FORWARD AND FUTURES CONTRACTS

6.2 FORWARD AND FUTURES CONTRACTS

A forward contract is an agreement to buy or sell an asset at a fixed date for a price determined today. The buyer agrees to buy the asset at the price F and sell it at the market price at maturity for a payoff S − F. The seller takes the opposite position and sells at the market price F and buys the asset at the market price S at maturity.

Forward contracts are thus an agreement between two parties or traders regard- ing the price, the delivery price, of a stock, a commodity or any another asset,

OPTIONS AND DERIVATIVES FINANCE MATHEMATICS

F (1) − S H

F (1)

F (1) −S L

Figure 6.5 Forward contract valuation.

an obligation to be maintained by the buyer and the seller at maturity. The party that has agreed to buy the forward contract is said to assume a long position while the party that agrees to sell is said to assume the short position. Such contracts allow for the parties to exchange the price risk at maturity. For example, a wheat farmer may be exposed to a fall of the wheat price when he brings it to market. He can then enter in a forward contract to sell his wheat today at the fixed price F. At maturity, he may sell wheat at a predetermined price and buy it at the spot rate S from the buyer (say, the baker) of the forward for a payoff of (F − S). The buyer (baker) takes the opposite position for a payoff of (S − F). Both sell and buy are in the market and their position is [(F − S) + S = F] and [(S − F) − S = F] respectively. The parties have therefore perfectly eliminated their wheat price risk as their payoffs are determined at the initiation of the contract. In this example, we evolved into a world where risk can be completely shifted away, which is also the risk-neutral world that conveniently discounts risky payoffs at the risk-free rate (under an appropriately defined probability measure). This transformation to the ‘risk-neutral world’ breaks down when a seller cannot find a buyer with the exact opposite hedging needs and vice versa. In this case, speculators are needed to take on the risk and a risk neutral world will no longer exist. Depending on whether excess hedging is in long or short forwards, the pressure will be upward or downward compared to the risk-neutral price.

To calculate the forward price at times t = 1 and t = 2, say F(1) and F(2) we proceed as follows. Consider the first period only, at which the gain can be either

F (1) − S H in case of a price increase or F(1) − S L in case of a price decrease (see Figure 6.5). Initially nothing is spent and therefore, initially we also get nothing. At present it is thus worth nothing. Assuming no arbitrage (otherwise we would

not be able to use the risk-neutral probability), and proceeding as in the previous section, we have:

0= ∗ [p ∗ (F(1) − S H )+q (F(1) − S L )]; p +q =1 1+R f which is an one equation in one unknown and where R f is an effective risk-free

annual rate. The forward price F(1) resulting from the solution of the equation above is therefore:

F (1) = [ p ∗ S H +q ∗ S L ] = S(1 + R f ) In other words, the one period forward price equals the discounted current spot

price. For two periods we note equivalently that when the spot price is S H or S L ,

143 then (from period 1 to 2):

FORWARD AND FUTURES CONTRACTS

∗ S HH +q ∗ S HL = (1 + R f )S H w.p. p ∗

F ˜ (2) = p ∗ S HL +q ∗ S LL = (1 + R f )S L w.p. q ∗

As a result, F(2) = E ∗ F ˜ (2) = p ∗ (1 + R f )S H +q ∗ (1 + R f )S L and therefore

F (2) = (1 + R f ) 2 S and obviously:

F (n) = (1 + R f ) n S

This means that the n periods forward price equals the n periods discounted current spot price (see also Figure 6.4). Of course, using the risk-neutral reasoning, since there is no initial expenditure at the time the forward contract is signed, while at time t, the profit realized equals the difference between the current price and the forward (agreed) on price at time zero which we write by F(n), we have:

E [S n − F(n)] and F (n) = E [S n ] Since under risk-neutral pricing,

we obtain at last the general forward price:

F (n) = S n 0 (1 + R f )

In practice, there may be some problems because decision makers may use forward prices to revalue the spot price. Feedback between these markets can induce an opportunity for arbitrage. Further, it is also necessary to remember that we have assumed a risk-neutral world. As a result, when traders use historical data, there may again be some problems, leading to a potential for arbitrage since the fundamental assumption of rational expectations is violated. For example, if the spot price of silver is $50, while the delivery price is $53 with maturity in one year, while interest rates equal 0.08, then the no arbitrage price is: 50(1 + 0.08) = $54. This provides an arbitrage opportunity since in one year there is an arbitrage profit of $1(=54 − 53) that can be realized.

A futures contract differs from a forward contract in that it is standardized, openly traded and marked to market. Marking to market involves adjusting an investor’s initial margin deposit by the change in the futures contract price each day. If the investor’s margin account falls below the maintenance margin, the trader asks the investor to fill the margin account back to the initial margin, posted in the form of interest-bearing T-bonds.

A futures price is determined as follows. The futures price one period hence

F (0, 1) at time t = 1 is set equal to the forward price for that time, since no cost is incurred. In other words, we have, F(0, 1) = F(1). Now consider the futures

price in two periods, F(0, 2). If the spot price increases to S H , the futures price turns out to equal the one-period forward price, or F H (1) (since only one more period is left till the exercise time). Similarly, if the spot price decreases to S L ,

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Marking to market

[ S HH − F H () 1 ]

[ F H () 1 − F (,) 02 ]

[ S HL − F H () 1 ]

[ F L () 1 − F (,) 02 ]

[ S HL − F L () 1 ] [ S LL − F L () 1 ]

Figure 6.6 Future price valuation.

the future price is now F L (1). As a result, cash flow payments at the first and second periods are given by Figure 6.6. Initially, the value of these flows is worth nothing, since nothing is spent and nothing is gained . Thus, an expectation of futures flows is worth nothing today. That is,

1 ∗ (F H 0= ∗ (1) − F(0, 2)) + q (F L (1) − F(0, 2)) 1+R f

1 p ∗2 (S HH −F H (1)) + p ∗ q ∗ (S HL −F H (1)) + + (1 + R f ) 2 p ∗ q ∗ (S HL −F L (1)) + q ∗2 (S LL −F L (1))

which is one equation and three unknowns. However, noting that for the one- period futures (forward) price, we have:

F H (1) = (1 + R f )S H , F L (1) = (1 + R f )S L Inserting these results into our equation, we obtain the futures price: F(0, 2) =

(1 + R f ) 2 S 0 which is equal to the forward price. This is the case, however, because the discount interest rate is deterministic. In a stochastic interest rate framework,

this would not be the case. A generalization to n periods yields:

F (0, n) = (1 + R f ) n S 0

Futures contracts are stated often in terms of a basis, measuring the difference between the spot and the futures price. The basis may be mis-priced, however, because of mismatching of assets (cross-hedged), because of maturity (forward versus futures) and the quality of related assets (options). There are some funda- mental differences between forward and futures contracts that we summarize in Table 6.1. These relate to the hedging quality of these financial products, their barriers to entry, etc. Further, although under risk-neutral pricing they have the same price, in practice (when interest rates are stochastic as stated above) they can differ appreciably. In many cases, futures contracts are preferred to forward contracts simply because they are more liquid and thereby more ‘tradable’.

Example

We compare the consequences of forward and futures contracts on a volume of 100 Dax shares each worth 77E over say five periods. We obtained the following

145 Table 6.1 Forward and futures contracts: contrasts.

RISK - NEUTRAL PROBABILITIES AGAIN

Exchange markets Standard contract

OTC (Private)

No

Yes

Barrier to entry

Substantial

Weak

Margin system Daily controls

Long–short Hedge quality

Inverse contract

Best

Problematic

results, pointing to differences in cash flow (Table 6.2). Calculations are performed as follows. The cash flows associated with a forward contract of five periods (denoted by ∗ ) and associated with a futures contract at period 2 (denoted by ∗∗ ) are given in Table 6.2.

Further, note that the sum of payments of mark to market are equal to the sum of payments of the forward since their initial prices (investment) were the same.

Table 6.2

T 1 2 3 4 5 DAX

7650 7730 Price forward

— 7730 Price future

7675 7730 Cash flow forward

— −7000(*) Cash flow future

−5500 +5500 (*) = (F[1;T] – F[0;T]) * volume = (7730 − 7800) * 100 = 7000

Say that K is the forward delivery price with maturity T while F is the cur- rent forward price. The value of the long forward contract is then equal to the

present value of their difference at the risk-free rate R f , or P L = (F − K ) e −R f T . Similarly, the value of the short forward contract is P S

f = −P T L = (K − F) e −R .

Example: Futures on currencies

Let S be the dollar value of a euro and let (R $ , R E ) be the risk-free rate of the local (dollar) and the foreign currency (euro). Then the relative rate is (R $ −R E )

(R and the future euro price T periods hence is: F = Se $ −R E )T .