6.1.4.1. Speculation and 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.

6.1.4.2. Future prices and storage decisions

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.

6.1.5.1. Hedging by traders

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 spot premiums for September between the years 1922 and 1952 are laid out along the x-axis. Here, the spot premiums represent the difference between the future price for a deadline of December and the spot price for the first working day in September in the same year;
• – the changes in spot premiums are listed on the y-axis. They represented the change in spot price between the first working day in September and the first working day in December.

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.

6.1.5.2. Hedging carried out by transformers

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.

6.1.5.3. The economic impacts of hedging

To conclude, Working refined and complemented the analysis of the influence hedging had on commodity markets. He stated various conclusions:

• – hedging makes it possible to reduce commercial risks (even if this is not necessarily the primary purpose of the operation). It reduces the risks resulting either from storage or from the commitments made in forward contracts, and it also reduces the risk of engaging in inadvisable commitments of storage or deferred delivery;
• – the large-scale use of futures markets tends to reduce the unjustified gaps on the spot market;
• – the possibility of turning to hedging facilitates storage by private organizations in periods where there is a surplus and facilitates the temporal adjustment of the storage.

6.2.1.1. The framework for the analysis

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.

6.2.1.2. The notations

• – x_i = number of physical units held on a market i;
• – a hedge of units is a position taken on the market j such that the price risk induced by holding x_i and is minimized (not just reduced);
• – is the variance in the price change (variance in return) on the market i and, therefore, represents the price risk resulting from holding one unit on the market i for the period thus represents the variance in financial yield resulting from holding x_i units on the market i;
• – on the market j, the price risk is measured by ;
• – cov_ij represents the covariance of price changes, or the covariance of return, between the markets i and j.

6.2.1.3. Writing out the equations

A combined position on the markets i and j presents the variance of return V (R), which can be expressed as follows:

[6.1]

This combined position offers an effective return R and an expected return E(R):

[6.2]

[6.3]

where:

• – B_i, B_j: the effective price changes that took place between t₁ and t₂;
• – u_i, u_j: the anticipated price changes at the time t₁ for the period [t₁; t₂].

Equation [6.1] allows us to determine the value , which minimizes the variance in the return from the combination x_i, :

[6.4]

Entering this value into equation [6.1], we have:

And thus:

where ρ is the correlation coefficient, ρ = cov_ij/σ_iσ_j.

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.

6.3.1.1. Holding hedged stocks

We use the following notations from Stein’s article:

• – u = expected unit gain from the holding of unhedged stock;
• – p^* = expected spot price at a later date; p^* is a stochastic variable;
• – p = current spot price “today”;
• – m = marginal carrying cost; carrying consists of buying stocks and storing them for later reselling.

Hence, u = p^* − p − m. p^* is a stochastic variable and thus the expected gain could, in fact, be a loss.

6.3.1.2. The holding of hedged stock

In addition to the above notations, we now use the following:

• – q = “Today’s” (current) price for the futures contract;
• – q^* = expected price for the futures contract at a later date.

The expected unit gain, h, for the holding of a hedged stock was thus expressed as:

[6.5]

[6.6]

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.

6.3.1.3. The optimal combination of hedged and unhedged stocks

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.

6.3.2.1. The demand for stocks

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:

1. 1) with p^* − p − m, the first term is the expected return of holding unhedged stock;
2. 2) with (p^* − q^*)+ (q − p) − m, the second term is the expected return of holding hedged stock.

The demand for stocks is given by the following equation:

[6.7]

where U is the demand for unhedged stock and H is the demand for hedged stock.

6.3.2.2. The supply of stocks

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.

6.3.2.3. Market equilibrium

When a market is equilibrated, the demand is equal to the supply, and thus the following equation must stand:

[6.8]

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:

[6.9]

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.

6.3.3.1. Supply of futures contracts

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:

6.3.3.2. The demand for futures contracts

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^*.

6.3.3.3. Equilibrium on the market of futures contracts

At equilibrium, the following equation must be respected:

[6.10]

Knowing that q = b + p:

[6.11]

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:

[6.12]

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.