3.3.1. Notations

Unless otherwise specified, 0 designates the point at which a transaction is initiated, T designates the maturity date of the futures contract under consideration and t designates a point in the future that precedes the maturity date: 0 < t < T. However, it must be noted that if 0 designates the point at which an initial position is taken by an agent, it moves over the course of time. In other cases, 0 designates the point at which we may begin negotiating a contract. In each case, the notations must be specified.

There is no stable set of notations in academic literature, nor in professional literature, and hence the reader must pay attention to context.

A futures contract links a buyer and a seller. The buyer has made the irrevocable commitment to buy, on the date T, an asset that is said to be the underlying entity of the contract. The seller has made the irrevocable commitment to deliver, on the date T, the underlying asset. The price that the buyer pays and at which the seller delivers the asset, at the time T, is fixed when the contract is drawn up at the time 0, predating T. Consequently, 0 then denotes the point in time at which the contract was drawn up. F_0,T designates the prices, stipulated on the date 0, for the delivery of the support on the date of maturity, T. More generally, F_t,T designates the futures price prevailing at the time t, for t ϵ [0,T] . For t ϵ [0,T], C_t represents the spot price of the underlying asset. We use “spot price”, “physical price” or “cash price”. In a very general manner, F_t,T ≠ C_t. The basis is thus defined as: B_t,T ≡ F_t,T − C_t. Apart from a few exceptions, arbitrage operations lead the market to a situation at maturity such that F_T,T = C_T ⇔ B_T,T = 0.

More generally, the basis B_t,T normally tends to zero when t tends toward T, as illustrated in the following graph: the explanation for the change resulting from arbitrage operations will be given later on in the text.

3.3.2. Gains and losses at the maturity of an elementary operation

The vocabulary that is ordinarily used to talk about futures markets may be misleading: at the date 0, the buyer buys nothing and only commits to buying a certain quantity of some entity on the date T and at the price F_0,T. In the same way, the seller sells nothing at the time 0. They only commit to selling a certain quantity of a certain underlying asset on the date T, at the price F_0,T. To avoid ambiguity, we should specify “the operator who has committed to sell” and “the operator who has committed to buy”. However, for convenience’s sake we use the terms “seller” and “buyer”. Readers who approach future markets will be advised to keep this in mind.

3.3.3. Closing a position before maturity

Let us take the example of a long position – that is the buyer’s position – for a contract that came into effect on the date 0 in the past; at the end of the day t, the operator sells a contract whose date of maturity is T at the price F_t,T. This is the same as closing their position, as the operator is simultaneously the buyer and seller for the same date of maturity T. The financial results following this closing of the position are presented in Table 3.4.

This table emphasizes that the position taken from the day t systematically generates two opposite flows, hence a zero sum, which is normal as the operator is simultaneously in the position of the buyer and the seller for the same support, the same price and the same date of maturity. It is then said that the initial long position has been cancelled or closed as the new position – which adds purchase and selling to the same contract – only generates null flows over the interval [t + 1; T]. The initial position taken, preserved over the interval [0; t], has thus produced a result equal to (F_t,T − F_0,T), then 0 + (F_1,T − F_0,T)+ (F_2,T − F_1,T)+ … + (F_{(t − 1),T} − F_{(t − 2),T})+ (F_t,T − F_{(t − 1),T}) = F_t,T − F_0,T.

This table clearly shows one of the key advantages of futures markets: at the time t, earlier than T, any player may liquidate the gains or losses resulting from the position taken earlier at the time t = 0. However, it must be noted that in order to make full use of this advantage, it is essential that the futures market be sufficiently liquid.

3.4.1. An elementary example of arbitrage through replication [BOS 02]

Three assets A, B and C are traded on a financial market. If A is purchased on the date 0, a flow F₁ is generated on the date t = 1 and a flow F₂ is generated on t = 2; if A is sold, these two flows must be paid; the holder of a financial asset B will receive a flow on t = 1 and the holder of C will receive a flow on t = 2. Table 3.5 gives the characteristics of these three assets; P₀ designates the price on the date t = 0:

1. a) the portfolio P_A = {1B ; 11C} replicates the flows resulting from A, as the holder of P_A receives 100 $ on t = 1 and 1,100 $ on t = 2, i.e. exactly what a holder of A would receive;
2. b) the price of P_A is 95 + (11 × 80) = 975 $. The price of P_A is lower than that of A, while both these assets have the same flows; an arbitrageur would make use of this opportunity by buying P_A, the underestimated asset and by selling A, the overestimated asset.

The initial flow F₀ generated by this arbitrage is equal to 25 $; consequently, the portfolio P_A generates a flow F₁ = 100 $, and then a flow F₂ = 1, 100 $, which is refunded to the holder of the asset A, which the arbitrageur has sold.

The series of flows for the arbitrageur is, therefore, {+25 $; 0 $; 0 $}; the opportunity for arbitrage is evident. It is in the arbitrageur’s interest to multiply this operation and they will, thus:

• – massively buy the portfolio P_A (that is the assets B and C);
• – massively sell the title A.

Having done this, the arbitrageur will then cause an increase in the prices of B and C and a reduction in the price of A; at a certain point, the arbitrage opportunity will disappear. This is a peculiarity of arbitrage operations: acting on arbitrage opportunities leads to the re-establishment of an NA situation.

3.4.2. A formal definition of NA [PON 96]

Let us consider an asset A, which will result in the flows F_A(t), in the future (t > 0) ; these flows may be determinist or random. A portfolio composed of m assets is characterized by an m-tuple (n₁, … n_k, … n_m), where n_k designates the number of assets k that make up the portfolio:

• – if n_k > 0, the asset k is held in the portfolio and results in inflows;
• – if n_k < 0, the holder of the portfolio is in the position of a seller for this asset (they hold the short position), which leads to outflows.

The portfolio P_A, characterized by the m-tuple (n₁, …, n_m), duplicates the asset A if:

We can also say that the portfolio P_A synthesizes or replicates the asset A. π(k) denotes the price of the asset k. When P_A = (n₁, … n_k, … n_m) replicates the asset A, the absence of an opportunity for arbitrage (NA) is translated by the equality between the price of any asset A and the price of any replication portfolio P_A:

where π(P_A) is the price of the portfolio P_A and π(A) is the price of the asset A.

3.5.1. An elementary example of hedging

Let us take the example of a milling wheat producer: between the time of sowing, in spring in the year N, and the time of sale, December, this agricultural producer based in Brie is subject to the risk of a drop in prices. There exists a futures market for milling wheat on Euronext Paris: a possible hedge that is very easy to implement consists of the agricultural producer committing to sell the wheat in December of the year N, at a price that is fixed when the contract(s) is/are signed. When this producer sows the wheat, the price of wheat is 200 €/ton; the producer finds this price satisfactory as it allows them to realize a sufficient margin. The risk is that the price may drop. By reading the listings on the future market, they can see that the contract expiring in December is being traded at 220 €/ton (we will see later why the price of the futures contract is assumed to be higher than the spot price). The producer then decides to take the position of the seller at this price; if they must sell 200 tons of wheat, and if each contract is for 50 tons, then they must sell four contracts. Let us recall that the following expression would be preferable in order to avoid all ambiguity: the agricultural producer commits to delivering four times 50 tons of wheat in December. This operation will be carried out through the intermediary of their cooperative.

In the sections that follow, we use a standardized presentation of the results of such an elementary hedge. The tables are constructed for 1 ton of the merchandise; to know the total financial flows, we would need to multiply these by the quantities covered in the contracts (200 tons in our example). However, textbooks usually present results for one unit of the underlying support and we use the same method here, even though n units of the underlying entity are covered by the contracts. Table 3.6 represents the position taken by the producer at the time of sowing.

This table requires some clarification: the sowing corresponds to taking a physical position as once this operation is realized, there is a definitive commitment to production. The sale of contracts corresponds to taking a financial position. The physical position is said to be the long position as the producer possesses the harvest at the time of reaping, without prejudging the volume of this harvest. The financial position is the short position as the producer has committed to selling something that they are not certain of possessing at the time when they must deliver it. The spot price at the time of sowing (200 €/ton) is only a piece of information that the producer includes in their decision-making process; no effective operation is realized by the producer at this price. On the other hand, the producer effectively commits to delivering wheat at 220 €/ton by end of December. If they do not close their financial position before this deadline, they must then deliver the wheat at this price. Let us now examine the result assuming that the producer has not modified their position.

It is assumed that the price of wheat has fallen and has stabilized at 160 €/ton by end of December.

At the time of maturity, the producer carries out two simultaneous operations.

In this example, we propose the very reasonable hypothesis that the miller is close to the agricultural site, thus, at a logistical level, this delivery is simple to carry out. On the financial market, the producer closes their position by taking the position of a buyer; let us recall that they took the position of a seller in the spring. Thus, they are simultaneously the seller and buyer of the same quantity of the underlying support, which is the same as saying that their position is now canceled (see section 3.3.3). This closing of the position generates a financial gain, as the position of the seller was taken committing to 220 €/ton, while the position of the buyer was taken at the price of 160 €/ton. At the cost of simplifying the language used, we can thus say that the producer sold something they had bought at 160 €/ton at 220 €/ton, which results in a gain of 60 €/ton.

This example reflects an essential property of hedges that can be carried out on the futures market: if we take the long position on the physical market and the value of the underlying entity decreases, the short position on the financial market leads to a gain. If the combination of the two positions is constructed correctly, then the gain on the short position can compensate for the losses from the long position. The complete results of the hedge are listed in Table 3.8:

• – as concerns the “real” wheat, the producer has carried out the sowing in spring and storage in July in the year N, before delivering it in December of the year N (to a miller close to the production site). By selling the wheat at 160 €/ton in December, the producer had a shortfall as the wheat was worth 200 €/ton at the time of sowing. It is in this sense that we consider there was a shortfall of 40 €/ton;
• – the producer sold their wheat at 160 €/ton on the physical market and realized a gain of 60 €/ton on the financial market. They thus realized total proceeds of 220 €/ton, the total proceeds being the sum of the proceeds from the sale and from the financial transaction;
• – the operator who contracted with the producer has lost 60 €/ton; as far as they are concerned, any price that is lower than 220 €/ton on the date of maturity translates into a loss, but any price higher than 220 €/ton would allow them to realize a gain.

We now assume that the price has risen and stabilized at 250 €/ton.

The results of the hedge are listed in Table 3.9.

In this case, the loss on the financial market reduces the gain on the physical market. We can see that the total proceeds = proceeds from the sale (+250 €/ton) + proceeds from the financial market (-30 €/ton) = 220 €/ton. If the basis cancels itself at the time of maturity (that is, if the price of the future contract is equal to the spot price), we can then easily verify, using the data from the example, that the total proceeds are always equal to 220 €/ton. In this case, the hedge is perfect, as the total proceeds are constant regardless of the spot price at the time of maturity. A more formal model makes it possible to define an optimal hedge in order to study the possibilities of guarding against the price risk by doing away with the consequences of volatility.

3.5.2. A model for an optimal hedge [PON 96]

The model presented below is heavily inspired by a landmark article by Leland Johnson [JOH 60], which will be discussed in detail in Chapter 6; in order to be consistent with the preceding sections, however, we will retain the notations adopted in the first part of this chapter [PON 96]. This model has two goals: on an operational level, it makes it possible to show that a good correlation between the futures market and the physical market is crucial for agents who wish to guard against price risk carrying out an efficient hedge; on a theoretical level, this model makes it possible to formalize operations on the futures market by using the classic framework of the expectation-variance model initially developed by John von Neumann.

Let the present date be t = 0; we consider a trader who has a stock of commodities that must be preserved until a date t = τ earlier than T . This stock is made up of m physical assets whose unit price is C_t at the point of time t ϵ [0,T]. C_t is interchangeably called the spot price, the physical price or the cash price. On the date τ, the value of the stock is ; according to a very commonly used convention, the tilde indicates random variables. The variance of , denoted by , is a measure of the risk of this position. For a trader taking the long position, the risk is a drop in the price of the commodity. To limit the risk involved in holding a stock, the trader – the hedger – creates a hedge on the date t = 0, by taking a financial position that complements their long position on the physical market. The trader carries out an operation involving x future contracts at the price F_0,T; if x > 0, we speak of a long position on the financial market, and if x < 0, we speak of a short position on the financial market.

The position of the hedge is, in fact, made up of the association of two simple positions: a position on the physical market and a position on the financial position. The position of the hedge can be written as:

with:

The value of the stock at the instant τ is represented by (value of the physical position), while x represents the gain or loss generated by the financial position if it is closed on the date t = τ . The price risk resulting from the hedge position is measured by ; a hedger who wishes to minimize this risk must resolve the following equation:

This expression is derived with respect to x and is made equal to 0: + 2mσ_CF = 0. We obtain the optimal hedge x^*:

We then examine the maximal reduction in variance that will be made possible by the hedge. This reduction is expressed in relative terms using the expression . We thus take the variance before the hedge, , and subtract from it the variance after the hedge, , and then divide by the variance before the hedge. denotes the minimal variance that we can arrive at; by inserting x^* in the expression for we obtain:

We introduce the expression for x^* and obtain:

We also know that , thus . By using this value in , we obtain:

is the square of the coefficient of correlation between and . If , this means that the hedge no longer presents price risks. The hedge is, therefore, perfect. Moreover, when ; in other words, the higher the correlation between and , the greater the reduction in the risk. If the correlation between the future price and the spot price is perfect, it is even possible to cancel the risk of the initial position by creating a hedge.

NOTE.– This point is very important in practice: in order for the futures markets to be effectively used for hedging, the spot prices and the futures prices must be well correlated.

3.6.1. Speculation and hedging, a model for the optimal position

The broad outlines for this model have also been inspired by Leland Johnson [JOH 60]. The unique feature of Johnson’s article is that the analysis of speculation and hedging was associated with the expectation-variance model, which was already well established on the theoretical level from the time of Harry Markowitz’s pioneering work [MAR 52].

In an expectation-variance model, an operator has two goals:

• – they will try to maximize a gain by reasoning through the expectation from a portfolio;
• – simultaneously, for a given expected return, the same operator will seek to minimize their risk by creating a portfolio that presents minimal variance.

More precisely, the optimal portfolio will be created through two strategies: the operator may prioritize fixing their expected return and will then search to minimize the risk. Alternatively, the operator may fix the maximal level of risk they are ready to take and then seek to maximize their expected return. Having compared the optimal choice between expected gain and risk to a component of the rational choice of portfolio, we can use the commonly used framework of analysis for expectation-variance of efficient portfolios. We return to the hedging model presented in section 3.5.2; we now assume that the operator is not only interested in minimizing the risk measured by , but that they are also simultaneously interested in the anticipated value of their holding at the time t = τ, that is E_V ≡ E{V_τ}, E being the expectation operator. The optimization program can then be written as:

Further, assuming that the first derivatives of f exist, we define the functions f₁ and f₂ [PON 96]:

thus, when the expected gain increases, the function f increases:

Therefore, when the variance increases, the function f decreases.

The expected gain from the hedge is equal to , E being the expectation operation, which implies:

E_∆F represents the expected gain resulting from the position taken on the futures market (the financial market).

Moreover, , thus:

The condition for optimality can thus be written as:

We finally obtain the optimal hedge x^**:

When an operator tries to maximize the expectation function and the variance of the terminal value of a position maintained over the period [0,τ ], the optimal position taken on the date 0 is made up of two components:

• represents the hedging component of the position; it is this component that leads to minimal variance;
• represents the speculative component of the position; the operator bets on the foreseeable gains on the financial market.

NOTE.– It should be noted that according to this model, a rational operator is simultaneously a hedger and a speculator.

The reader must keep in mind that the strategy presented above is very basic as the operator takes a position on the date 0 and then waits for the results from this position on date τ; in practice, participants in the future markets regularly adjust their positions based on the changes on the spot market and the futures market.

3.8.1. The bases of the Black model

The spot price of a commodity at the instant t is denoted by c_t. x_t,T, or more simply x, denotes the price at which a future contract with the maturity date T is traded at the time t, with t ≤ T ; x is the price upon which at least two operators have agreed. It is, thus, a market price resulting from a negotiation. In general:

[3.1]

Equation [3.1] must be read as a limit case: if we commit today to a future contract maturing immediately, rationality imposes that this be done at the market price as it would be aberrant for either the seller to commit to a lower price or for a buyer to commit to a higher price than the price that prevails at that point in time. This general relationship is all the more important when t = T as it signifies that the basis cancels itself out, a necessary condition for the optimal hedge.

The value of a forward is denoted by v and the value of a futures is denoted by u. The word “value” here signifies that a forward or a futures will lead to positive or negative financial flows for the operators involved in these contracts. The flows resulting from a contract are in opposite directions for the buyer and seller of this contract. Black ignores commissions and taxes. To simplify the discussion, u and v will be considered as the value of the contract for the buyer, that is the operator taking the long position, to use the jargon of the commodity markets. The interest rates are assumed to be zero and it is, therefore, not necessary to update the financial flows. Using x to denote the price at which a futures contract is concluded, we can write that the value of a forward is:

[3.2]

Black’s notation, which is extremely concise, must be explained: the pair (x, t) of the quadruplet (x, t, x, T) designates the fact that a futures contract was concluded at a price x at the time t; the pair (x, T) from the quadruplet (x, t, x, T) designates the fact that a forward contract with the maturity date T was concluded at the price x. Equation [3.2] must thus be read as: v (x, t, x, T) = (x, t) − (x, T) = (price of the futures) - (price of the forward). In concrete terms, this formula states that a buyer who has committed to a forward contract has made a good deal if the price of the futures contract becomes higher than that of the forward. This is because in this case the buyer has committed to a lower price than the market price.

At the time that it is concluded, the price of a forward is always equal to the price of a future. Thus, the difference in values is zero. If this were not the case, an easy arbitrage opportunity would immediately emerge. For example, if a forward contract could be concluded for delivery at a price c < x, it would be enough to take the long position on the futures contract and the short position on the forward contract to yield certain gain.⁵ Symmetric arbitrage can clearly be formulated in the case where c > x. Equation [3.2] also reflects a reality of derivative markets, namely that a futures can be used as a forward by taking a position on any date t, maintaining it till maturity and proceeding with a physical delivery upon maturity. It is, thus, normal that when a forward contract is concluded, it is concluded at the same price as a futures contract, because this can be used as a forward.⁶

c denotes the price that the buyer of a forward contract commits to paying for the underlying asset at the time T. Of course, the seller has committed to delivering the underlying entity at this same price, c, however, let us recall that for convenience’s sake, the value of the contract is studied solely from the buyer’s point of view. As soon as the market price changes and thus differs from c, the value of the forward contract increases for one party and decreases for the other. For the buyer of a forward, the value changes as follows:

[3.3]

This system of equations can be rewritten as:

The value of a forward is measured by the difference between the price at which it was concluded and the price of the futures contract. The price at which the forward was concluded is constant; on the contrary, the price of the futures contract changes very frequently and, thus, the value of the forward contract, as defined by Black, changes at the same rate as the price of the futures contract. Formally, the value of the forward depends directly on x and, therefore, on c_t (the price of the underlying support) as x_t,t ≡ c_t. It also depends on c, the price at which the delivery must take place. At the time of maturity, T :

[3.4]

In summary, at the time that it is concluded, a forward contract has a null value. Consequently, depending on the change in the price of the support, the contract will gain or lose value, but it is only at the time of maturity that it results in a financial flow, which may be positive or negative, depending on the position taken by the operator. This point can be easily understood. However, the next part concerning futures is more difficult. We reproduce here, almost verbatim, a crucial passage from Fischer Black: the contract is rewritten each day with a new exercise price, c, equal to x, the price of the corresponding futures. A futures contract can be compared to a series of forward contracts. Every day when the markets close the contract signed the previous day is executed and a new contract is concluded with an exercise price that is equal to the price of the futures contract with the same maturity date.

Equation [3.2] shows that a forward contract concluded at the price of a futures contract is of null value at the time that it is signed. Moreover, a futures contract results in a margin call every day, payable or receivable, depending on the change in price with respect to the previous day’s closing price. Once the margin call is collected or paid, the value of a futures is reset to zero. It becomes possible to compare a futures contract to a series of forward contracts rewritten after each closing of the market, with these forward contracts being of null value at the instant that they are rewritten. In other words, the value of a futures is reset to zero on a daily basis following the payment of the margin calls. This can be summarized in the following equation:

[3.5]

Let us highlight the fact that equation [3.5] is valid only for the closing of the market, after the margin calls have been paid. Before the closing, the futures contract gains or loses value like the corresponding forward. The paradox in equation [3.5] can be easily explained: every day, after the margin calls are paid, a future has null value, but since the operator has taken a position and has not closed this position, they have accumulated margin calls that translate into a gain or a loss over the duration of the position being held. In other terms, gains or losses accumulate over the period that a position is maintained: even if the value of a futures is reset to zero every day after the margin calls, the position that the futures allowed the operator to take generates results.

The Black model formalizes what we presented in section 3.3.2, but also goes further. This model makes it possible to highlight the economic uniqueness of futures and forwards: the value of these financial instruments change constantly depending on the changes in the price of the underlying support. More surprisingly, every day after the payment of the margin calls that follow the closing of the markets, a futures contract has a null value until the market reopens again. This result is surprising because expression [3.5] only takes into account the financial flows that have already been paid; it does not take into account the flows expected by the operators who sign a future contract. Another remarkable result, to which we will return in section 4.1.2 can be deduced from this expression: futures markets would not exist in the absence of uncertainty. It is uncertainty that creates the need for futures markets.

3.8.2. The dynamic of futures prices

The manner in which futures prices form and evolve is a central question in financial economics and we will return to it repeatedly in the remaining sections of the text. We will turn here to the broad outlines of the analysis that Black proposed, as this analysis yields several important results.

Black hypothesized that it was possible to work within the framework of the Capital Asset Pricing Model, which meant that the value of any asset included in an investment portfolio could be characterized with the help of a few simple parameters. Futures contracts make up one class of assets among others and can, therefore, be characterized in the same generic manner as any other financial asset [DUS 73]. Within the framework of the CAPM, the expected return from an asset i is given by the expression:

[3.6]

In this expression is the return on the asset i, expressed as a fraction of its initial value; R is the short-term rate of interest and R_m is the return from the market portfolio that contains all of the assets. R is, in fact, a risk-free rate, while represents the return that we may expect if we invest in the totality of risky assets. Taxes and transaction costs are assumed to be zero. The coefficient β_i is defined as:

[3.7]

If β_i > 1, the variations in the asset i are greater than the variations on the market; we then consider that investing in the asset i is riskier than investing across the market as a whole. Conversely, if β_i < 1, the asset i is considered to be less risky but the expected returns will be lower.

Equation [3.6] cannot be directly used for a futures contract because its initial value is null. Thus, the concept of return is meaningless here. Black rewrites this equation so that it can be expressed in monetary units and not in percentage. If the asset i, with a non-zero initial price, does not result in any financial flow over an investment period, then its return is: the terminal price, denoted by , minus the initial price , the whole divided by the initial price.

Equation [3.6] is rewritten as:

[3.8]

Upon multiplying by P_i,0 we have:

[3.9]

At the beginning of the period, a futures has a value of zero, thus P_i,0 = 0; at the end of the period, and before paying the margin calls, the value of a contract is equal to the change in price of the futures contract over the period. The variation of the futures price over the period is denoted by . Hence:

[3.10]

By now using a value β^*, defined as the “dollar beta” [BLA 76], we obtain:

[3.11]

This equation shows that the expectation of variations in a futures price may be positive, negative or null. It is zero notably when . This result is very important: the variations in price may be the source of a financial return, thus, futures contracts may be investment supports, not only hedging tools. From a theoretical point of view, Black opens up the path to what we today call the financialization of futures markets, which will be discussed in detail in Chapter 4.

3.9.1. Definition and example

In finance, a swap is used to designate a contract through which two parties exchange two series of financial flows. In a commodity swap, one of the operators (Bank B) commits to paying, at n regular intervals, a financial flow that depends on the price observed on a market for each maturity date. In return, the other contractor (Company A) will pay a fixed flow, defined at the time the contract is signed. In this example, Company A will win if the series of spot prices is such that the variable flows they pay are, on average, lower than the fixed flows they will receive.

A fixed flux is usually represented by a single-headed arrow and a variable flux by an arrow with two heads. A very important point: each flow is repeated at regular intervals for the full duration of the swap. A commodity swap does not usually lead to a physical delivery. To be on the safe side, we will qualify that we have never heard of such a contract. Having said this, the contractor enjoys quite a bit of freedom and thus, a swap with a physical delivery is not impossible. A few clarifications must be made: first of all, the variable price of the commodity is called the floating price and the fixed price is called the constant flow, paid for the full duration of the swap. Further, a swap is signed for a limited period, thus the number of flows and their periodicity are part of the terms of the contract. It can be easily understood that, depending on the changes in the support, the swap will gain in or lose value for one or the other of the parties. For example, if the price of the commodity rises, then the party paying the fixed flow is in a winning position, while the party paying the variable flow loses.

Commodity swaps are very widely used. In practice, they are proposed by specialized actors, often banks. An operator who seeks to hedge, an airline company, for instance, that must regularly buy jet fuel, will easily find a range of propositions for swaps. The negotiation of a swap is, in fact, a rapid process as the contracts are heavily standardized and the terms of the contract are not negotiated line by line. Consequently, if a price is suitable to one hedger, a few “clicks” suffice for the contract to be concluded and to come into effect. Once the swap is signed, the hedger is protected against variations in price, while the seller of the hedge becomes exposed to this risk. In a very general manner, the seller of the swap will themselves engage in another swap that will allow them to diminish or cancel out their risk. Of course, a bank simultaneously engages in many swaps only if the revenue from the flows that it receives and pays out allows it to realize a margin that is deemed sufficient. Indeed, banks – or other specialized operators – normally take few risks by combining diverse operations where the revenue from the flow is positive and near-certain.

In this case, Bank B pays a variable flow to Company A, but it receives another variable flow, indexed on the same commodity, from Bank C. The two fixed flows are known. B is, therefore, no longer exposed to the risk of fluctuations in the price of the commodities. Of course, B will only engage in this double operation if the arithmetic sum of the flows is positive. Nevertheless, the operation is not, strictly speaking, completely risk free for B as one or the other of its counterparties may default. As this is an OTC operation, a completion of the transactions is not guaranteed by a CH. We will see, in Chapter 8, that these incompressible risks resulting from OTC operations present regulation problems.

This very simple example also makes it possible to highlight another crucial point: it is almost certain that Bank C will itself be engaged in other swaps. We can thus see that swaps constitute a sort of chain in which multiple actors are involved; if one of these actors defaults, the consequences may affect all actors along the chain, resulting in the problems regulators face.

3.9.2. Pricing a swap

A commodity swap generates random financial flows. Both on the theoretical level as well as the operational terms that aid in decision making, it is necessary to evaluate a contract before signing it. The basic principle of pricing a swap is extremely simple and can be summarized in an expression as follows:

[3.12]

V₀ designates the value of the swap at the time it was concluded, on the date t = 0, T is the maturity date of the swap, c is the fixed flow amount, is the random flow and r is the discount rate. In this formula, V₀ is estimated from the point of view of the party that receives the fixed flows and pays the variable flows. The value of the swap for the seller is, clearly, the opposite of V₀.⁷