Is there a fully-revealing trap in commodity futures

Is there a fully-revealing trap in commodity futures markets? Etienne Borocco

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January 31, 2016

Abstract The usual models with an exogenous spot price is exogenous do not help to understand the relationship between the futures and spot prices. Well, this issue became burning with the financialization of the commodity markets. In this article, a model of a futures market with an endogenous spot price and asymmetric information is presented. We decided to extend Ekeland et al. (2014) which presents a model with endogenous prices. Therefore, the futures price can convey the revealed information to the spot price. The core of both models is the same so the descriptions are very similar. There are different categories of agents. This heterogeneity allows different groups to fulfill different functions in the market. Processors, storers and informed speculators have a private information about the supply. While the uninformed speculators have not any signal at all. Our model highlights the effects of the physical constraint and information on the future price. We show that informational frictions can offset the physical effect on the future price. This effect is very strong if an erroneous information drives the futures price up in contango. Financial purposes will increase the volatility of the spot prices through the storage channel. Thus, regulators have to be careful if they want to send a unique signal to every agent. Prices can be drove in the wrong direction if you the sent signal is misleading. Last, the futures price can be biased even if the risk premium is null.

Contents 1 Introduction

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2 Literature review

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3 The model 3.1 Description of the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Biased or unbiased consensus? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4 Main equations 4.1 industrial hedging . . . . . . . . . . . . . . . . . . 4.1.1 Processor’s hedging . . . . . . . . . . . . . 4.1.2 Storer’s hedging . . . . . . . . . . . . . . . 4.1.3 Hedging pressure . . . . . . . . . . . . . . . 4.2 Determination of the Bayesian linear equilibrium . 4.3 Characterization of the Bayesian linear equilibrium 4.4 Positions . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Market-clearing . . . . . . . . . . . . . . . . . . . . ∗

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LEDa, Université Paris-Dauphine, Place du Maréchal de Lattre de Tassigny, 75016 Paris. Email: [email protected]

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5 Definition of the equilibrium 5.1 Existence of a quasi-equilibrium . 5.2 Equilibrium . . . . . . . . . . . . 5.2.1 Region 1 . . . . . . . . . . 5.2.2 Region 2 . . . . . . . . . . 5.2.3 Region 3 . . . . . . . . . . 5.2.4 Region 4 . . . . . . . . . . 5.3 The equilibrium is fully revealing

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6 Equilibrium analysis 6.1 The new frontier . . . . . . . . . . . . . . . . . . . . . 6.1.1 The stochastic nature of the new frontier . . . 6.1.2 The informational effect: a probabilistic bias in 6.2 The factors of the informational effect . . . . . . . . .

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7 The impact of speculation 7.1 An Increasing weight of the uninformed speculators . . . 7.2 Increasing of the informed speculation . . . . . . . . . . 7.2.1 Effect on the future price: a fully-revealing trap? 7.2.2 Effect on the variances . . . . . . . . . . . . . . . 7.2.3 Impact on the utilities . . . . . . . . . . . . . . . 7.3 Toward a partially-revealing equilibrium . . . . . . . . .

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8 Conclusion

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A Properties of the Quasi-equilibrium and the equilibrium A.1 Quasi-equilibrium . . . . . . . . . . . . . . . . . . . . . . . . A.2 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2.1 Region 1 . . . . . . . . . . . . . . . . . . . . . . . . . A.2.2 Region 2 . . . . . . . . . . . . . . . . . . . . . . . . . A.2.3 Region 3 . . . . . . . . . . . . . . . . . . . . . . . . . A.2.4 Region 4 . . . . . . . . . . . . . . . . . . . . . . . . . A.3 Distribution which supports the equilibrium . . . . . . . . .

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B Impact of informed speculation B.1 Region 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.2 region 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.3 region 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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C Discussion C.1 Region C.2 Region C.3 Region C.4 Region

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1

Introduction

The price of the crude oil barrel was never higher than $40 before mid 2004. In July 2008, it rocketed at $145 and at the end of 2008, it fell at $30. Then it rose again until $110 in 2011. The 29th January 2015, it was lesser than $45. Speculation is one of the potential explanation of this high volatility. The CNUCED pointed out the financialization of the commodities markets these last ten years. Commodities thus become assets that institutional investors buy to diversify their portfolios. Speculation in commodities markets is a burning issue since the end of the nineteenth century. Speculators are accused of numerous evils(Weiner, 2002) since a long time: « For example, during the Gulf Crisis of 1990-91, complaints about price volatility from oil consumers were often disguised claims that prices were too high relative to fundamentals. In an example from the other side, crude oil producers attributed the price decline of the late 1870s to the rise of speculation in crude oil futures on the Oil City and New York exchanges. The Petroleum Producers’ Union, a 19th-century trade group, listed among the causes of low prices “the manipulation of stocks by speculators and buyers to depress price to suit their purposes, which are always adverse to the interests of producers” [Petroleum Producers’ Union 1878]. » According to Weiner (2002), the accusations against speculators fall into three categories. First, speculators would drive away the prices from the fundamentals, generating bubbles. Second, they manipulate the markets by spreading false news. Third, they would be poorly informed so it would be why, they would be more inclined toward herding behaviors and following-trend strategies. This old issue became burning again after the world commodity price spike of 2008. In this context, the Agricultural Market Information System (AMIS) has been launched by the G20. The AMIS crops are wheat, maize, rice and soybeans. One of the aim of the AMIS is to improve agricultural market information. To fulfill this purpose, the AMIS provides analysis and short-term supply and demand forecasts at both national and international levels.The AMIS wants to address the critics of speculation described by Weiner (2002). In this article, we will focus on the futures only among the derivatives products because t is the most used kind of contracts in commodities markets. A future contract is a standardized agreement between two counterparts. They are negotiated in an organized market under the supervision of a clearing house. As it was noticed by Lautier and Simon (2009): A consequence of the standardization is that the transactions on futures contracts rarely lead to a physical delivery. Futures contracts are purely financial instruments The derivatives markets have important economic functions. It is a crucial tool in risk management because it makes possible for the traders to share risks. Moreover, there is an important role of price discovery because the prices for different maturities will give information about them(Lautier and Simon, 2009). The first issue is to analyze how asymmetric information affects the functions of the derivative markets. More broadly, we can look at the functions of the commodity markets in their whole. An important dimension is the storage. The prices have a direct influence on the storage. If the forward curve is upward sloping, the level of storage will be high. Because, it is profitable to hold stocks to sell them later. Inversely, if the the forward curve is downward sloping. The forward curve reveals the anticipations of the market traders. The anticipations depends on the available information heavily. Indeed, an information is advantage face to the other actors. It is possible to exploit it to know about the future value of the fundamentals. The price revelation of the information is a key feature of efficient markets. A market is efficient if all the information is revealed including private information. However, Grossman and Stiglitz (1982) shows that a Fully-Revealing Rational Expectations Equilibrium (FRREE) is not implementable with information cost. Indeed, no one will buy any information if the price reveals it to the other ones for free. Therefore, this equilibrium is not implementable. The equilibrium have to be partially revealing (PRREE) thus. Implementable equilibrium are not efficient. A PRREE exist when private heterogeneity prevents 3

the price to reveal the average signal fully. Moreover, the information can be noisy, therefore it can generates frictions disconnected to fundamentals. These informational frictions can even offset the physical constraints. Therefore, we can write our problematic: How does asymmetric information affect the functions of the commodity markets? First, we will describe the model. Then, we will solve the main equations to determine the marketclearing conditions. In the same time, we will show that a Bayesian linear equilibrium exists. More broadly, we will determine the conditions of existence of the equilibrium. At last, we will analyze this equilibrium to understand the effects of the asymmetric information on the commodity market that we study

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Literature review

Some speculators can be informed. For example, Hau (2001) studied the electronic trading system Xetra of the German Security Exchange which provides data source on the equity trades of 756 professional traders located in 23 different cities and eight European countries. He showed that traders located outside Germany in non-German-speaking cities show lower proprietary trading profit in comparison to local German traders. In commodity markets, there is the same kind of concentration in cities like Geneva, Singapore or Houston. Thus, we can suppose that informed speculators exist. This the point of view of Khoury and Martel (1989) who made the assumption that the speculators are more informed than the hedgers: « These speculators can exploit economies of scale and of specialization in order to have access to a continuous flow of additional relevant information concerning future supply and demand conditions in the spot ,and futures markets at a much lower cost than hedgers (Khoury and Martel 1985, 1986) [...] Moral hazard and/or the loss of lucrative trading opportunities by speculators hamper the transfer of the needed information to hedgers; and the cost (in terms of time and money) to hedgers of acquiring this information on their own generally exceeds the benefit to them. » The both authors claim their model provides an alternative explanation to the convenience yield (which is the benefit associated with holding a physical good, Kaldor (1939))to explain storage even when the future price is lower than the spot price. In this is situation, the market is said in "backwardation". This statement is controversial and others authors have made different assumptions. The involved agents in the physical commodity trading as the storers or the processors can exploited their information for speculation (Cheng and Xiong, 2014): « One possibility is that industrial hedgers may attempt to exploit informational advantages by trading against speculators. For example, commercial firms may exploit informational frictions in spot markets, as they may have better knowledge of local physical market conditions. » According to Vives (2010), the informed speculators are the producers while the processors are uninformed hedgers: « The private information of producers cannot help the production decisions because it comes too late, but allows the, to speculate in the futures market where uninformed speculators (market makers) and other hedgers operate. This will tend to diminish the hedging effectiveness of the futures market and consequently diminish the output of risk-adverse producers (since they will be able to hedge less of their production). The adverse selection is aggravated with more precise information. Adverse selection is eliminated if the signal received by producers is made public. However, more public information may decreases production because it destroys the insurance opportunities ». 4

The last effect is the "Hirshleifer effect". When all the information is released, it can leads to a no-trade situation so the utility of the agents can decrease. Empirically, the information asymmetry on the futures markets has been studied by Knill et al. (2006). They noticed that speculators can get information about oil and gas producers through analysts: « The uninformed cannot resolve the information asymmetry from price movements, she can rely on quasi-private channels, namely analyst forecasts and their error, in predicting oil and gas companies’ performance. Extant evidence shows that analysts often err optimistically in their forecasts of corporate earnings ». In their empirical analysis, "a measure of aggregate earnings surprise for the industry" is used as a proxy for information asymmetry. The later is considered as proportional to the former. Their results show a "large degree of information asymmetry" on the futures markets for oil and gas. Moreover, they remind that the errors are not distributed identically and independently among speculators. The analysts’ forecasts can be biased in the same direction. Therefore, we decided to endow hedgers with signals in our model because they know the field thanks to their physical activity. Moreover, we will make a distinction between informed and uninformed speculators because they are not equal in their ability to get information. As highlighted by Biais and Foucault (2013), the high-frequency trading did not change the practices described by Weiner. What changed is that trades are faster, which is obvious. They noticed that the informed speculators who use high-frequency trading pass market orders while marketmakers make limit orders(Biais and Foucault, 2013). The effect of the information integration by the high-frequency trading are quite ambiguous and remained unclear. The study of asymmetric information matters to understand to what extent the market is efficient. If the private information is included in the price, the equilibrium is fully revealing and then it is strongly efficient. Otherwise, it is partially revealing and the efficiency of the market is semi-strong. Vives (2010) elaborated a two-period model of asymmetric information in a futures market. There are on one hand speculators who are whether informed or uninformed. The informed speculators have a signal on the future spot price at the next period. In the Vives model, the spot price is exogenous to the future price. On an other hand, there are hedgers who are informed about an individual endowment shock. This endowment is the quantity that the hedger, who is a processor, will sell on the manufactured good markets. The model is quite complicated to solve with strong limitations such as the exogeneity of the spot price and of the quantities of manufactured goods. We will show that it is possible to have a very tractable model with endogenous spot prices and quantities of manufactured goods with an asymmetry on the harvest. Thus our approach is different of Perrakis and Khoury (1998) where their spot price is also exogenous in their asymmetric information model. In their model, the spot price is a martingale process. Thus, they test their model on three commodities markets of the Winnipeg Commodities Exchange’s (WCE), the Canadian futures market of commodities, between 1982 and 1994. At this time, the WCE was the only futures market for canola and barley. As a benchmark, they tested their model on the wheat market. Wheat is exchanged in Chicago too. They found non significant results for wheat markets but they got a significantly Fully-Revealing Rational Expectations Equilibrium (FRREE) for the barley and canola markets. The authors acknowledge that the assumption of an exogenous price is reasonable for small markets like canola and barley at the time of the WCE. However, it looks like less consistent for financialized markets. Because, the activities of hedger present on both markets may have consequences on the spot market. It is true in particular for storers. If the futures price is high, they will buy more spot. Therefore, the demand increases on the spot market so the spot price increases. Whatever the spot price is endogenous or exogenous, the expected price by an agent is conditional to one’s private signal. When the spot price is exogenous, the signal directly informs about the spot price in the next period. When it is endogenous, the signal is about a component of the spot price equation. Therefore, it can be an element of the demand or of the supply. On the demand side, the signal can reveal information about a productivity shock of the processors, which are end-users, for 5

example (Leclercq and Praz, 2014). Our approach is quite similar to Sockin and Xiong (2015) who studies informational frictions. However, the information asymmetry is about the demand side. Each good producer observes a private signal about a common productivity factor. The authors have a macroeconomic approach. The productivity shock of end-users is a macroeconomic factor. In this context, to study information asymmetry is interesting because it makes possible to see how informed agents convey their information about the macroeconomic situation to the commodity prices. While in our article, the information asymmetry is about the supply side. Our article follows a microeconomic approach which aims to study the interactions between the physical and futures market. We study how traders convey the information about the commodity production to the commodity market prices. Moreover, the different groups of agents have different signals. There is a no a lone informed group. An other big difference is that they refuse normal distribution for the parameters. However, their variables are log-normal and at the end, the informational effect can offset fundamental values. Their message is the following: « While the trading of financial traders does not have any direct effect on commodity supply and demand, it affects the futures price, through which it can further impact commodity demand and the spot price. » We have exactly the same issue in this model. Here, the financial traders can influence the future price which does affect the hedging pressure and thus the stored quantities and the processors’ demand. The Sokhin-Xiong’s approach and ours are complementary (Sockin and Xiong, 2015). The first one tackles informational frictions which offset the cost effect on price. The ours is about the informational effect which offsets the hedging pressure. Both are models with a rationalexpectations equilibrium (REE) under asymmetric information as defined by Grossman (1981). The agents have rational expectations, i.e traders know how the economy works. They gather all the available information and they can compute the state of the economy through the set of prices directly. We thus make the assumption that the traders use the correct model. By asymmetric information, we mean that the traders have a diverse information. This private information is revealed to traders through signals which are known only by the recipients. A model of information asymmetry on the supply side has been proposed by Goldstein and Yang (2015). Although our model is a pure extension of Ekeland et al. (2014), it has a very similar structure to Goldstein and Yang (2015). The farmers of the authors’ model are equivalent to the storers in our article. They have the same maximization program (their linear cost parameter plays the same role than the spot price in the period 1 for storers). Moreover, when the futures price increases under the influence of speculators, it will give incentives to the farmers or the storers to increase the supply in the next period. Therefore, a higher futures price drives down the spot price in particular in contango. It is interesting to notice the incompleteness of the markets in Goldstein and Yang (2015). The technological uncertainty generates more risks than tools to hedge it. Therefore, there is not an unique equilibrium. The authors found a price interval which depends positively of the risk aversion. It is consistent with the bid-ask spread defined by Carr et al. (2001) in incomplete markets. This spread corresponds to an absence of acceptable opportunity. An acceptable opportunity is an investment opportunity that is attractive to a sufficiently large group of investors so that the prices of the involved assets will be affected. The price interval for the equilibrium in Goldstein and Yang (2015) corresponds to an absence of acceptable opportunity. At least, their model rely on a strong assumption. The information among speculators is iid in Goldstein and Yang (2015). It implies a non-biased consensus among speculators. Therefore, an increasing weight of the speculators improves the quality of the average signal. It is not true anymore if there is a common error among all the individuals of the same group as in our model.

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3

The model

3.1

Description of the model

This article is an extension of Ekeland et al. (2014). It introduced the additional feature of asymmetric information. Therefore, the description of the model is largely inspired by the one in the article which is quoted above. The model is based on two time periods. There is one commodity, a numéraire, and two markets: the spot market at times t = 1 and t = 2 and a futures market in which contracts are traded at t = 1 and settled at t = 2. The model allows for short positions on the futures market. When an agent sells (resp. buys) futures contracts, his or her position is short (resp. long), and the amount of f he or she holds is negative (resp. positive). On the spot market, short positions are not allowed. In other words, the futures market is financial, while the spot market is physical. The model introduces asymmetric information in Ekeland et al. (2014) where there is speculation in a two-period equilibrium with endogenous spot prices and quantities of manufactured goods. A first difference with Ekeland et al. is the number of agents which is four instead of three. The three agents in Ekeland et al. are the processors, the storers and the speculators. • Processors (P ), or industrial users, use the commodity to produce other goods that they sell to consumers. Because of the inertia of their own production process and/or because all of their production is sold forward, they decide at t = 1 how much to produce at t = 2. They cannot store the commodity, so they have to buy all of their input on the spot market at t = 2.They also trade on the futures market. • Storers (I for inventory) have storage capacity and can use this capacity to buy the commodity at t = 1 and release it at t = 2. They trade on the spot market at t = 1 and at t = 2. We thus separate the roles of the processor and the storer, although in reality, processors can also hold inventory. The storers also operate on the futures market. • Speculators, or money managers, use the commodity price as a source of risk to make a profit out of their positions in futures contracts. They do not trade on the spot market. Here speculators are divided in two groups, informed ones (B ) and uninformed ones (S ). As noted by Vives (2010), the uninformed speculators play a role of liquidity providers. They bring liquidity in the futures market. However, market-making is a source of benefits for the both categories of speculators. The difference is that the informed speculator will exploit one’s private information to make a profit. The informed speculators, the storers and the processors are informed. Therefore,P they have a ¯ σθ ) which sθ is the signal, we have: private signal which is unbiased. For a parameter θ ∼ N (θ, sθ = θ˜ + ε

with ε ∼ N (0, σε2 )

˜ θ ) = ξθ sθ + (1 − ξθ )θ E(θ|s ˜ θ ) = (1 − ξθ )σ 2 V ar(θ|s θ

(1) (2) (3)

cov(θ,sθ ) θ) Where ξθ = cov(θ,s V ar(sθ ) . As Vives (2010) noticed, ξθ = V ar(sθ ) is the regression coefficient of the parameter θ on the private signal by the ordinary least squares. It is also equals to the square of the correlation coefficient. We made the assumption that there is no information cost. It allows us to study the fully-revealing rational expectations equilibrium (FRREE) which is not implementable with information cost. The partially-revealing rational expectations equilibrium (PRREE) does not change the analytic solutions of the equilibrium. We get an unknown distribution of the prices with a PRREE so it is tractable to study the stochastic aspects of the model. However, the mechanisms are still the same. For example, we will still get confidence interval but their shape will be unknown because the distribution of the prices is unknown as well. It is we will study the FREE mainly because we can still highlight the main mechanisms even if it is a very strong assumption. Further, the futures and spot markets operate in a sort of partial equilibrium framework: in the

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background, there are other sellers of the commodity, and processors as well. These additional agents are referred to as spot traders, and their global effect can be described by a demand function. At time t = 1, the (net) demand is µ1 − mP1 , and it is µ2 − mP2 at time t = 2. The Pt is the spot price at time t and the (net) demand can be either positive or negative; the superscript indicates a random variable. All traders make their decisions at time t = 1, conditionally on the information available for t = 2. The timing is as follows: • For t = 1, the commodity is in total supply ω1 , and the spot and the futures markets are open. On the spot market, there are spot traders and storers on the demand side, and the price is P1 . On the futures market, the processors, the storers, and the speculators all initiate a position, and the price is F . However, the storers have to decide simultaneously how much to buy on the spot market and what position to take on the futures market. • For t = 2, the commodity is in total supply ω ˜ 2 , to which one has to add the inventory carried by the storers from t = 1, and the spot market is open. The processors and the spot traders are on the demand side, and the price is P˜2 . The futures contracts are then settled. We assume that there is a perfect convergence of the basis at the expiration of the futures contract. Thus, at time t = 2, the position on the futures market is settled at price P˜2 that is prevailing on the spot market. There are NP processors, NB informed speculators, NS uninformed speculators, and NI storage companies. We assume that all agents (except the spot traders) are risk averse, inter-temporal utility maximizers. To make their decisions at time t = 1, they need to know the distribution of the spot price P˜2 at t = 2. Uncertainty is modeled by a probability space (Ω, A, P). The ω ˜2 , µ ˜2 , P˜2 are random variables on (Ω, A, P). At time t = 1, their realizations are unknown, but their distributions are common knowledge. We assume that µ ˜2 , ω ˜ 2 are normal and cov(˜ µ2 , ω ˜ 2 ) = 0, then µ ˜2 , ω ˜ 2 are independent. It is a strong restriction compared to Ekeland et al. but it is necessary to get a linear bayesian equilibrium. (εj )j∈{B,I,P } is the difference between the private signal (sj ) about the spot supply and the expected value of the spot supply (ω2 ) such that εj = sj − ω2 . Like in (1), εj ∼ N (0, σj2 ). Before we proceed, some clarifications are in order. • Production of the commodity is inelastic: the quantities ω1 and ω ˜ 2 that reach the spot market at times t = 1 and t = 2 are exogenous to the model. Traders know ω1 and µ1 , and share the same prior about ω ˜ 2 and µ ˜2 . However, they have different posteriors according to their private signals. That is the main difference with Ekeland et al. (2014) which is also the ˜ θ )) is the starting point of this article. In a Bayesian setting, the posterior distribution (P (θ|s following: ˜ (θ) fsθ |θ˜(sθ |θ)P ˜ P (θ|sθ ) = (4) R f ˜dθ˜ sθ | θ

˜ is the prior distribution. In this model, all agents share the same one for one parameter. P (θ) It means that all agents have the same belief about the distribution of any parameter. • A negative spot demand equals extra spot supply. If for instance P1 > µm1 , then the spot price at time t = 1 is so high that additional means of production become profitable, and the global economy provides additional quantities to the spot market. The number µ1 (demand when P1 = 0) is the level at which the economy saturates that induces spot traders to demand quantities larger than µ1 ,that is, the traders offer a negative price P1 < 0 for the commodity. The same situation occurs at time t = 2. • We set the risk-free interest rate at 0.

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3.2

Biased or unbiased consensus?

A key feature of this model is the common error for each group. It can be interpreted as a biased consensus. The literature highlighted the existence of an optimistic biased consensus among analysts (Knill et al., 2006). An explanation with rational agents has been brought (Lim, 2001). Analysts in an uncertain information environment and who are reliant on the management access as primary source are more likely to make optimistic bias forecasts about the companies’ earnings. However, our model relies on the assumption that the whole group is unbiased because of tractability purposes. In the normal-linear quadratic model, it is not possible to have a biased error. Moreover, the signal is not about earnings but the harvest in the next period. It is hard to say if forecasts about harvests are biased. For example, the USDA forecasts errors about harvests of have been associated mostly with structural changes. there is no evidence that they are biased systematically toward leniency or pessimismIsengildina-Massa et al. (2013). All the individuals in an informed group will receive the same signal. Individually, the agents are biased because their individual error is the group error. It is a "herding bias". The consequence is very important because when a group’s weight increases, the group’s bias in the average signal increases as well. For example, if the informed speculators have a very erroneous signal, the average signal will be distorted. There is only three different signals in the model so the distortion effect will be very important. It is the big difference with Goldstein and Yang (2015) where the agents have an unbiased individual signal. Unlike our model, an increasing weight of the speculators population increases the number of signals. Because the signals are unbiased and iid, the average signal will become closer of the true value of the fundamental. Moreover, if the speculators weight leans toward infinite, the average signal will lean toward the true value of the fundamental because of the central limit theorem. It is possible to get the result in our model by adding groups of speculators with different signals. Therefore, we must be aware that the convergence to the true value of the fundamental to next period is a consequence of the increasing number of signals. It is not a consequence of the increasing weight of the informed speculators population. The convergence described by Goldstein and Yang (2015) does not hold with the "herding bias" in our model. The average signal is distorted by the increased weight of the group’s error. The effects are ambiguous and difficult to analyze because it depends in which direction the average signal is pushed.

4

Main equations

In this section, we describe the main equations of the model and we show that a Bayesian linear equilibrium exists. This condition is sufficient to prove that a rational-expectations equilibrium (REE) exists.

4.1

industrial hedging

The behavior of the industrial hedgers is identical to the ELV model. In this framework, all agents have the possibility to undertake speculative operations in the futures market. First, they hedge 100 percent of their physical positions, then they adjust this position according to their expectations. The separation of the physical and the futures decisions was derived from Danthine (1978). Then, it is important to notice that the speculative positions depend on the private information while the hedging positions does not. The hedge are determined only with prices which are public information. The storers will compare the future price to the spot price in the period 1. The processors will do the same between the future price of the asset which is their input and the forward price of their output. The interests of the two categories are opposite. If the future price increases, the storers have a stronger incentive to increase their hedge while the processors would wish to decrease their one.

9

4.1.1

Processor’s hedging

The processor maximizes the following profit function: β 2 y )Z − y P˜2 + fP (P˜2 − F ) 2 β is the parameter of the quadratic function of production. Z is our convention for the forward price of the output. fP is the position of the processor on the futures market. Therefore, the processor’s optimal decisions are: π ˜P = (y −

Y ∗ = max(Z − F, 0) Y∗ y∗ = βZ E(P˜2 |F, sP ) − F fP = y ∗ + αP V ar(P˜2 |F, sP )

(5) (6) (7)

The futures market is also used by the processor to plan his or her production, particularly if the price of his or her input F is below that of his or her output Z. The position on the futures market can be decomposed into two elements: a hedge position y ∗ (the processor goes long on futures contracts in order to protect himself against an increase in the spot price) and a speculative position. αP and sP respectively are the risk aversion and the signal of the processor. 4.1.2

Storer’s hedging

The processor maximizes the following profit function: 1 π ˜ (x) = x(P˜2 − P1 ) + fI (P˜2 − F ) − Cx2 (8) 2 C is the parameter for the quadratic storage cost function. The storers can decide to buy a quantity x on the spot market at the first period to sell it in the second period(Ekeland et al., 2014). We define also the optimal hedge position x∗ : X ∗ = max(F − P1 , 0) X∗ x∗ = C

(9) (10) (11)

So the futures position of the storer is: fI = −x∗ +

E(P˜2 |F, sI ) − F αI V ar(P˜2 |F, sI )

(12)

αI and sI respectively are the risk aversion and the signal of the storer. 4.1.3

Hedging pressure

Like Ekeland et al. (2014), we define per-unit inventories X ∗ and per-unit demand Y ∗ . We will use synthetic numbers of processing units (nP ) and storing units (nI ) when it is relevant: NP βZ NI nI = C The hedging pressure (or the unbalance of hedging positions) is represented by nP

=

HP = nI X ∗ − nP Y ∗

(13) (14)

(15)

It is important to notice the hedging pressure is a weighted sum of the hedging positions. Therefore, the hedging pressure is public information because it relies on the prices which are known by everyone. 10

4.2

Determination of the Bayesian linear equilibrium

We will take the definition as given by Vives (2010): This equilibrium is the unique one in games (such as Cournot or Bertrand Games) where the strategy space of players is a noncontingent strategy like an output or the price. [...] In models where the strategy space is a functional, a supply or a demand function like in rational expectations models, then typically we can only show that there is a unique equilibrium in the linear class. The approach that we will follow is very similar to Vives (2010, section 4.4). The informed traders of Vives have the same behavior as the processors and the hedgers are very similar to the storers. However, the storers have private information in our model and the spot price is endogenous. Moreover, there is no private shocks in our model. Definition 4.1 Bayesian linear equilibrium The market-clearing conditions are met in a situation of asymmetric information. Moreover, all the endogenous prices are linear functions of the fundamentals. The existence of a such equilibrium implies the existence of a rational-expectations equilibrium. However, it is not because this equilibrium exists that it means this equilibrium is implementable. In our model, the spot prices in both period and the future price will be linear functions of the supply (ω2 ) and demand (µ2 ) parameters.

4.3

Characterization of the Bayesian linear equilibrium

All the agents have a mean-variance program and the received signal is the same for all the agents of the same informed group. For example, all the processors have the same signal. Likewise the storers have an other signal which is identical for all the population in this group. It is the same situation for the informed speculators as well. For an agent of the informed type i, the signal has the following shape: si = ω ˜ 2 + i (16) With i ∼ N (0, τi−1 ) . τ is the inverse of the variance of the error that we define as the precision of the signal. The conditional expectation of the variable according to the signal has been defined in the equation (2). We can write the regression coefficient of ω ˜ 2 on si as follows: ξi =

τi τω2 + τi

(17)

Higher is the precision of an agent, closer ξi is of 1. It means that the quality of the regression improves. When the precision of an agent is better than the others, he has a greater informational advantage. As we will see below, the trader will use this advantage to take a greater position. According to (10) and (6), we get: fS = fB = fP

=

fI

=

E(P˜2 − F |F ) αS V ar(P˜2 |F ) E(P˜2 − F |F, sB ) αB V ar(P˜2 |F, sB ) E(P˜2 − F |F, sP ) Y∗ + αP V ar(P˜2 |F, sP ) β Z E(P˜2 − F |F, sI ) X ∗ − C αI V ar(P˜2 |F, sI )

(18) (19) (20) (21)

si and αi for i = B, I, P respectively stands for the private signal which is unbiased and for the risk aversion. On the spot market, there is a physical constraint on the market-clearing condition. 11

Only positive quantities are allowed. Thus, the supply has to be equal to the demand. At the period 2, the harvest (˜ ω2 and the quantities sold by the storers (nI x∗ ) are on the supply side. The ∗ input level (nP y ) of the processors and the demand of the spot trader (˜ µ2 ) are on the demand side. The market-clearing condition is thus(Ekeland et al., 2014): ω ˜ 2 + nI X ∗ = nP Y ∗ + µ ˜2 − mP˜2

(22)

We take the conditional expectations and variance of P˜2 to the futures price from Ekeland et al. (2014). Thus : E(P˜2 |F ) = V ar(P˜2 |F ) =

1 X∗ Y∗ (µ2 − ω2 − NI + NP ) m C βZ V ar(µ˜2 ) + V ar(˜ ω2 ) m2

(23) (24)

As Vives (2010), we have to define a Bayesian linear equilibrium. Moreover, normal expectations have affine expression. We try to guess linear forms such as: X∗ Y∗ + NP − F m) C βZ Y∗ X∗ = mΨB (µ2 − ω2 − NI + NP − F m) − aB (sB − ω2 ) C βZ X∗ Y∗ Y∗ = mΨP (µ2 − ω2 − NI + NP − F m) − aP (sP − ω2 ) + C βZ βZ ∗ ∗ X Y X∗ = mΨI (µ2 − ω2 − NI + NP − F m) − aI (sI − ω2 ) − C βZ C

fS = mΨS (µ2 − ω2 − NI

(25)

fB

(26)

fP fI

(27) (28)

ai with i = B, I, P is a reaction coefficient to the spread between the signal and the expected value of the supply. We have the following market-clearing condition from Ekeland et al.: Σi={B,I,S,P } Ni fi = 0 ⇔ mΥ(µ2 − ω2 − HP ) − Υ F m2 − Σj={B,I,P } Nj aj (sj − ω2 ) − HP = 0

Where Υ = Σi={B,I,S,P } Ni Ψi . Thus, we get: F =

Σj={B,I,P } Nj aj (sj − ω2 ) 2 (µ2 − ω2 ) 1 − (1 + 2 )HP − m Υ m m Υ m2 Υ

(29)

Now, we have to identify the following coefficients (ai , Ψi ) for i = {B,I,S,P}. From the market clearing condition, we know that: NS

E(P˜2 |F ) − F E(P˜2 |F, sB ) − F E(P˜2 |F, sP ) − F Y∗ E(P˜2 |F, sI ) − F X∗ + NB + NP ( + ) + NI ( − ) C αS V ar(P˜2 |F ) αP V ar(P˜2 |F, sP ) αP V ar(P˜2 |F, sP ) β Z αI V ar(P˜2 |F, sI ) ∗

= NS

∗

m(µ2 − ω2 − NI XC + NP βY Z ) − F m2 αS (V ar(˜ µ2 ) + V ar(˜ ω2 )) ∗

+Σj={B,I,P } Nj

∗

m(µ2 − ω2 − ξj (sj − ω2 ) − NI XC + NP βY Z ) − F m2 αj (V ar(˜ µ2 ) + (1 − ξj )V ar(˜ ω2 ))

+ NP

Y∗ X∗ − NI βZ C

Then, we notice immediately: Υ = NB ΨB + NS ΨS + NI ΨI + NP ΨP ΨS = (αS (V ar(˜ µ2 ) + V ar(˜ ω2 ))) Ψj

= (αj (V ar(˜ µ2 ) + (1 − ξj )V ar(˜ ω2 ))) 12

(30)

−1

(31) −1

, j = B, I, P

(32)

Ψi is the inverse of the product of the risk aversion and of the conditional variance of the price to the signal of the agent i. It is the risk-adjusted information advantage (Vives, 2010). More correlated is the signal to a fundamental element of the price (i.e it is more informative), the bigger is risk-adjusted information advantage. If it is very high, Ψi leans toward infinite. At the opposite, the lower bond is ΨS which depends of the unconditional variance. Bigger is this advantage, higher the position of the trader will be. Thus: m2 Υ F

1 X∗ Y∗ Y∗ X∗ (µ2 − ω2 − NI + NP )) + NP − NI m C βZ βZ C −Σj={B,I,P } Nj Ψj ξj (sj − ω2 )

= mΥ(

Then we get: F =

Σj={B,I,P } Nj Ψj ξj (sj − ω2 ) 1 1 1 (µ2 − ω2 ) − ( + 2 )HP − m m m Υ m2 Υ

∗

(33)

∗

Where HP = NP βY Z − NI XC and Υ = Σi={B,I,S,P } Ni Ψi . Υ is related to the market depth, when it leans toward infinite, the reaction to the private signals becomes 0 and the sensitivity to the hedging pressure reaches its lower bound. As Ekeland et al. (2014), we defined φ as the sensitivity of the demand to the hedging pressure: φ=1+

1 mΥ

(34)

As we cans see, the lower bound of the demand sensitivity is 1.

4.4

Positions

The position of the uninformed speculator on the future market is: fS = mΨS (µ2 − ω2 − HP − F m)

(35)

. For the informed agents, we add aj (sj − ω2 ). For the commercial traders, we add the physical positions to cover. We guess immediately the coefficients of reaction to the difference between the private signals and the mean of the random variables in (29) and (33): aj = ξj Ψj

(36)

We can notice this coefficient is the product of the OLS-regression coefficient (ξj which measures the reliability of the signal) and the risk-adjusted information advantage (Ψj ). We get thus a weighting coefficient of the deviation of the group’s signal (sj ) in comparison to the expected value of the spot supply (ω2 ). We can say now that a Bayesian linear equilibrium exists, thus we have a rational expectations equilibrium which exists.

4.5

Market-clearing

Eventually, we have the market-clearing equations (Ekeland et al., 2014). They correspond to the equilibrium on the spot and on the future markets. On the spot, the supply is equal to the demand. The short positions are forbidden. While on the futures market which is financial, the equilibrium

13

is met when the sum of the positions is null. Therefore, we get:  NP  nP = β,Z       nI = NCI    ∗    X = max(F − P1 , 0)   Y ∗ = max(Z − F, 0)  HP = nI X ∗ − nP Y ∗     1   P1 = m (µ1 − ω1 + nI X ∗ )     1  P˜2 = m (˜ µ2 − ω ˜ 2 − nI X ∗ + nP Y ∗ )     F = 1 (µ − ω ) − ( 1 + 1 )HP − Σj={B,I,P } Nj aj (sj −ω2 ) 2 2 m m m2 Υ m2 Υ

(37)

The difference with the equilibrium with symmetric information is an additional term in the price of the future (F). It is the asymmetric information term that we will write as follows: η=

Σj={B,I,P } Nj aj (sj − ω2 ) m2 Υ

(38)

We can notice that the hedging pressure depends of the synthetic weight of the hedgers while the asymmetric information term depends of the nominal weight.

5

Definition of the equilibrium

Above, we have defined a sufficient condition for the existence of a rational-expectation equilibrium. Now, we define the necessary conditions.

5.1

Existence of a quasi-equilibrium

If the system (37) is satisfied, we have a quasi-equilibrium (Ekeland et al., 2014). A quasiequilibrium is a family (X ∗ , Y ∗ , P1 , F, P2 ) such that all prices, except possibly P˜2 , are non negative; processors, storers, and speculators act as price-takers; and all markets clear. We get a slightly different system of two non linear equations for two variables as Ekeland et al. (2014) from the market-clearing conditions (37) : mP1 − nI X ∗ = µ1 − ω1 Σj={B,I,P } Nj aj (sj − ω2 ) 1 mF + (1 + )(nI X ∗ − nP Y ∗ ) + = µ2 − ω2 mΥ mΥ

(39) (40)

Thus we have the four regions of Ekeland et al. (2014) but their size will vary because of the term for asymmetric information.

5.2

Equilibrium

To have an equilibrium, as Ekeland et al.(2014) defined the following condition from the necessary positivity of the price at the period 2 (P˜2 ) such as: P˜2

≥0

⇔ µ ˜2 − ω ˜ 2 ≥ HP Now, we can write the modified additional conditions of the theorem 2 of Ekeland et al.(2014) which states under which condition an equilibrium exists. These additional conditions at the existence of a quasi-equilibrium are proper to each region. We will write the conditions of existence by using the notation for the demand sensitivity φ as defined in (34).

14

Figure 1: Physical and financial decisions in space (P1 , F ): the four regions defined by Ekeland et al. (2014)

5.2.1

Region 1

The hedging demand from both processors and storers is positive so X ∗ > 0 and Y ∗ > 0. We thus get: µ ˜2 − ω ˜2 ≥ m

−ω2 mnI ( µ2m −

µ1 −ω1 m )

−ω2 − nP (m + nI )(Z − µ2m ) − η(nI nP + m(nP + nI )) m(m + nI + (nI + nP )φ) + nI nP φ

(41)

The difference with Ekeland et al. (2014) is this term for the asymmetric information: η(nI nP + m(nP + nI )) 5.2.2

Region 2

The hedging demand from processors is null while the storers’ one is positive so X ∗ > 0 and Y ∗ = 0. We thus get: −ω1 −ω2 − µ1m ) − η) m(nI ( µ2m (42) µ ˜2 − ω ˜2 ≥ 1 + nI (1 + φ) In this region, the difference with Ekeland et al. (2014) is just the asymmetric information term at the numerator. 5.2.3

Region 3

The hedging demand is null so X ∗ = 0 and Y ∗ = 0. So: µ ˜2 − ω ˜2 ≥ 0

(43)

There is no difference with Ekeland et al. (2014) because in this very particular case, the spot price is completely exogenous. That fact shows us the very interest to relax the hypothesis of exogenous spot prices to study futures markets. It is true only if the hedging pressure is null, which is not always true in reality. 5.2.4

Region 4

The hedging demand from storers is null while the processors’ one is positive so Y ∗ > 0 and X ∗ = 0. We thus get: −ω2 + η) mnP (Z − µ2m µ ˜2 − ω (44) ˜2 ≥ − nP φ + m 15

The difference with Ekeland et al. (2014) in the region 4 is the same as in the region 2. Because the expectation η is null, the expectation of µ2 − ω2 does not change in comparison to Ekeland et al. Thus, we can apply their theorem 3 directly about the conditions on the values of µ2 − ω2 per region.

5.3

The equilibrium is fully revealing

The equilibrium of this model is a fully revealing rational expectations equilibrium (FRREE) as defined by Grossman (1976). It means that the private signals are revealed by the future price. Mathematically, the future price is a linear function of a weighted average of the signals by the sizes and the risk-adjusted information advantages of the groups of traders. Thus, it is possible to invert this function to get this weighted average. If all the private signals were equal to this weighted average, the equilibrium would not be modified. Thus, the future price fulfills its function of price discovery1 . Proposition 5.1 In a future market, under the assumption that the private signal is the only source of private noise, a FRREE is equivalent to a future price which is a linear function of the weighted-average of the signals. Proposition 5.2 The equilibrium of this model is a FRREE It is important to notice the difference with Perrakis and Khoury (1998). In their model specifications, the market is in a PRREE if the spot supply is random. In our model, we get a FREE while there is a random spot supply. The difference between the spot supply at the two periods is: 1 P˜2 − P1 = (˜ µ2 − ω ˜ 2 − µ1 + ω1 − 2nI X ∗ + nP Y ∗ ) m

(45)

With X ∗ = M ax(0, F − P1 ) which is the basis at the first period when it is positive and Y ∗ = M ax(0, Z − F ). The coefficient of the basis is hard to interpret in Perrakis and Khoury (1998). In our model, it is a non positive coefficient which is in absolute value two times the negative weight of the storers. Indeed, greater is the basis, higher is the incentive for storers to buy commodities to sell them in the next period. Therefore, it raises the spot price in the initial period and it decreases it in the second period. While in Perrakis and Khoury (1998), the sign is non negative. Moreover, the authors of this article add that a null coefficient for the basis means a symmetric information. It is not true when the spot price is endogenous. A null coefficient means that the storer are not hedging. Moreover, if hedging is null for storers but positive for processors, the equilibrium is only partially revealing if there are heterogeneous technologies among processors. We will deal later with this issue after we analyzed the equilibrium.

6

Equilibrium analysis

In this section, we will show how the privates signals make stochastic the regions defined by Ekeland et al. (2014). The belonging of a point in the physical space is not deterministic any more. Asymmetric information makes it probabilistic. Then, we will study the factors of the informational effect.

6.1

The new frontier

Contrary to Ekeland et al., the sign of the hedging pressure does not say any more if the future price is negatively or positively biased. It means that the futures price does not reflect the hedging 1

proof in the annex C

16

pressure fully. The asymmetric information term,η, shifts the border of future-price equation (33). Thus inside the region 1 we get a new frontier, ∆ = F − E[P˜2 ] = nI X ∗ − nP Y ∗ + Σj={B,I,P } Nj aj (sj − ω2 ) = 0

(46)

This new frontier has a very strong implication: the asymmetric information term can completely offset the dynamics of the physical space which are described in the Ekeland-Lautier-Villeneuve model. Without asymmetric information, the borders of the different regions are deterministic. One point in the physical space will always belong to the same region. It is not true any more with informed agents because the borders are probabilistic. Therefore, one point in the physical space will belong to a region according to a probability. If the point remains in the deterministic region, the physical effect is stronger. However, if the point belong to a new region, it means that the informational effect is stronger. The informational effect is the new feature of the ELV model with asymmetric information. The hedging pressure is not only determined by the physical constraints but also by the private signals of the informed agents. The separation of the region 1 in two subregions is thus stochastic as well. 6.1.1

The stochastic nature of the new frontier

If we look at the asymmetric information term in the new frontier (46), we have a sum of of normal variables by taking in account the non biased aspect of the signal and the equation (36) : Σj={B,I,P } Nj aj (sj − ω2 ) = Σj={B,I,P } Nj ξj Ψj (εj + ω ˜ 2 − ω2 )

(47)

Therefore, the equation (47) is a sum which is weigthed by the nominal weight, the reliability of τε the signal and the risk-adjusted information advantage. We have ξj = τε +τj ω with τx = σx−1 . j

2

According to (32), we can write the risk-adjusted information advantage as follows: ψj =

1 τµ (τεj + τω2 ) αj τεj + τω2 + τµ N

τ

(48) τε

Thus, Σj={B,I,P } Nj ξj Ψj εj ∼ N (0, Σj={B,I,P } ( αjj τε +τωµ +τµ )2 τωj (τεj + τω2 )). 2 2 j For lightening the notation, we will use the following one for the variance: V = Σj={B,I,P } ( 6.1.2

τε τµ Nj )2 j (τεj + τω2 ) αj τεj + τω2 + τµ τω2

(49)

The informational effect: a probabilistic bias in the future price

The asymmetric information term is random so the spot price and the future price are random as well. Thus, they are not determined by the hedging pressure. We can distinguish two effects, the market-making effect and the informational effect. The first one is deterministic and the second one is probabilistic. We can study the probability of the sign of the bias of the future price in two spaces: the price space and the physical space. In the price space, the sign of the hedging pressure is associated to a probability. Its value varies according to the private signals of the informed agents. First, for HP>0, which would mean normally a positive future bias in the frame of Ekeland et al, we will get the following probability

17

of negative bias in the future price: F − E[P˜2 ] ≥ 0 ⇔ −(

NI ∗ NP ∗ X (F − P1 ) − Y (Z − F )) − Σj={B,I,P } Nj ξj Ψj (εj + ω ˜ 2 − ω2 ) ≥ 0 c βZ ⇔ Σj={B,I,P } Nj ξj Ψj (εj + ω ˜ 2 − ω2 ) ≤ −HP ⇔

Σj={B,I,P } Nj ξj Ψj (εj + ω ˜ 2 − ω2 ) √ V

HP ≤ −√ V

HP ⇔ P (Σj={B,I,P } Nj ξj Ψj (εj + ω ˜ 2 − ω2 ) ≤ −HP ) = 1 − Φ( √ ) V We can generalize the result to get a probability of offsetting: |HP | P (|Σj={B,I,P } Nj aj (sj − ω2 )| ≥ |HP |) = 1 − Φ( √ ) V

(50)

Then, we get the following probability of conservation by symmetry: |HP | P (|Σj={B,I,P } Nj aj (sj − ω2 )| ≤ |HP |) = Φ( √ ) V Thus, we can establish a conservation ratio of the hedging pressure, C = 1 2

(51) |HP √ |. V

If C->0, the

probability will lean toward and if C rises, the probability will increase. To get the probability of conservation, we just need to look at the table for the standard normal law. 0.4 0.3 0.2 0.1

−2

0

2

4

Figure 2: The probability of conservation according to the conservation ratio embodied by the red dashed lines. The range of values of conservation ratio is positive numbers. The area under the curve at the left of the dashed line is the probability of conservation. At the right, it is the probability of offsetting. We can notice that the conservation ratio is a local informativeness ratio for a given hedging pressure. We can distinguish two different effects: • The physical effect: If the hedging pressure is high under the physical constraint ceteris paribus, the future price will be likely biased in function of the parameters for the demand (µ2 ) and for the supply (ω2 ) • The informational effect: If the variance of the sum of the weighted signal deviations to the unconditional expected value of the harvest is high, the future price will be likely offset. The informational effect is quite similar to Sohkin and Xiong (2014) even it is about the commodity cost for producers (Sockin and Xiong, 2015). The informational effect will be stronger when the precision of the supply parameter and of the signal errors is low. 18

We can traduce this phenomenon graphically in the Ekeland-Lautier-Villeuve (ELV) map of price biases from the equations (39) and (40):

F Z

2

A 1U

3

M

Δ

* 0 Y > * −n P * < 0 X n I * n PY − n IX

4

1L

45°

O

P1

Figure 3: The modified ELV map with a random frontier

Ekeland et al. (2014) defined 4 regions: • The region 1 where F > P1 and Z > F so both kind of industrials are hedging. • The region 2 where F > P1 and Z < F so storers are hedging only. • The region 3 where F < P1 and Z < F so no one is hedging. • The region 4 where F < P1 and Z > F so processors are hedging only. The first region is divided into two subregions, the region 1U where the hedging pressure is positive and the region 1L where the hedging pressure is negative. The red dashed line is for a frontier shift upward, thus the information term is negative. It means that the informed agents have lower signals (sj ) than the expected value of the supply parameter (ω2 ). Therefore, the conditional expectations of the informed agent are in average lower than the unconditional expectation of the spot supply. It drives up the futures price and so the hedging pressure. Thus, the futures-price bias (which is the difference between unconditional expectation and the futures price) decreases. Contrary to Ekeland et al. (2014) where the frontier divides only the first region in two subregions, the new frontier crosses the region 2. Therefore, a part of the region 2 are in a situation of a negative bias in the futures price. In these parts at the right of the red dashed lines, the informational effect dominates the physical effect. In the region 3, where the physical effect is null, the informational effect dominates automatically. Thus, the region 3 will be in a situation of a negative bias in the future price automatically. It shows that when the informed traders anticipate a negative supply shock, they push the futures price up. It is the opposite when the frontier is shifted downward, it is embodied by the green dashed lines. Higher expectations from the informed traders about the spot supply will drive down the futures price, the hedging pressure and the futures-price bias. A part of the region 1L ( area with a negative hedging pressure in the region 1) and of the region 4 will be in a situation of positive bias in the future price. Ultimately, the private signals are integrated in the price. 19

We can determine exact values for the probability in the physical space. we represent the equations (39) and (40) in the space (ξ1 , ξ2 ):

Figure 4: In the space (ξ1 , ξ2 ), The borders between the regions will be stochastic as well. They will be shifted upward if η > 0 (green arrows) or downward if η < 0 (red arrows). The quasi-equilibrium is more detailed in the appendix A.1.

6.2

The factors of the informational effect

As we saw above, the variance of the informational term in the future price is: V = Σj={B,I,P } (

τε Nj τµ )2 j (τεj + τω2 ) αj τεj + τω2 + τµ τω2

The precision is greater than zero so the variance of the asymmetric information term depends positively of the precision of each group signal. It is consistent with Vives (2010), there is more volatility because there is more information about fundamentals. The effect of the precision of the harvest is more ambiguous. the first-order condition is a eighth-order polynomial so it is hard to solve it.

7

The impact of speculation

First, we must distinguish the effects according to the information of the speculator. The consequences will not be the same if the increasing weight of speculators is informed or not.

7.1

An Increasing weight of the uninformed speculators

If there is an increasing of uninformed speculation, there is no difference with the results of Ekeland et al. The increasing of the weight (NS ) or of the risk-adjusted information advantage (ψS ) of uninformed speculators increase the term υ. As Vives reminded it, this term is related to market depth (Vives, 2010). The direct consequence is that the price of the future leans toward the expected price at period 2. In the same time, volatility is increasing. Vives explained that the spot price at period 2 becomes more volatile because it becomes more informative about fundamentals. Region 4 is an exception because storers are not active in this zone, thus the shock is not transferred. It is also why the spot price remains constant in period 1.

20

Figure 5: The effects of the increasing of uninformed speculation as described in the ELV model: Impact of speculators on prices and quantities. Legend: % variable increases; & variable decreases; 0 variable is null; ←→ no impact on variable.

The difference between the informed and the uninformed speculators is the informational effect. To measure the effect of the increasing of informed speculation on the future price, we must take in account both the market-making effect and the informational effect.

7.2 7.2.1

Increasing of the informed speculation Effect on the future price: a fully-revealing trap?

The variation of the future price affects the hedging pressure. The reduction of the difference between the future price and the expected price at period 2 increases the disequilibrium of the ∂F hedging pressure. For any region k (k ∈ [[1, 4]]), we can reduce the study of ∂N to the following B condition: reg

regk

βψB Z(NI + Cm)(GB k (sB − ω2 ) − AP

regk

(sP − ω2 ) − AI

(sI − ω2 ) − J(µ2 − ω2 ) + J 0 (µ1 − ω1 ) + $Z) < 0

The coefficients (Ai )i={B,I,P } , GB , $, J and J’ are strictly non negative. Moreover, the coefficients (Ai )i={B,I,P } and GB depend on the risk-adjusted information advantage of the agents2 . AB and GB are also linear in ξB , thus it is possible to deduce lower bounds for ξB corresponding to the conditions above. This last coefficient is the regression coefficient of the supply parameter (ω2 ) on the private signal (sB ) of the informed speculators by the ordinary least squares. The future price will increase if: reg

µ 2 − ω2 >

regk

GB k (sB − ω2 ) − AP

regk

(sP − ω2 ) − AI J

(sI − ω2 ) + J 0 (µ1 − ω1 ) + $Z

(52)

Contrary to the ELV model, we have a different condition for the positive bias in the future price: reg

µ 2 − ω2 >

regk

AB k (sB − ω2 ) − AP

regk

(sP − ω2 ) − AI J

(sI − ω2 ) + J 0 (µ1 − ω1 ) + $Z

(53)

If the conditions (52) and (53) hold, an increasing weight of informed traders will increase the future price while the bias is positive. In a FRREE, the future price is a linear function of the weighted average of the signals. An increased weight of informed speculators thus increases their weight in the average. If the speculators have very different expectations of the hedgers, the informational effect will reinforce the bias between the future price and the expected spot price for the next period. We can notice that this dynamics is the consequence of a strongly efficient market. Moreover, there is only three different signals in this model. Therefore, a very heavy group can distort the average signal toward its group’s signal. It can be a problem if the signal is very 2

More precision in the appendix B about the computations of the conditions

21

erroneous. The average signal will push the price in a direction which is not related to the true value of the spot supply in the next period. Well, the average signal is fully revealed so the price information depends of it very much. Therefore, a very heavy group with an erroneous signal will distort the average signal and so the price information will be wrong. Therefore, it will modify the hedgers’ strategies. For example, if the average signal underestimates the harvest, the futures price will increase. If it generates or increases a situation of contango, stocks will rise. Therefore, spot prices will rise in the first period and they will diminish in the second period. In this particular case, the inter-period volatility on the spot market increases. We even may have a "fully-revealing trap" if |˜ ω2 − ω2 | < |η|. It implies that the precision-adjusted weighted deviation (η) is greater than the initial spread between the harvest and the expected value. It means that the signals lead the traders to a misleading information set. However, the precision-adjusted weights limit the influence of the less precise signals in the weighted average signal. It is interesting to notice that the oil term structure switched in contango in December 2014 and that one year later, the market is flooded by a high-level of storage which keeps the prices low. Since the end of 2014, speculators were stashing high quantities of crude oil on tankers. The signals are independent and we know from the market-clearing conditions 37, that the future price (F) depends negatively of the asymmetric information term(η). Therefore, the spot price depends negatively of it in period 1 (P1 ) and positively in the period 2 (P2 ). So we can study ∂η the effect of an increasing informed speculation on the prices by studying the sign of ∂N directly: B ξB >

(NB ψB + NI ψI + NP ψP + NS ψS )(NI ψI ξI (sI − ω2 ) + NP ψP ξP (sP − ω2 )) NB ψB

(54)

We have determined a lower bound for the confidence of the informed speculators. We can deduce a constraint on the weight of informed speculators (NB ). To summarize, the informational effect is ambiguous because it depends of the weight of the informed speculators in the weighted sum of the demeaned signals. 7.2.2

Effect on the variances

The existence of informed trading increases the variance of the prices. It adds a new source of randomness. So, the variance of the prices increases according to the spread of the asymmetric information term. The variance of the signals appears first in the future price. It is spread by the hedging demand from the storers (X ∗ ) to the spot price in the period 1 (P1 ) and by the hedging pressure in the period 2 (P˜2 ). Therefore, the variance of the spot price in the period 1 will be higher in the region 1 and 2 but not in the region 3 and 4 in comparison to the ELV model. Moreover, there is no difference with ELV in the variance of the spot prices in the region 3. However, the big difference with ELV is the presence of the asymmetric information term in the future-price bias (E[P˜2 ] − F ) so its variance increases. According to (49), we have: V ar(η) =

Σj={B,I,P } (Nj Ψj )2 (V ar(εj ) + V ar(ω2 )) m4 (NB ψB + NI ψI + NP ψP + NS ψS )2

(55)

So if we derive by NB , we get the following conditions to have an increasing variance with an increasing number of informed speculators: NB φB >

NI2 φ2I (V ar(εI ) + V ar(˜ ω2 )) + NP2 φ2P (V ar(εP ) + V ar(˜ ω2 )) (V ar(εB + V ar(˜ ω2 ))(NI φI + NP φP )

(56)

The effect of an increasing number of informed speculators depends of a threshold. The variance of the asymmetric term is decreasing until this threshold. The liquidity effect dominates so an increasing number of informed speculator has the same effect as an increasing of the uninformed speculation. Once the threshold is reached, the informational effect is stronger so the 22

variance increases. Greater is the number of speculator, greater will be the variance of the asymmetric information term (η). The risk-adjusted information advantage has a similar effect and it decreases the threshold for the numbers of informed speculators. It is thus equivalent to study the nominal weight or the risk-adjusted information advantage of the speculators. To complete the analysis about the liquidity effect and the informational effect, it would be interesting to study the covariance between the speculators’ positions and the futures price (Goldstein and Yang, 2015). Unfortunately, the first-order conditions are too hard to be computed for the signal precision (τεj ) and the nominal weight (NB ) of the speculators. 7.2.3

Impact on the utilities

In all the utilities, there is a speculative component. For any agent, the speculation benefits are originally from the spread between one’s expectation of the spot price and the future price, i.e from the future-price bias. The hedgers have a benefit from the hedging activity. As summarized by Ekeland et al. (2014): « For all agents, we see a clear separation between the two components of the indirect utilities. The speculative component is associated with the level of the expected basis. The hedging component changes with the category of agent considered. For the storers, it is positively related to the current basis F - P1, and for the processors, it rises with the margin on the processing activity Z - F. » We have the following indirect utilities: US = UB = UI

=

UP

=

E[(P˜2 − F )2 |F ]

(57)

ar(˜ ω2 ) 2αS V ar(˜µ2 )−V m2 E[(P˜2 − F )2 |F, sB ] ω2 ) B )V ar(˜ 2αB V ar(˜µ2 )−(1−ξ m2 E[(P˜2 − F )2 |F, sI ] ω2 ) I )V ar(˜ 2αI V ar(˜µ2 )−(1−ξ m2 E[(P˜2 (P˜2 − F )2 |F, sP ] ω2 ) P )V ar(˜ 2αP V ar(˜µ2 )−(1−ξ m2

(58) +

X2 2C

+

Y2 2β Z

(59) (60)

The denominator is the inverse of the risk-adjusted information advantage. A Bigger risk-adjusted information advantage is equivalent to a smaller denominator. Therefore, the position is greater. An increasing number of informed speculator has two effects. First, it makes the market deeper exactly like for an increasing uninformed speculation. Second, it increases the weight of their signal in the weighted-average of the signals. Therefore, we can distinguish two different impacts: 1. The impact on the speculative benefits will have different effects according to the variation of the spread between the future price and the conditional expectation to the signal and to the price of the different groups (E[P˜2 |sj , F ]). 2. The impact on the hedging benefits is ambiguous. If the market-making effect is not offset by the informational effect then the utility of the hedger category which has the biggest hedging position. Inversely, if the informational effect goes in an opposite direction and is stronger than the informational effect (the condition (52) is fulfilled), the hedging benefits decreases for this same category.

7.3

Toward a partially-revealing equilibrium

The futures market is ruled by a FRREE. Yet, this equilibrium is not implementable, as explained by Pennacchi (2008): 23

However, as shown by Sanford Grossman and Joseph Stiglitz (Grossman and Stiglitz 1980), this fully revealing equilibrium is not robust to some small changes in assumptions. Real-world markets are unlikely to be perfectly efficient. For example, suppose each trader needed to pay a tiny cost, c, to obtain his private signal, si . With any finite cost of obtaining information, the equilibrium would not exist because each individual receives no additional benefit from knowing si given that they can observe s¯ from the price. In other words, a given individual does not personally benefit from having private (inside) information in a fully-revealing equilibrium. In order for individuals to benefit from obtaining (costly) information, we need an equilibrium where the price is only partially revealing. For this to happen, there needs to be one or more additional sources of uncertainty that add “noise” to individuals’ signals, so that other agents cannot infer them perfectly. The noisy rational expectations equilibrium was introduced by Grundy and McNichols (1989). They introduced endowment shocks in the distribution of shares of the risky asset among traders. Thus, the equilibrium is partially revealing. Vives (2010) has chosen an analogous solution in its model of futures market. In his model, the hedgers’ positions on the futures market are subject to an endowment shock. The initial endowments are distributed randomly among among the hedgers and they have a future random value. This one is correlated to the spot price. The equilibrium in the futures market thus becomes partially-revealing. The direct consequence is that the function of price discovery is inflected but the Grossman-Stiglitz paradox (Grossman and Stiglitz, 1982) is solved. The endowment shocks remain private information. As noticed by Ekeland et al. (2014), the heterogeneity of costs do not modify closed-form solution of the spot price and the futures price in the period 1 for the region 1: P1 = F

=

−ω1 2 −η m(m + (nI + nP )φ) µ1m + mnI µ2 −ω + nI nP φ Z m m(m + (nI + nP )φ) + mnI nI nP φ

(61)

−ω1 2 −η mnI φ µ1m + m(m + nI ) µ2 −ω + (m + nI )nP φ Z m mnI φ + m(m + nI )nP φ(m + nI )

(62)

The only difference is the modification of the synthetic weights. We have nP = Z1 i β1i and P nI = i C1i . Therefore, with an heterogeneity in the costs, we have a partially-revealing equilibrium in the region 1. Because we have three unknowns value (nI , nP and η). The region 4 will be partially-revealing as well because the spot price is not informative about the futures price. Indeed, the storage level is null. Moreover, the forward price for the final good is exogenous. It would be different if it was endogenous. It is discussed in the appendices. On the contrary, the region 3 and 2 will be fully-revealing. As we can see in the equations (62) and (61), the spot (P1 ) and futures price (F) in the period 1 −ω1 2 −η are convex combination of µ2 −ω , µ1 m and Z. The convex set is slightly different from Ekeland m µ2 −ω2 et al. (2014) because we added −η to m . With asymmetric information, the convex set is random because of η. Without a cost heterogeneity, the equilibrium is fully-revealing because the convex combinations are known. Therefore, it is possible to compute the missing point of the convex set. With cost heterogeneity, it is fully revealing too if there is one unknown value in the convex combination as in the region 2. In the region 2, the storers only are hedging. However, the equilibrium is not normal-linear because the prices are not linear in the random variables any more. Therefore, there is no partially-revealing equilibrium which is linear-normal. The convex combination and the convex set are both random when there is both asymmetric information and cost heterogeneity. Moreover, it is not possible to draw confidence interval as in the section 6.1.2 for the frontier in the region 1 when the equilibrium is fully-revealing. P

The normal distribution is not realist for the commodity prices. The cost heterogeneity makes the model more realist but less tractable. An alternative is to adopt log-normal distribution 24

to keep the properties of a bayesian linear equilibrium for the logarithm values of the parameters. This method has been adopted by Sockin and Xiong (2015). Moreover, they introduced an uncertainty in the cost function of the producer. With this uncertainty in the supply, the equilibrium is only partially revealing.

8

Conclusion

The boundaries of the equilibrium map are stochastic. The asymmetric information introduced a random term in the definition of the equilibrium. The boundaries move according to the value of the supply parameter in the next period and the errors of the signal. The informed agents adjust their speculative position according to two parameters. First, they consider their risk-adjusted information advantage to take a greater position. Last, they subtract the difference between their signal and the expectation of the supply parameter in function of the signal reliability to the initial uninformed speculative position. The stochastic nature of the region boundaries make more difficult to analyse the impact of the informed speculation on the utilities. The variation of the utilities depends of the errors of the informed agents which are random. Last, the Bayesian linear equilibrium implies the separation the hedging position, the uninformed speculative position and the informed speculative position. The assumptions on the distribution make it possible. The model shows that the increasing of the informed speculation decreases volatility until a threshold. Firstly, the liquidity effect dominates so the volatility decreases. Secondly, the informational effect outweighs and the volatility rises once the threshold is reached. It is important to notice that there is no difference with Ekeland et al. (2014) about the uninformed speculation. It still increases liquidity. Therefore, the results of this model are very similar to Sockin and Xiong (2015) . There is a difference between the providers and the consumers of liquidity. We have to distinguish between strategies which consumes liquidity and strategies which provides liquidity. It is consistent with Cheng and Xiong (2014). The increasing of informed speculation can distort the average signal if the speculators’ one is very erroneous. This "herding bias" can drive the futures prices for motives which are not relative to fundamentals. Therefore, the storage level is affected. If the futures prices are drove up in a contango situation by an erroneous signal, it will decrease the spot price in the first period and it will increase it in the second period. This feedback on the spot prices exists because of the storage channel. Storers will adjust their behavior to a futures price which reveals a poor information set. The partially-revealing equilibrium is not normal-linear. It is possible to introduce private shocks in the cost parameters of the processors and the storers to get a partially revealing equilibrium in the region 1 and 4. The equilibrium is implementable but it is not a linear Bayesian equilibrium because of the cost heterogeneity. However, if a second futures market for the final good is introduced, the equilibrium will be fully-revealing3 . Moreover, the separation of the physical and the futures decisions would still hold (Anderson and Danthine, 1983). It is possible to get then a partially revealing equilibrium with a stochastic demand.

3

The introduction of private shocks is discussed in annex C

25

List of Figures 1 2 3 4 5

The decisions in the price space . . . . . . . . . . . . The conservation probability of the hedging pressure Stochastic Frontier . . . . . . . . . . . . . . . . . . . The decisions in the physical space . . . . . . . . . . effects of an increased uninformed speculation . . . .

26

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

15 18 19 20 21

References Anderson, R. W. and Danthine, J.-P. (1983). Hedger Diversity in Futures Markets. Economic Journal, 93(37):370–89. Biais, B. and Foucault, T. (2013). High-frequency trading and market quality. In Bachelier, I. L., editor, High frequency trading, liquidity and stability, volume 2. Jean-Michel Beacco, Paris. Carr, P., Geman, H., and Madan, D. B. (2001). Pricing and hedging in incomplete markets. Journal of Financial Economics, 62(1):131–167. Cheng, I.-H. and Xiong, W. (2014). Financialization of commodity markets. Annual Review of Financial Economics, 6(1):419–441. Danthine, J.-P. (1978). Information, futures prices, and stabilizing speculation. Journal of Economic Theory, 17(1):79–98. Ekeland, I., Lautier, D., and Villeneuve, B. (2014). Speculation in commodity futures markets: A simple equilibrium model. Goldstein, I. and Yang, L. (2015). Commodity financialization : Risk sharing and price discovery in commodity futures markets. Grossman, S. (1976). On the efficiency of competitive stock markets where trades have diverse information. The Journal of Finance, 31(2):573–585. Grossman, S. and Stiglitz, J. (1982). On the impossibility of informationally efficient markets: Reply. The American Economic Review, 72(4):875. Grossman, S. J. (1981). An introduction to the theory of rational expectations under asymmetric information. The Review of Economic Studies, 48(4):pp. 541–559. Grundy, B. D. and McNichols, M. (1989). Trade and the revelation of information through prices and direct disclosure. Review of Financial Studies, 2(4):495–526. Hau, H. (2001). Location matters: An examination of trading profits. The Journal of Finance, 56(5):1959–1983. Isengildina-Massa, O., Karali, B., and Irwin, S. H. (2013). When do the usda forecasters make mistakes? Applied Economics, 45(36):5086–5103. Kaldor, N. (1939). Speculation and economic stability. The Review of Economic Studies, 7(1):pp. 1–27. Khoury, N. T. and Martel, J. (1989). A supply of storage theory with asymmetric information. Journal of Futures Markets, 9(6):573–581. Knill, A., Minnick, K., and Nejadmalayeri, A. (2006). Selective hedging information asymmetry, and futures prices. The Journal of Business, 79(3):1475–1501. Lautier, D. and Simon, Y. (2009). Energy finance: The case for derivatives markets. In Macmillan, P., editor, The New Energy Crisis: Climate, Economics and Geopolitics, pages 231–255. JeanMarie Chevalier, Basingstoke. Leclercq, E. and Praz, R. (2014). Equilibrium commodity trading. Lim, T. (2001). Rationality and analysts’ forecast bias. The Journal of Finance, 56(1):369–385. Pennacchi, G. (2008). Theory of Asset Pricing, chapter Equilibrium with private information, pages 459–482. Addison-Wesley series in finance. Pearson/Addison-Wesley. 27

Perrakis, S. and Khoury, N. (1998). Asymmetric information in commodity futures markets: Theory and empirical evidence. Journal of Futures Markets, 18(7):803–825. Sockin, M. and Xiong, W. (2015). Informational frictions and commodity markets. The Journal of Finance, page forthecoming. Vives, X. (2010). Information and Learning in Markets: The Impact of Market Microstructure. Princeton University Press. Weiner, R. J. (2002). Sheep in wolves’ clothing? speculators and price volatility in petroleum futures. The Quarterly Review of Economics and Finance, 42(2):391–400.

A

Properties of the Quasi-equilibrium and the equilibrium

Note that nI =

NI C ,

Np =

NP βZ

and:

φ = 1+ η =

1

(63)

m(Σi={B,I,S,P } Ni Ψi )

Σj={B,I,P } Nj ξj Ψj (sj − ω2 ) m2 Σi={B,I,S,P } Ni Ψi )

(64)

ΨS = (αS (V ar(˜ µ2 ) + V ar(˜ ω2 )))−1 Ψj

= (αj (V ar(˜ µ2 ) + (1 − ξj )V ar(˜ ω2 )))

(65) −1

, j = B, I, P

(66)

The region is determined by (ξ1 , ξ2 ), and the final expressions of equilibrium prices are as follows. A remark for all subsequent calculations. Starting from Region 1, setting nP = 0 gives expressions for Region 2; setting nI = 0 gives expressions for Region 4.

A.1

Quasi-equilibrium

We describe here the modified theorem 1 of Ekeland et al. (2014). We examine the images by ϕ of Regions 1 to 4. In Figure 5.1, we denote by O the origin in R2+ , by A the point (0, Z ), and by M the point (Z, Z ). In Region 1 (triangle OAM ), we have: ϕ(P1 , F ) =

mP1 − nI (F − P1 ) mF + φ(nI (F − P1 ) − nI (Z − F )) + η

!

This map is composed by the equations (39) and (40). The images ϕ(A), ϕ(M )andϕ(0) are easily computed: ϕ(0) = (0, −nP + η), ϕ(A) = (−ZnI , Z(m + φ nI F ) + η), ϕ(M ) = (mZ, mZ + η) From this, one can find the images of all four regions. The image of Region 1 is the triangle ϕ(0)ϕ(A)ϕ(M ). The image of Region 2 is bounded by the segment ϕ(A)ϕ(M ) and by two infinite half-lines: one of which is the image of {P1 = 0, F ≥ Z}, the other being the image of {P1 = F, F ≥ Z}. In Region 2, we have: ϕ(P1 , F ) =

mP1 − nI (F − P1 ) mF + φ(nI (F − P1 ) + η

!

The first half-line emanates from ϕ(A) and is carried by the vector (−Z NCI , Z(m + φ NCI F ) + η). The second half-line emanates from ϕ(0)ϕ(M ) and is carried by the vector (mZ, mZ + η). Both 28

of them (if lengthened) go through the origin. The image of Region 4 is bounded by the segment ϕ(M ) and by two infinite half-lines, one of which is the image of F = 0, the other being the image of {P1 ≥ F, F = Z}. In Region 4, we have: ϕ(P1 , F ) =

mP1 mF − nP (Z − F )) + η

!

so the first half-line emanates from ϕ(0) and is horizontal with vertical coordinate − NβP + η, and the second emanates from ϕ(M ) and is horizontal. The image of Region 3 is entirely contained in R2+ where it is the remainder of the three images we described. The only difference is the asymmetric information term in the coordinates of the points. The borders will thus be translated upward (η > 0) or downward(η < 0). The sign of η is equivalent to the sign of the sum of the weighted bias of the informed groups. Here is the bias of an informed group is equal to si − ω2 for i = {B, I, P }.

A.2

Equilibrium

A.2.1

Region 1

P1 = F P˜2 =

=

−ω1 2 −η m(m + (nI + nP )φ) µ1m + mnI µ2 −ω + nI nP φ Z m m(m + (nI + nP )φ) + mnI + nI nP φ

(67)

−ω1 2 −η mnI φ µ1m + m(m + nI ) µ2 −ω + (m + nI )nP φ Z m mnI φ + m(m + nI ) + (m + nI )nP φ

(68)

µ ˜2 −˜ ω2 m

+

−ω1 2 −η − ((m + nI )nP + mnI ) µ2 −ω + (m + nI )nP Z mnI µ1m m mnI φ + m(m + nI ) + (m + nI )nP φ

(69)

All the denominators are equal. They are written in different ways only to show that P1 and F are −ω1 µ2 −ω2 −η convex combinations of µ1m , and Z. We can notice we have a random convex set because m η is a normal-distributed variable. Thus: E[P˜2 ] − F =

−ω2 1 −ω1 ) (φ − 1)(mnI µ2 −ω2 −(µ + (m + nI )nP (Z − µ2m )) + (m + nI )η m 2 mnI + m + mnI φ + mnP φ + nI nP φ

(70)

Quantities: X∗ = Y

∗

=

−ω2 1 −ω1 ) m(m + nP φ) µ2 −ω2 −(µ + mnP φ(Z − µ2m ) − mη m mnI + m2 + mnI φ + mnP φ + nI nP φ

(71)

−ω2 1 −ω1 ) + m(m + nI (1 + φ))(Z − µ2m ) + (nI + m)η mnI φ µ2 −ω2 −(µ m 2 mnI + m + mnI φ + mnP φ + nI nP φ

(72)

We can see directly that the hedging pressure (HP = nI X ∗ − nP Y ∗ ) depends negatively of the asymmetric information term (η). It makes sense because this term affects also negatively the futures price while the hedging pressure depends positively of it. It reflects the negative effect of the higher conditional expectations of the spot supply (in comparison to the unconditional expectation) on the futures price.

29

A.2.2

Region 2

Here nP = 0, thus: P1 = F

=

P˜2 = X∗ =

−ω1 2 −η m(m + nI φ) µ1m + mnI µ2 −ω m mnI + m2 + mnI φ −ω1 2 −η mnI φ µ1m + m(m + nI ) µ2 −ω m mnI + m2 + mnI φ −ω1 2 −η nI ( µ1m − µ2 −ω ) µ ˜2 − ω ˜2 m +m m nI + m + nI φ µ2 −ω2 −(µ1 −ω1 ) m

−η nI + m + nI φ

Y∗ = 0

(73) (74) (75) (76) (77)

We know that X ∗ > 0 thus the following condition holds in the region 2: µ2 − ω2 − (µ1 − ω1 ) >η m

(78)

Therefore: 1 −ω1 ) (φ − 1)nI µ2 −ω2 −(µ + (m + nI )η m ˜ E[P2 ] − F = >0 nI + m + nI φ η nI µ2 − ω2 − (µ1 − ω1 ) >− ⇔ m + nI m (φ − 1) nI ⇔ (µ2 − ω2 − (µ1 − ω1 )) > −Σj={B,I,P } Nj ξj Ψj (sj − ω2 ) m + nI

A.2.3

Region 3 P1 =

A.2.4

µ 1 − ω1 ˜ µ ˜2 − ω ˜2 µ2 − ω2 ; P2 = ;F = − η; E[P˜2 ] − F = η m m m

(79)

Region 4

P1 =

2 −η 2 −η m µ2 −ω + nP φ Z ˜ ) ∗ µ 1 − ω1 µ ˜2 − ω ˜ 2 nP (Z − µ2 −ω m m ;F = ; P2 = + ; X = 0; m m + nP φ m m + nP φ −ω2 m(Z − µ2m )+η Y∗ = m + nP φ

(80) (81)

Therefore: −ω2 (φ − 1)nP (Z − µ2m )−η >0 m + nP φ µ2 − ω2 ⇔ nP (Z − ) > Σj={B,I,P } Nj ξj Ψj (sj − ω2 ) m

F − E[P˜2 ] =

A.3

Distribution which supports the equilibrium

Ekeland et al. determined a distribution which supports the equilibrium. For any region, the condition is the following: E[P˜2 ] > 0

30

Well, E[η] = 0 so the conditions are exactly the same as in the ELV model: µ2 − ω2 m µ2 − ω2 m µ2 − ω2 m µ2 − ω2 m

B

µ −ω

>

nI 1m 1 +(m+nI )nP Z m(m+(φ−1)nP )+nI (mφ+(φ−1)nP )

in region 1

(82)

>

µ1 −ω1 nI − m+n m Iφ

in region 2

(83)

>

0

in region 3

(84)

>

nP Z mφ+(φ−1)nP

in region 4

(85)

Impact of informed speculation

If we differentiate the future price by the number of informed speculators, we get:

B.1

Region 1

∂F ∂NB A

B

=

A B (β ψB Z(NI + Cm)(NI NP sB ξB − NI NP ω2 ξB − Cm3 µ2 NP + Cm3 NP ω2

−

m2 µ2 NI NP + Cm4 NP Z + m2 NI NP ω2 + m3 NI NP Z + CmNP sB ξB − CmNP ω2 ξB

+

β m3 µ1 NI Z − β m3 µ2 NI Z − β m3 NI ω1 Z + β m3 NI ω2 Z + β mNI sB ξB Z

−

β mNI ω2 ξB Z + mNI2 NP ψI sB ξB + mNI NP2 ψP sB ξB − mNI2 NP ψI sI ξI − mNI NP2 ψP sP ξP

−

mNI2 NP ψI ω2 ξB − mNI NP2 ψP ω2 ξB + mNI2 NP ψI ω2 ξI + mNI NP2 ψP ω2 ξP

+

Cm2 NP2 ψP sB ξB − Cm2 NP2 ψP sP ξP − Cm2 NP2 ψP ω2 ξB + Cm2 NP2 ψP ω2 ξP + 2β m2 NI2 ψI sB ξB Z

−

2β m2 NI2 ψI sI ξI Z − 2β m2 NI2 ψI ω2 ξB Z + 2β m2 NI2 ψI ω2 ξI Z

+

mNI NP NS ψS sB ξB − mNI NP NS ψS ω2 ξB + Cm2 NI NP ψI sB ξB − Cm2 NI NP ψI sI ξI

+

Cm2 NP NS ψS sB ξB − Cm2 NI NP ψI ω2 ξB + Cm2 NI NP ψI ω2 ξI − Cm2 NP NS ψS ω2 ξB

+

β Cm3 NI ψI sB ξB Z + β Cm3 NP ψP sB ξB Z − β Cm3 NI ψI sI ξI Z + β Cm3 NS ψS sB ξB Z

−

β Cm3 NP ψP sP ξP Z − β Cm3 NI ψI ω2 ξB Z + β Cm3 NI ψI ω2 ξI Z − β Cm3 NP ψP ω2 ξB Z

+

β Cm3 NP ψP ω2 ξP Z − β Cm3 NS ψS ω2 ξB Z + 2β m2 NI NP ψP sB ξB Z

+

2β m2 NI NS ψS sB ξB Z − 2β m2 NI NP ψP sP ξP Z − 2β m2 NI NP ψP ω2 ξB Z + 2β m2 NI NP ψP ω2 ξP Z

−

2β m2 NI NS ψS ω2 ξB Z))

=

(m(NI NP + CmNP + β mNI Z + mNI2 NP ψI + mNI NP2 ψP

+

Cm2 NP2 ψP + 2β m2 NI2 ψI Z + mNB NI NP ψB + mNI NP NS ψS + Cm2 NB NP ψB + Cm2 NI NP ψI

+

Cm2 NP NS ψS + β Cm3 NB ψB Z + β Cm3 NI ψI Z + β Cm3 NP ψP Z + β Cm3 NS ψS Z

+

2β m2 NB NI ψB Z + 2β m2 NI NP ψP Z + 2β m2 NI NS ψS Z)2 )

=

−

Now, we can study the sign of -A. If A

J 0 (µ1 − ω1 ) + $Z J

This condition can be easily simplified. We can notice it only depends on fundamental parameters: the forward price (Z), the cost parameters (β,C) and the numbers of the different groups. In the presence of asymmetric information, this condition becomes: reg

µ2 − ω2 >

reg

reg1

GB 1 (sB − ω2 ) − AP 1 (sP − ω2 ) − AI J

(sI − ω2 ) + J 0 (µ1 − ω1 ) + $Z

The coefficients (Ai )i={B,I,P } and GB depend on the risk-adjusted information advantage of the agents. That is a fundamental difference with the ELV model. It can be explained by the fact that the coefficient of reaction to the difference between the signal and the expected value of the supply is the product of the risk-adjusted information advantage (ψi ) and of the coefficient of regression of the supply parameter on the signal (ξi ). If we compare with F − E(P˜2 ) < 0 which is equivalent to: (mNI NP ω2 − mµ2 NI NP − Cm2 µ2 NP + Cm2 NP ω2 + Cm3 NP Z + m2 NI NP Z + β m2 µ1 NI Z − β m2 µ2 NI Z − β m2 NI ω1 Z + β m2 NI ω2 Z − NI2 NP ψI sI ξI − NI NP2 ψP sP ξP + NI2 NP ψI ω2 ξI + NI NP2 ψP ω2 ξP − NB NI NP ψB sB ξB + NB NI NP ψB ω2 ξB − CmNP2 ψP sP ξP + CmNP2 ψP ω2 ξP − CmNB NP ψB sB ξB − CmNI NP ψI sI ξI + CmNB NP ψB ω2 ξB + CmNI NP ψI ω2 ξI − 2β mNI2 ψI sI ξI Z + 2β mNI2 ψI ω2 ξI Z − β Cm2 NB ψB sB ξB Z − β Cm2 NI ψI sI ξI Z − β Cm2 NP ψP sP ξP Z + β Cm2 NB ψB ω2 ξB Z + β Cm2 NI ψI ω2 ξI Z + β Cm2 NP ψP ω2 ξP Z − 2β mNB NI ψB sB ξB Z − 2β mNI NP ψP sP ξP Z + 2β mNB NI ψB ω2 ξB Z + 2β mNI NP ψP ω2 ξP Z)/(m2 (NI NP + CmNP + β mNI Z + mNI2 NP ψI + mNI NP2 ψP + Cm2 NP2 ψP + 2β m2 NI2 ψI Z + mNB NI NP ψB + mNI NP NS ψS + Cm2 NB NP ψB + Cm2 NI NP ψI + Cm2 NP NS ψS + β Cm3 NB ψB Z + β Cm3 NI ψI Z + β Cm3 NP ψP Z + β Cm3 NS ψS Z + 2β m2 NB NI ψB Z + 2β m2 NI NP ψP Z + 2β m2 NI NS ψS Z)) < 0 We can write this condition like this: reg

reg

reg1

AB 1 (sB − ω2 ) − AP 1 (sP − ω2 ) − AI J = M2mNB ψB ξB

µ 2 − ω2 > reg1

AB

(sI − ω2 ) + J 0 (µ1 − ω1 ) + $Z

∂F We notice that when the private signals are uninformative, F − E(P˜2 ) < 0 ⇔ ∂N > 0 which B is a property of the ELV model. Thus, this property does not hold any more with asymmetric information.

Region 2 ∂F ∂NB

= −ψB (NI + Cm)(2mµ1 NI − 2mµ2 NI − 2mNI ω1 + 2mNI ω2 + NI sB ξB − NI ω2 ξB

−

4mNI2 ψI sI ξI + 4mNI2 ψI ω2 ξI + 2mNI NP ψP sB ξB + 2mNI NS ψS sB ξB

−

4mNI NP ψP sP ξP − 2mNI NP ψP ω2 ξB + 4mNI NP ψP ω2 ξP − 2mNI NS ψS ω2 ξB

+

Cm2 NP ψP sB ξB − 2Cm2 NI ψI sI ξI + Cm2 NS ψS sB ξB − 2Cm2 NP ψP sP ξP

+

2Cm2 NI ψI ω2 ξI − Cm2 NP ψP ω2 ξB + 2Cm2 NP ψP ω2 ξP − Cm2 NS ψS ω2 ξB )

/

(m(NI + 4mNB NI ψB + 2mNI NP ψP + 2mNI NS ψS + 2Cm2 NB ψB + Cm2 NP ψP + Cm2 NS ψS )2 )

32

Thus: ∂F >0 ∂NB reg reg reg ⇔ 2mNI (µ1 − ω1 − (µ2 − ω2 )) + GB 2 (sB − ω2 ) − AI 2 (sI − ω2 ) − AP 2 (sP − ω2 ) < 0 With: reg2

= (NI (1 + 2m(NP ψP + NS ψS )) + Cm2 NS ψS )ξB

AI

reg2

= (4mNI + 2Cm2 )NI ψI ξI

reg AP 2

= (4mNI + 2Cm2 )NP ψP ξP

GB

The difference with the region 1 is that all the factors linked to the hedging demand of the processors disappear. However, we can write a similar condition to the region 1 to see when prices decrease while the future price is negatively biased.

B.2

region 3 ∂F >0 ∂NB ⇔ (NI ψI + NP ψP + NS ψS )ξB (sB − ω2 ) − NI ψI ξI (sI − ω2 ) − NP ψP ξP (sP − ω2 ) < 0

It is also a big difference with ELV where the derivative is 0 because there is no term for the asymmetric information. The future price will evolve according to the demeaned signals, the weight of the different groups and the precision of their signals.

B.3

region 4

We get: ∂F ∂NB

= −(βψB Z(m2 NP Z − mµ2 NP + mNP ω2 + NP sB ξB

−

NP ω2 ξB + mNP2 ψP sB ξB − mNP2 ψP sP ξP − mNP2 ψP ω2 ξB + mNP2 ψP ω2 ξP

+

mNI NP ψI sB ξB − mNI NP ψI sI ξI + mNP NS ψS sB ξB − mNI NP ψI ω2 ξB

+

mNI NP ψI ω2 ξI − mNP NS ψS ω2 ξB − β m2 NI ψI sB ξB Z − β m2 NP ψP sB ξB Z

+

β m2 NI ψI sI ξI Z − β m2 NS ψS sB ξB Z + β m2 NP ψP sP ξP Z + β m2 NI ψI ω2 ξB Z

−

β m2 NI ψI ω2 ξI Z + β m2 NP ψP ω2 ξB Z − β m2 NP ψP ω2 ξP Z + β m2 NS ψS ω2 ξB Z))

/

(NP + mNP2 ψP + mNB NP ψB + mNI NP ψI + mNP NS ψS − β m2 NB ψB Z

−

β m2 NI ψI Z − β m2 NP ψP Z − β m2 NS ψS Z)2

Thus: ∂F >0 ∂NB reg reg reg ⇔ mNP (m − (µ2 − ω2 )) + GB 4 (sB − ω2 ) − AI 4 (sI − ω2 ) − AP 4 (sP − ω2 ) < 0 With: reg4

= (NP (1 + m(NP ψP + NI ψI + NS ψS ) + β Zm2 (NI ψI + NP ψP + NS ψS ))ξB

reg4

= (mNP + β Zm2 )NI ψI ξI

reg4

= (mNP + β Zm2 )NP ψP ξP

GB AI

AP

The region 4 is a particular case of the region 1 like the region 2. Symmetrically to the region 2, the terms linked to the storage costs disappear. 33

C C.1 F

Discussion on the fully-revealing equilibrium Region 1 = (Z(m NI NP + C m2 NP + β mµ1 NI − β m NI ω1 + C m3 NP2 ψP + m2 NI2 NP ψI + m2 NI NP2 ψP + β m2 µ1 NI2 ψI + β m2 µ2 NI2 ψI − β m2 NI2 ψI ω1 − β m2 NI2 ψI ω2 + C m3 NB NP ψB + C m3 NI NP ψI + C m3 NP NS ψS + m2 NB NI NP ψB + m2 NI NP NS ψS − β NI2 ψI sI ξI + β NI2 ψI ω2 ξI + β C m3 µ2 NB ψB + β C m3 µ2 NI ψI + β C m3 µ2 NP ψP + β C m3 µ2 NS ψS − β C m3 NB ψB ω2 − β C m3 NI ψI ω2 − β C m3 NP ψP ω2 − β C m3 NS ψS ω2 + β m2 µ1 NB NI ψB + β m2 µ2 NB NI ψB + β m2 µ1 NI NP ψP + β m2 µ2 NI NP ψP + β m2 µ1 NI NS ψS + β m2 µ2 NI NS ψS − β m2 NB NI ψB ω1 − β m2 NB NI ψB ω2 − β m2 NI NP ψP ω1 − β m2 NI NP ψP ω2 − β m2 NI NS ψS ω1 − β m2 NI NS ψS ω2 − β NB NI ψB sB ξB − β NI NP ψP sP ξP + β NB NI ψB ω2 ξB + β NI NP ψP ω2 ξP − β C m NB ψB sB ξB − β C m NI ψI sI ξI − β C m NP ψP sP ξP + β C m NB ψB ω2 ξB + β C m NI ψI ω2 ξI + β C m NP ψP ω2 ξP )) /

(m(NI NP + C m NP + β m NI Z + m NI2 NP ψI + m NI NP2 ψP + C m2 NP2 ψP

+ 2β m2 NI2 ψI Z + m NB NI NP ψB + m NI NP NS ψS + C m2 NB NP ψB + C m2 NI NP ψI + C m2 NP NS ψS + β C m3 NB ψB Z + β C m3 NI ψI Z + β C m3 NP ψP Z + β C m3 NS ψS Z + 2β m2 NB NI ψB Z + 2β m2 NI NP ψP Z + 2β m2 NI NS ψS Z)) We can see that the future price is fully revealing. The future price is linear in the signals. Indeed, there is no signal ((si )i={B,I,S} ) in the denominator and at the numerator, all the signals are just multiplied by a scalar. So it is possible to deduce from the price the following weighted average (¯ s) of the signals: P i={B,I,P } Ni ψi ξi si (86) s¯ = P i={B,I,P } Ni ψi ξi Indeed, F Y

Z (Y + JF0 (µ1 − ω1 ) + JF (µ2 − ω2 )) + χ(¯ s − ω2 )) Π X = mNP (NI + m(C + (Cm + NI ) Ni ψi )

=

i={B,I,P,S}

JF0

X

= β mNI (1 + m

Ni ψi )

i={B,I,P,S}

JF

X

= m2 β(Cm + NI )

Ni ψi

i={B,I,P,S}

χ = β(NI + Cm)

X

Ni ψi ξi

i={B,I,P }

Π = m(NI NP + m(C NP + β NI Z) +

X

Ni ψi (m NI NP + C m2 NP + 2β Zm2 NI + β ZC m3 )

i={B,I,P,S}

such as we have a function F (¯ s) which is linear in s¯ so it is possible to compute s¯ = F −1 (F ). By construction s¯ is a sufficient statistics for the private signals of the traders. It implies that the future price is fully revealing as defined by Grossman (1976). As consequence, the market is strongly efficient. We can notice that the future price fulfills it price-discovery function here. According to the market clearing conditions, we have: P1 =

1 NI (µ1 − ω1 + (F − P1 )) m C 34

(87)

So we have both spot prices(P1 ) and future (F) prices, which are both function of C,β and s¯. If there are private shocks in the both cost parameters (C and β), the equilibrium will be partially revealing (Grundy and McNichols, 1989). However, if privates shocks are introduced in only one of the parameters, the equilibrium will be still fully revealing. It is because, we would get a linear system with two equations and two unknown variables. In this model, as in ELV, the forward price (Z) is exogenous. If we make it endogenous, we would have four equations for four unknown parameters, so the equilibrium will be fully revealing.

C.2 F

Region 2 = (µ1 NI − NI ω1 + mµ1 NI2 ψI + mµ2 NI2 ψI − mNI2 ψI ω1 − mNI2 ψI ω2 − NI2 ψI siξI + NI2 ψI ω2 ξI + mµ1 NB NI ψB + mµ2 NB NI ψB + mµ1 NI NP ψP + mµ2 NI NP ψP + mµ1 NI NS ψS + mµ2 NI NS ψS − mNB NI ψB ω1 − mNB NI ψB ω2 − mNI NP ψP ω1 − mNI NP ψP ω2 − mNI NS ψS ω1 − mNI NS ψS ω2 − NB NI ψB sB ξB − NI NP ψP sP ξP + NB NI ψB ω2 ξB + NI NP ψP ω2 ξP + Cm2 µ2 NB ψB + Cm2 µ2 NI ψI + Cm2 µ2 NP ψP + Cm2 µ2 NS ψS − Cm2 NB ψB ω2 − Cm2 NI ψI ω2 − Cm2 NP ψP ω2 − Cm2 NS ψS ω2 − CmNB ψB sB ξB − CmNI ψI siξI − CmNP ψP sP ξP + CmNB ψB ω2 ξB + CmNI ψI ω2 ξI + CmNP ψP ω2 ξP )/(m(NI + 2mNI2 ψI + 2mNB NI ψB + 2mNI NP ψP + 2mNI NS ψS + Cm2 NB ψB + Cm2 NI ψI + Cm2 NP ψP + Cm2 NS ψS ))

According to the market clearing conditions, we have: P1 =

1 NI (µ1 − ω1 + (F − P1 )) m C

(88)

There is no hedging demand from the processors so private shocks on their costs (β) will not influence how the future price is revealing. However, privates shocks in the storage costs (C) will not make the equilibrium partially revealing. Indeed, we have a linear system with two equations and two unknown variables. It is logical if we are considering the functions of commodities markets: storage, risk sharing and information discovery. The future market provides information about the expectations of the traders. It is exactly what we can observe. The existence of the futures markets provides information about storage costs and private information. Moreover, risk sharing is improved because speculators can enter in the market and correct the hedging pressure. The hedging costs thus are lowered.

C.3

Region 3

The equilibrium is always fully revealing because the spot and future prices are not dependent of the cost parameters.

C.4

Region 4

The spot price (P1 ) does not reveal information about the processors’ costs (β) because the hedging demand of the storers is null. Indeed: P1 =

1 (µ1 − ω1 ) m

(89)

However, as we said about the region 1, the forward price (Z) is exogenous. With an endogenous forward price, we would get back to a linear system with two equations and two unknown variable. Thus, the equilibrium would be fully revealing.

35