# Forward Futures

of 8

## Content

Contents
[hide]
• • • •

• • •

1 Payoffs 2 How a forward contract works 3 Example of how forward prices should be agreed upon 4 Spot - forward parity o 4.1 Investment assets o 4.2 Consumption assets o 4.3 Cost of carry 5 Relationship between the forward price and the expected future spot price 6 Rational pricing o 6.1 Extensions to the forward pricing formula 7 Theories of why a forward contract exists

• • • •

 Payoffs
The value of a forward position at maturity depends on the relationship between the delivery price (K) and the underlying price (ST) at that time.
• •

For a long position this payoff is: fT = ST − K For a short position, it is: fT = K − ST

 How a forward contract works

actually need Canadian dollars at a future date such as to pay a debt owed that is denominated in Canadian dollars. Other times, the party opening a forward does so, not because they need Canadian dollars nor because they are hedging currency risk, but because they are speculating on the currency, expecting the exchange rate to move favorably to generate a gain on closing the contract. In a currency forward, the notional amounts of currencies are specified (ex: a contract to buy \$100 million Canadian dollars equivalent to, say \$114.4 million USD at the current rate—these two amounts are called the notional amount(s)). While the notional amount or reference amount may be a large number, the cost or margin requirement to command or open such a contract is considerably less than that amount, which refers to the leverage created, which is typical in derivative contracts.

 Example of how forward prices should be agreed upon
Continuing on the example above, suppose now that the initial price of Andy's house is \$100,000 and that Bob enters into a forward contract to buy the house one year from today. But since Andy knows that he can immediately sell for \$100,000 and place the proceeds in the bank, he wants to be compensated for the delayed sale. Suppose that the risk free rate of return R (the bank rate) for one year is 4%. Then the money in the bank would grow to \$104,000, risk free. So Andy would want at least \$104,000 one year from now for the contract to be worthwhile for him - the opportunity cost will be covered.

 Spot - forward parity
Main article: Forward price See also: Cost of carry and convenience yield Spot-forward parity provides the link between the spot market and the forward market. It describes the relationship between the spot and forward price of the underlying asset in a forward contract. While the overall effect can be described as the cost of carry, this effect can be broken down into different components, specifically whether the asset:
• • •

pays income, and if so whether this is on a discrete or continuous basis incurs storage costs is regarded as o an investment asset, i.e. an asset held primarily for investment purposes (e.g. gold, financial securities); o or a consumption asset, i.e. an asset held primarily for consumption (e.g. oil, iron ore etc.)

 Investment assets

For an asset that provides no income, the relationship between the current forward (F0) and spot (S0) prices is F0 = S0erT where r is the continuously compounded risk free rate of return, and T is the time to maturity. The intuition behind this result is that given you want to own the asset at time T, there should be no difference in a perfect capital market between buying the asset today and holding it and buying the forward contract and taking delivery. Thus, both approaches must cost the same in present value terms. For an arbitrage proof of why this is the case, see Rational pricing below. For an asset that pays known income, the relationship becomes:
• •

Discrete: F0 = (S0 − I)erT Continuous: F0 = S0e(r − q)T

where is the present value of the discrete income at time t1 < T, and q%p.a. is the continuous dividend yield over the life of the contract. The intuition is that when an asset pays income, there is a benefit to holding the asset rather than the forward because you get to receive this income. Hence the income (I or q) must be subtracted to reflect this benefit. An example of an asset which pays discrete income might be a stock, and example of an asset which pays a continuous yield might be a foreign currency or a stock index. For investment assets which are commodities, such as gold and silver, storage costs must also be considered. Storage costs can be treated as 'negative income', and like income can be discrete or continuous. Hence with storage costs, the relationship becomes:
• •

Discrete: F0 = (S0 + U)erT Continuous: F0 = S0e(r + u)T

where is the present value of the discrete storage cost at time , and u%p.a. is the storage cost where it is proportional to the price of the commodity, and is hence a 'negative yield'. The intuition here is that because storage costs make the final price higher, we have to add them to the spot price.

 Consumption assets
Consumption assets are typically raw material commodities which are used as a source of energy or in a production process, for example crude oil or iron ore. Users of these consumption commodities may feel that there is a benefit from physically holding the asset in inventory as opposed to holding a forward on the asset. These benefits include the ability to profit from temporary shortages and the ability to keep a production process

running,[1] and are referred to as the convenience yield. Thus, for consumption assets, the spot-forward relationship is:
• •

Discrete storage costs: F0 = (S0 + U)e(r − y)T Continuous storage costs: F0 = S0e(r + u − y)T

where y%p.a. is the convenience yield over the life of the contract. Since the convenience yield provides a benefit to the holder of the asset but not the holder of the forward, it can be modelled as a type of 'dividend yield'. However, it is important to note that the convenience yield is a non cash item, but rather reflects the market's expectations concerning future availability of the commodity. If users have low inventories of the commodity, this implies a greater chance of shortage, which means a higher convenience yield. The opposite is true when high inventories exist.[1]

 Cost of carry
The relationship between the spot and forward price of an asset reflects the net cost of holding (or carrying) that asset relative to holding the forward. Thus, all of the costs and benefits above can be summarised as the cost of carry, c. Hence,
• •

Discrete: F0 = (S0 + U − I)e(r − y)T Continuous: F0 = S0ecT, where c = r − q + u − y.

 Relationship between the forward price and the expected future spot price
Main articles: Normal backwardation and Contango

The market's opinion about what the spot price of an asset will be in the future is the expected future spot price.[1] Hence, a key question is whether or not the current forward price actually predicts the respective spot price in the future. There are a number of different hypotheses which try to explain the relationship between the current forward price, F0 and the expected future spot price, E(ST). The economists John Maynard Keynes and John Hicks argued that in general, the natural hedgers of a commodity are those who wish to sell the commodity at a future point in time.[3][4] Thus, hedgers will collectively hold a net short position in the forward market. The other side of these contracts are held by speculators, who must therefore hold a net long position. Hedgers are interested in reducing risk, and thus will accept losing money on their forward contracts. Speculators on the other hand, are interested in making a profit, and will hence only enter the contracts if they expect to make money. Thus, if speculators are holding a net long position, it must be the case that the expected future spot price is greater than the forward price. In other words, the expected payoff to the speculator at maturity is: E(ST − K) = E(ST) − K, where K is the delivery price at maturity Thus, if the speculators expect to profit, E(ST) − K > 0 E(ST) > K E(ST) > F0, as K = F0 when they enter the contract

This market situation, where E(ST) > F0, is referred to as normal backwardation. Since forward/futures prices converge with the spot price at maturity (see Basis), normal backwardation implies that futures prices for a certain maturity are increasing over time. The opposite situation, where E(ST) < F0, is referred to as contango. Likewise, contango implies that futures prices for a certain maturity are falling over time.[5]

 Rational pricing
If St is the spot price of an asset at time t, and r is the continuously compounded rate, then the forward price at a future time T must satisfy Ft,T = Ster(T − t). To prove this, suppose not. Then we have two possible cases. Case 1: Suppose that Ft,T > Ster(T − t). Then an investor can execute the following trades at time t: 1. go to the bank and get a loan with amount St at the continuously compounded rate r; 2. with this money from the bank, buy one unit of stock for St; 3. enter into one short forward contract costing 0. A short forward contract means that the investor owes the counterparty the stock at time T. The initial cost of the trades at the initial time sum to zero. At time T the investor can reverse the trades that were executed at time t. Specifically, and mirroring the trades 1., 2. and 3. the investor 1. ' repays the loan to the bank. The inflow to the investor is − Ster(T − t); 2. ' settles the short forward contract by selling the stock for Ft,T. The cash inflow to the investor is now Ft,T because the buyer receives ST from the investor. The sum of the inflows in 1.' and 2.' equals Ft,T − Ster(T − t), which by hypothesis, is positive. This is an arbitrage profit. Consequently, and assuming that the non-arbitrage condition holds, we have a contradiction. This is called a cash and carry arbitrage because you "carry" the stock until maturity. Case 2: Suppose that Ft,T < Ster(T − t). Then an investor can do the reverse of what he has done above in case 1. But if you look at the convenience yield page, you will see that if there are finite stocks/inventory, the reverse cash and carry arbitrage is not always possible. It would depend on the elasticity of demand for forward contracts and such like.

 Extensions to the forward pricing formula
Suppose that FVT(X) is the time value of cash flows X at the contract expiration time T. The forward price is then given by the formula:

The cash flows can be in the form of dividends from the asset, or costs of maintaining the asset. If these price relationships do not hold, there is an arbitrage opportunity for a riskless profit similar to that discussed above. One implication of this is that the presence of a forward market will force spot prices to reflect current expectations of future prices. As a result, the forward price for nonperishable commodities, securities or currency is no more a predictor of future price than the spot price is - the relationship between forward and spot prices is driven by interest rates. For perishable commodities, arbitrage does not have this The above forward pricing formula can also be written as: Ft,T = (St − It)er(T − t) Where It is the time t value of all cash flows over the life of the contract. For more details about pricing, see forward price.

 Theories of why a forward contract exists
Allaz and Vila (1993) suggest that there is also a strategic reason (in an imperfect competitive environment) for the existence of forward trading, that is, forward trading can be used even in a world without uncertainty. This is due to firms having Stackelberg incentives to anticipate their production through forward contracts

## Recommended

#### Futures

Or use your account on DocShare.tips

Hide