value of the contract to the long party. It is a zero-sum game i.e. the sum of the value to the long party and the short party is always equal to zero.
We can make an adjustment to the value of the forward contract for interim benefits and costs.
Vt(T) = St - (γ -θ)*(1+r)t - F0(T)*(1+r)-(T-t)
The above equation can be easily worked out and kept as an exercise for the candidate. All you need to do is to take the present value of the differential between the forward price for the contract initiated at time t and the forward price initiated at time 0.
Check your concepts:
(58.7) Which of the following best describes the value of a short forward position at expiration?
(a) Price of underlying minus the forward price
(b) Forward price minus the price of underlying
(c) Forward price plus the price of underlying
(58.8) What will be the impact on the forward price if the risk-averseness of the investor decreases?
(a) Increase in forward price
(b) Decrease in forward price
(c) No change in forward price
Solutions:
(58.7) Correct Answer is B: For a long position in the forward contract, the value of forward contract equals the price of the underlying minus the forward price. For the short position, the value is minus of the value for the long position. The value for short forward position equals forward price minus the price of the underlying.
(58.8) Correct Answer is C: The forward price is based on the assumption of risk neutrality. Hence, the risk-averseness of an investor will have no impact on the forward price.
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