Of what use is hedging? We have chosen three authors to help illustrate the “classic” responses to this question. We will first speak of classic literature because contemporary work is carried out within the broad outlines created by these authors:
We will approach the theories developed by these three authors through a detailed study of articles that have been abundantly cited by those who came after them. This examination is presented in chronological order of publication of these articles.
For Working’s theory, we will examine a text that was the source for a wide swathe of literature [WOR 53a]. Working’s principal objective was to characterize, as precisely was possible, the nature of operators and operations carried out on futures markets. He considered that the simplified presentation that was usually given led to a poor comprehension of these markets and risked resulting in erroneous public policy.
The trading of futures contracts is traditionally defined using the properties of these contracts (as was done in Chapter 2). The alternative representation proposed by Working starts out with a definition of the trading of futures contracts based on the operations carried out by the traders:
“Futures trading in commodities may be defined as trading conducted under special rules and conventions, more restrictive than those applied to any other class of commodity transaction, which serve primarily to facilitate hedging and speculation by promoting exceptional convenience and economy of the transaction”.
By choosing this definition, Working explicitly considered that futures contracts were instruments that enable the execution of operations that were unique to this type of market, while deferred delivery operations were common on a large number of markets. Thus, it is not the interval between various operations that characterizes futures markets, but indeed the very nature of the operations, especially hedging and speculation. What remains, in order to understand the dynamic of the futures market, is to determine the respective roles played by speculation and hedging:
“It is possible to think of trading in futures contracts solely from the operator’s desire to speculate; it would appear, however, that this trading would not be durable unless certain conditions appeared, such as the emergence of a speculative risk, and that certain operators were led to transfer these risks to others through hedging”.
Working offered a series of examples to illustrate the fundamental points of the analysis below: “A major source of misunderstanding regarding hedging results from the habitual practice of introducing it using a hypothetical example in which the price of the futures contract used for hedging is assumed to increase or decrease by the same amount as the cash price. Instead, let us study hedging in a realistic manner using the prices that are observed in practice”.
The following examples are based on real data from the futures market for wheat in Kansas City; the prices are those of the first day of trading for each month in which the contracts expire for the year 1951–1952.
EXAMPLE 6.1.–
Table 6.1, reproduced from Working’s article, shows the possibility of gain through a hedge.
Table 6.1. Opportunity for gain provided by a hedge [WOR 53a]
Price | Date and price | Gain or loss | |
July 2 | September 4 | ||
Hard wheat no. 2 (lowest in the day) | 229.25 | 232.25 | ….. |
Futures contract expiring in September | 232.50 | 233.50 | ….. |
Spot premium | -3 | -1 | +2 (gain) |
Let us take the case of an operator who carries out two simultaneous operations on July 2, 1952: they buy wheat on the spot market at a price of 229.25 $/bushel and sell futures contracts expiring in September at a price of 232.25 $/bushel. On September 4, 1952, they resell the wheat at the spot price of 232.50 $/bushel and realize a profit of 3.25 $/bushel; on the futures market, they dissolve their position by taking the position of the buyer at 233.50 $/bushel and suffer a loss of 1.25 $/bushel. The financial result of the operation amounts to (+3.25 − 1.25), that is, a gain of 2 $/bushel.
Indeed, the operator has gained by betting on a narrowing of the basis, which decreased from 3 to 1 between July and September.
EXAMPLE 6.2.–
Another hedge makes it possible to realize a gain between September and December.
A trader carries out two simultaneous operations on September 4, 1952: they buy wheat on the spot market at a price of 232.50 $/bushel and sells futures contracts expiring in December at a price of 238.25 $/bushel. On December 1, 1952, this trader resells the wheat at the spot price of 252 $/bushel and realizes a gain of 19.50 $/bushel; on the futures market, they dissolve their position by taking up the position of a buyer at 252 $/bushel and make a loss of 13.75 $/bushel. The financial result of the operation amounts to (+19.50 − 13.75), that is, a gain of 5.75 $/bushel.
Table 6.2. Opportunity for a gain opened up by a hedge [WOR 53a]
Price | Date and price | Gain or loss | |
September 4 | December 1 | ||
Hard wheat no. 2 (lowest of the day) | 232.50 | 252.50 | ….. |
Future contract expiring in September | 238.25 | 252.50 | ….. |
Spot premium | -5.75 | 0 | +5.75 (gain) |
Here again, the trader made a gain by betting on a narrowing of the basis, which narrowed from 5.75 to 0 between July and September.
EXAMPLE 6.3.–
A third example, similar to this, describes a possible operation between December 1952 and May 1953.
A trader conducts two simultaneous operations on December 1, 1952: he buys wheat on the spot market at a price of 252 $/bushel and sells future contracts expiring in May 1953 at a price of 251 $/bushel. On May 1, 1953, he resells the wheat at the spot price of 247.25 $/bushel and suffers a loss of 4.75 $/bushel; on the futures market, he dissolves his position by taking the position of a buyer at 238.25 $/bushel and collects a gain of 12.75 $/bushel. The financial result of the operation amounts to (− 4.75 + 12.75), that is, a gain of 8 $/bushel.
Table 6.3. Opportunity for gain opened up by a hedge [WOR 53a]
Price | Date and price | Gain or loss | |
December 1 | May 1 | ||
Hard wheat no. 2 (lowest in the day) | 252.00 | 247.25 | ….. |
Future contract expiring in September | 251.00 | 238.00 | ….. |
Spot premium | +1 | +9 | +8 (gain) |
The third example is particularly interesting: the prices on the spot market and on the futures market varied in the same direction (decreased), but the slightly negative basis (-1) became quite strongly negative (-9). The gain was possible only by betting on this kind of an evolution of the basis.
EXAMPLE 6.4.–
This fourth example discusses an operation that results in a loss.
On May 1, a trader pays in cash for wheat, at 247.25 $/bushel, and takes up the position of a seller at a price of 229.25 $/bushel, with the contract maturing in July. On July 1, he resold his stock at 218.50 $/bushel and suffered a loss of 28.75 $/bushel. On the futures market, he dissolved his position by buying a contract at 225 $/bushel; he thus realized a gain of 4.25 $/bushel. The financial result of the operation amounted to (−28.75 + 4.25), that is, a loss of 24.50 $/bushel.
From these examples, Working developed three propositions related to operations carried out on futures markets.
NOTE.– Finally, according to Working, the objective of reducing the risk through hedging has been highly overvalued in literature on economics. Hedging consists, first and foremost, of taking advantage of changes in the basis; the reduction in risks induced by the fluctuating prices of the support is only an incidental benefit.
After having developed his central theory on hedging, conceived of as arbitrage or a speculation on the basis, Working asks the classic question: does the existence of futures markets have a stabilizing or destabilizing effect on cash price dynamics?
Working studied operations carried out by speculators who intervened massively as counterparties for industrialists and traders who hedge their physical positions by taking positions on derivative markets. Working was specifically interested in day traders who made up the largest category of speculators. As their name indicates, day traders take various positions in the same way on contracts and choose to collect on their gains or cash their losses every day. These players make a profit if they have more profitable days than days that end in losses. In other words, efficient day traders are able to produce daily anticipations in price fluctuations and these are more often validated than not. Working could access information shared by various traders on the details of the positions they took up. These data will not be reproduced here; it reflects the fact that perennial speculators realize, on average, higher gains than losses, which is the result of a capacity to quite reasonably anticipate very short-term price fluctuations, sometimes over the span of a few minutes. The gains from a successful operation are low, but there are many operations: speculators do not earn by carrying out “large operations”, but by increasing the number of positions taken that generate small gains. These positions consist of benefiting from an increase by taking the position of a vendor or by profiting from a decrease by taking the position of a buyer. Day traders thus have a smoothing effect on the movements of short-term prices.
By supplying simultaneous information on cash prices and anticipated prices, the futures market considerably facilitates storage policies. This information is used not only by operators on these markets, but also by all those concerned with production, trading or the transformation of commodities. The case of farmers is emblematic in this respect: only a minority of these operators directly use futures markets, but almost all – if not all – of them take future prices into consideration when making decisions related to cultivation. The informational content of anticipated prices thus appears as a positive externality of the functioning of futures markets.
In a second article, titled “Hedging reconsidered” [WOR 53b], Working complements the analysis that we studied in the earlier sections. We will discuss the outlines of this new article here. We know the standard definition of the hedger: a player who protects themselves by carrying out a financial operation to compensate the losses they may suffer on a physical market. Working closely studied the motivations of players on derivative markets. Based on repeated observations, he arrived at the quite radical conclusion that these players did not really act as per the scheme by now considered the classic explanation.
Working was first interested in the rationale used by stockers and developed a major idea: anticipations on wheat prices were very uncertain, but on the contrary, the evolution of the basis was not very difficult to anticipate, especially for a professional. A corollary to this was that the efficacy of the hedging associated with the storage depended on the differences in the change in future price and the change in spot prices associated with the relative possibility of foreseeing these differences. To demonstrate the relative ease of anticipating the evolution of the basis, Working used two examples, which were based on the real data from the Kansas City futures market. The first example consisted of crossing two variables:
The correlation coefficient between the two variables is equal to 0.839.
A second, analogous example – for the period May–July – presents a similar profile with a coefficient correlation of 0.975. These high correlation scores make it possible to affirm that experienced players will have a rather good possibility of anticipating changes in the basis.
Working then went on to analyze the behaviors of millers. A detailed study of the available data led him to formulate several propositions. Out of these, we have chosen one that seems very important: storage behaviors among the millers who use futures markets are very variable and follow no easily identifiable patterns. Working concluded from this that hedging offered a great flexibility in behavior in order to adapt to its own constraints. The losses industrialists suffered due to an absence of hedging was lower than for a trader. Working concluded from this that the frequent use of futures markets was due to the commercial ease and opportunities for gain that these markets offered.
To conclude, Working refined and complemented the analysis of the influence hedging had on commodity markets. He stated various conclusions:
In a very frequently cited article, Johnson disagrees to some extent with Working’s conclusions. According to him, contrary to Working’s declaration:
There was hence a need to develop a new theory of hedging.
Johnson studied the usual situation of an operator subject to a price risk on a physical market and who hedged by taking a position on a futures market.
We will use the notations used by Johnson in his article. We will study a hedge carried out between t1 and t2, while the deadline for the future is t3, to hedge a stock of x units bought on t1 and resold on t2. S1 and S2 denote the spot price, and F1 and F2 denote the price of the future on the dates t1 and t2, respectively. On the physical market, the operator buys at the price S1 and sells at the price S2. From this, we have a result equal to x(S2 − S1); on the futures market, the operator takes up a position of seller at the price F1 and a position of buyer at the price F2, which gives a result equal to x(F1 − F2). The operator will realize an overall gain or loss whose value is x[(S2 − S1)+ (F1 − F2)]. This can be rewritten as x[(S2 − S1) − (F2 − F1)]. The hedge is perfect if [(S2 − S1) − (F2 − F1)] = 0, which signifies that the gains on the financial market perfectly compensated the losses on the physical market.
A combined position on the markets i and j presents the variance of return V (R), which can be expressed as follows:
This combined position offers an effective return R and an expected return E(R):
where:
Equation [6.1] allows us to determine the value , which minimizes the variance in the return from the combination xi, :
Entering this value into equation [6.1], we have:
And thus:
where ρ is the correlation coefficient, ρ = covij/σiσj.
The higher the correlation between the markets i and j, the more effective the hedge will be. If the correlation is 1, the hedge may even prove to be perfect.
In this reformulated version, both the price risk and the efficacy of the hedge are processed quite independently of the effects of the effective price changes. The trader keeps the price risk in mind from the instant t1 onwards. Formally, this is the variance of the probability distribution of the financial return, a priori1.
The Johnson model deserves to be studied in its entirety. The reader is thus advised to go through the complete original article. We present, below, the points we find most important. Figure 6.1 represents the indifference curve for a trader.
The optimum situation corresponds to a position that generates E(R) and and such that the trader achieves the highest indifference curve. This optimum is determined graphically by jointly using Figures 6.2 and 6.3.
This set of points of contact make up the ray OZ. These points are marked in Figure 6.2 and make up the ray OW . The optimal position for the trader is represented by K, which is the point of contact with the highest utility curve that we can achieve. We then map the coordinates for the point K in Figure 6.3 and obtain the point L with the coordinates (X1; - Y1). -Y1 thus represents a financial hedge with respect to the physical position X1; in other words, the financial position -Y1 makes it possible to reduce the risks resulting from the physical position X1. Based on this scheme, Johnson then analyzed various strategies that traders could develop. According to the author, the advantages of the model could be summarized in this way2.
Jerome Stein opened up a new avenue of research. His objective was to show the conditions under which equilibrium prices could be formed simultaneously on the spot and future markets. Stein thus studied the properties of equilibrium on the spot market and on the futures market, while Working and Johnson were primarily interested in the motivations and strategies of hedgers. Stein used the framework of expected utility to carry out this study.
A holder of stocks has two possible alternatives: they can sell them at a known price, on the spot market, or by using a forward contract where they could conserve their stocks and later sell at an uncertain price. In the second case, the unhedged stocks induce a risk; the operator can then choose to conserve their unhedged stocks and hedged stocks in such a proportion that its expected utility will be maximal.
We use the following notations from Stein’s article:
Hence, u = p* − p − m. p* is a stochastic variable and thus the expected gain could, in fact, be a loss.
In addition to the above notations, we now use the following:
The expected unit gain, h, for the holding of a hedged stock was thus expressed as:
The hedge makes it possible to limit the losses that may result from holding stock. We can observe that we can rewrite h = u + (q − q*), which shows that the expectation of gain from a hedged position is equal to the sum of the expectation of gain from the physical position and the expectation of gain from the financial position. We thus find the elements described in Chapter 2, which can be written in a new form.
The proportion of unhedged stocks varies from zero to 100%, and thus the expected return varies from h to u. For convenience’s sake, Stein uses the variance of expected return as a measure of risk. He also assumes that the density function is symmetrical. The holder of one unit of unhedged stock faces a risk equal to the variance of u; p and m being known, the variance of u is equal to the variance of p*. The holder of a unit of hedged stock faces a risk equal to the variance of h. We know p, m and q, and thus, we can write var(h) = var(p*)+var(q*) − 2cov(p*q*). The proportion of unhedged stock varies from 0 to 100%, and thus the risk of a position made up of hedged and unhedged stocks lies between [var(p*)+ var(q*) − 2cov(p*q*)] and var(p*).
The graph in Figure 6.4 illustrates this information; the line HU represents the possibilities open to an agent who holds 100 units of stock. At the point H, the stock is entirely hedged; at the point U, it is entirely unhedged. The unhedged stocks present a higher expectation of gain as well as higher risk in comparison to hedged stocks, resulting in the positive slope for HU. The indifference curves between risk and expected return are convex. The curve I2 represents a utility that is higher than that of curve I1. When we know the line of opportunity HU and the indifference curves, we can determine the point P, which represents the optimal combination between hedged and unhedged stocks. If q, the current price of the futures contract, increases, then with all other things being equal, the new line of opportunity is H′U and the optimum becomes Q.
There is a willingness to stock if the expected utility of the storage exceeds the utility resulting from a sale on the spot market. This willingness to stock is here considered as a demand for stocks. As the risks are known, that is, as var(h) and var(u) are known, the demand for stock increases:
The demand for stocks is given by the following equation:
where U is the demand for unhedged stock and H is the demand for hedged stock.
The quantity of available stocks is equal to the initial quantity of stock – denoted by S−1 – plus the difference between production and current consumption, which is denoted by X(p, a). The quantity X(p, a) represents excess supply resulting from current production; p is the spot price and a is a parameter. An increase in a translates to a change “toward the right” of the curve for excess supply from current production.
When a market is equilibrated, the demand is equal to the supply, and thus the following equation must stand:
The two variables to be determined are the spot price p and the basis b, that is, the difference between the future price and the cash price. The basis b may be negative, but p is necessarily positive.
[6.8] is differentiated with respect to p and we obtain ∂b/∂p:
with:
An SS curve is the set of pairs (p, b) that must prevail in order for the supply and demand of stocks to be equal. It can be intuitively understood that an SS curve must be ascending, as a rising spot price would lead to an increase in the quantities produced and a decrease in the quantities consumed. Consequently, the basis b must increase in order to incentivize players to stock, as an incentive for stocking is an increasing function of the basis. In order for equilibrium to be established, the basis b must increase when the spot price p increases.
When we hedge one unit of stock, we supply a contract; thus, the demand for hedged stocks is equal to the supply of contracts. We can deduce from this that the supply of future contracts is:
The quantity of futures contracts demanded by speculators is an increasing function of the anticipated profit:
This profit anticipated by speculators depends on the anticipation that they form for q', the price that the contract will attain in the future; q' may differ from q*.
At equilibrium, the following equation must be respected:
Knowing that q = b + p:
Using equation [6.11], we can trace a curve, FF, describing the relationship that must exist between b and p in order for the supply and demand of future contracts to be equal. Upon differentiating equation [6.11] with respect to p, we obtain:
FF thus has a negative slope. Intuitively, if the spot price p increases, while the basis b does not vary, this means that q, the price of the futures contracts, is increasing. This is because b = q − p. The increase in the price of the futures will reduce the demand for futures as the gap q' − q shrinks. On the other hand, the supply in the quantity of futures will remain unchanged as the expected profit from holding the stocks is unchanged, as the basis is unchanged. The excess supply of futures will lead to a reduction in q and, therefore, to a reduction in the basis b. To sum up, when the spot price increases, the basis must decrease to maintain the equilibrium between the supply of and demand for futures contracts. Hence, FF has a negative curve.
There are two conditions required for a simultaneous equilibrium: the supply of and demand for stocks must be equal (the curve SS), and the supply of and demand for futures must be equal (curve FF). This simultaneous equilibrium is, for example, achieved at the point (p0, b0), as shown in Figure 6.5, which led Stein to declare that the “simultaneous determination of spot and futures prices has been demonstrated”.
This representation then makes it possible to develop an analysis in terms of comparative statistics. This is done by studying the impact of the variations in current production and then the variations in anticipation of the price.
The pioneering work carried out by Working, Johnson and Stein established a veritable theoretical tradition in literature in economics dedicated to commodity derivative markets. Moreover, their influence was not limited to the academic field: contemporary reflections on the regulation of physical and financial markets continue to use the conceptual frameworks built by these researchers. Financial mathematicians, especially Black, Scholes and Merton also exercise an important influence in complementing these exceptional theories that have left their mark on the conceptual framework of economics.
Chapter written by Joël PRIOLON.
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