Strategic futures trading in oligopoly


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1 Strategic futures trading in oligopoly Jhinyoung SHIN Department of Business Administration, Yonsei University, Seoul, Korea Kit Pong WONG School of Economics and Finance, University of Hong Kong Pokfulam Road, Hong Kong July 7, 2003 Keywords: Commitment; Futures contracts; Observability; Speculation JEL classification: D82; G4; L3 We wish to acknowledge helpful comments of Sudipto Dasgupta, Tim Adam, Changhyun Yoon, and seminar participants at Hong Kong University of Science and Technology, Seoul National University, Korea University, and Korea Advanced Institute of Science and Technology. All remaining errors are entirely our own. Corresponding author. Department of Business Administration, Yonsei University, Seoul, Korea. Phone: , School of Economics and Finance, University of Hong Kong, Pokfulam Road, Hong Kong. Tel.: , fax: ,
2 2 Abstract This paper investigates the incentives for firms to engage in futures trading in the context of an oligopolistic product market and strategic trading in a futures market. If the producing firms are risk neutral and have perfect information about product market conditions, they can use futures trading strategically to improve their positions in the product market competition or/and to gain trading profits from their private information. It is demonstrated that if the futures positions taken by the producing firms are observable as they make their product market decisions, their motivation of adopting futures trading as a credible commitment device for the product market competition outweighs the incentive to earn trading profits from informationbased trading, and they producing firms expect to lose money from futures trading. However, if the futures transactions of the producing firms are not observable, the effectiveness of futures trading as a commitment device in the product market competition disappears, and they trade in the futures market just to earn trading profits. A couple of policy implication on the welfare effect on consumers and producers, and the stability of futures market are drawn from the model.
3 . Introduction It is well known that investors participate in futures markets for various reasons: (i) to hedge against risk and uncertainty, (ii) to gain arbitrage profits by taking advantage of any mispricing in futures and/or spot markets, (iii) or to earn trading profits from speculation. Allaz (992) and Allaz and Vila (993) find that riskneutral producers use futures contracts strategically to improve their competitive positions in an oligopolistic spot market. In the context of a twostage model of futures contracting followed by production in an oligopolistic spot market, they demonstrate that in the absence of any uncertainty, futures contracts are employed by riskneutral firms as a credible strategic device in an attempt to commit themselves to more aggressive output decisions later in the production stage. Hughes and Kao (998) show that observability of futures positions is crucial for futures contracts to function as an effective commitment device, and in case of unobservable futures contracts, risk aversion of firms or cost uncertainty are called for to restore the strategic usefulness of futures contracts. Allaz (992), Allaz and Vila (993), and Hughes and Kao (998) all assume perfect competition of a futures market, where the futures price is always an unbiased estimator of the spot price at which futures contracts will be settled. This paper considers a futures market where strategic trading is conducted by informationbased traders including producing firms. 2 Unlike the models of Allaz (992), Allaz and Vila (993), and Hughes and Kao (998) in which firms decides their optimal futures positions taking the futures price as given, we extend the strategic trading models of Kyle (985) and Admati and Pfleiderer (988), and analyze the optimal futures Allaz and Vila (993) extend the basic twostage model to repeated rounds of futures trading. 2 Broll and Wong (2002) consider an imperfect futures market wherein producing firms are risk averse and uninformed.
4 2 trading strategies adopted by producing firms which take the effects of their futures trading on the transaction price of futures contracts as well as the spot price at which the futures contracts will be settled. 3 If producing firms are risk neutral and have perfect information about product market conditions, risk reduction cannot be the rationale for their participation in a futures market. There are two reasons that prompt the producing firms to trade futures contracts. First, as shown in Allaz (992), Allaz and Vila (993), and Hughes and Kao (998), the producing firms can use futures trading strategically to improve their positions in the product market competition. Second, they participate in the futures market as informationbased traders to gain trading profits. What sets this paper apart from the standard market microstructure framework is that the producing firms not only have access to the private information on the spot price at which futures contracts will be settled, but also they can directly affect it since their production decisions determine the spot price in the product market. 4 Depending on whether the futures positions taken by the producing firm are observable or not, qualitatively different results emerge. If the futures positions taken by the producing firms are observable, their motivation of adopting futures trading as a credible device to commit themselves to more aggressive product market strategies outweighs their incentives to earn positive trading profits from futures trading. In fact, the producing firms expect to lose money from futures trading although they 3 Hughes, Kao and Williams (2002) also adopt strategic trading model of Kyle (985) to analyze the equilibrium of futures market. What is demonstrated by them is that if information asymmetry on the demand condition of the product market exists between producing firms, and therefore only one firm can be engaged in informationbased futures trading, mandate disclosure of futures position forces firms to share their private information, and thereby alters the strategic nature of product market competition. This paper assumes information symmetry among producing firms, and there is no information sharing effect due to the mandate disclosure of futures positions. 4 Kyle (985), Admati and Pfleiderer (998), and other market microstructure models typically assume that orders submitted by traders only affect transaction prices, not the fundamental payoff of the security.
5 3 possess utmost advantage against other participants in the futures market. However, if the futures transactions are not observable, the effectiveness of futures trading as a commitment device in the product market competition disappears, and the producing firms trade in the futures market just to earn trading profits. Unlike Hughes and Kao (998) in which firms are not engaged in any futures trading in case of unobservability, we show that the producing firms still participate in the futures market to earn trading profits even when the usefulness of futures position as a commitment device in the product market disappears due to unobservability. The issue of accounting disclosure of derivative positions taken by firms has caught public attention as the draft proposal of Statement of Financial Accounting Standard No. 33 (SFAS No. 3) entitled Accounting for Derivative Instruments and Hedging Activities. 5 This paper intends to analyze implication of public disclosure of futures positions taken by producing firms on the equilibrium of both product and futures market as well as the welfare of consumers and producers. This paper also raises the possibility of nonexistence of linear rational expectations equilibria in futures markets. This is mainly due to the producing firms ability of affecting the transaction price at which futures contracts are traded as well as the spot price at which futures positions are settled. As the transaction price of futures contracts becomes less sensitive to orders submitted by traders due to sufficiently large number of informationbased traders including the producing firms, substantial noise in the futures market, or competition among market makers, the producing firms might adopt extremely aggressive trading strategies in the futures market to take full advantage of their positions in affecting both futures and spot prices. 6 5 Please refer to Hughes, Kao and Williams (2002) on details on this proposal and financial executives reaction to it. 6 This paper provides explanation on nonexistence of futures market of certain commodities such as semiconductors.
6 4 The rest of this paper is organized as follows. In the next section, we delineate the basic model of a twostage imperfect information game. Sections 3 derives the rational expectations equilibrium in which the futures positions taken by the producing firms are observable. Sections 4 characterizes the rational expectations equilibrium in which the futures positions taken by producing firms are unobservable. We then compare and contrast the two equilibria. The final section concludes the paper. 2. The model Consider a homogeneous commodity which has a futures market and a spot market. There is one period with two dates, 0 and. At date, the commodity is produced by k firms, indexed by i =,, k, and is sold in the spot market according to the following inverse industry demand function: k p = a + θ q i, i= where p is the spot market price of the commodity, q i is the output of firm i, a is a positive constant, and θ is an industrywide demand shock normally distributed with mean zero and variance σθ. 2 The k firms compete in quantities and produce at constant marginal costs which are normalized to zero. 7 The futures market for the commodity opens at date 0. Infinitely divisible futures contracts, each of which calls for delivery of one unit of the commodity at date, are traded in this market at a prespecified futures price schedule, p f, which is a functions of net aggregate orders submitted by traders. There are four types of riskneutral agents populated in the futures market: the k producing firms, n informed traders (indexed by j =,, n), a cohort of noise traders who trade for exogenous 7 The results of this paper do not change at all if positive constant marginal cost is assumed.
7 5 reasons, and a competitive market maker. The trading mechanism is that of Kyle (985) and Admati and Pfleiderer (988) in which the k firms, the n informed traders, and the group of noise traders submit market orders to the market maker who takes a position that balances supply and demand. Prior to trading in the futures market at date 0, the k firms and the n informed traders receive a private signal which perfectly reveals the date demand parameter, θ. 8 Based on this private signal, firm i and informed trader j submit market orders to purchase (sell if negative) x i and y j futures contracts to the market maker, respectively. Noise traders trade for liquidity reasons. They, as a group, submit a market order to purchase (sell if negative) u futures contracts to the market maker, where u is a normally distributed random variable independent of θ, with mean zero and variance σu. 2 The aggregate order flow, z, is thus given by z = k i= x i + n j= y j + u. Each participant in the futures market observes only his or her own market order. Following the prespecified price schedule, the market maker sets the futures price, p f, conditioned on the aggregate order flow, z, such that she earns zero expected date profits (i.e., the futures market is semistrong form efficient). The market maker observes only the aggregate order flow, z, but not individual market orders, x i, y j, and u. The properties of the random variables and the numbers of the producing firms and the informed traders are common knowledge. The setup is a twostage imperfect information game. The first stage of the game occurs at date 0 wherein the four types of agents trade in the futures market and the futures price is determined by the market maker. The second stage of the game occurs at date. We consider two different scenarios in the second stage: (i) The futures positions of the k firms are publicly revealed (the observable case), and (ii) the 8 Making the private signal imperfectly reveal θ affects none of the qualitative results.
8 6 futures positions of the k firms remain private information (the unobservable case). In either case, the k firms compete in a Cournot fashion from which the spot market price of the commodity is determined. Futures contracts are then settled at this price level. 3. The observable case In this section, we consider the observable case where the futures positions of the k firms become publicly known at date prior to the firms making their production decisions. As usual, we use backward induction to solve for a rational expectations equilibrium. In the second stage at date, for a given set of futures positions of the k firms, (x,, x k ), the futures price, p f, and the realized demand parameter, θ, firm i s decision problem is given by max q i ( a + θ k q h )q i + h= ( a + θ k q h p )x f i, () h= where the first term is the firm s profits from selling in the spot market, and the second term is the gain or loss due to the firm s futures position. 9 Using the results in Allaz (992), the CournotNash equilibrium is given by q i = ( a + θ kx i + x h ). (2) k + h i Substituting equation (2) into the inverse industry demand function yields the spot market price of the commodity: p = ( a + θ + k + 9 We assume cash settlement of futures position for tractability of model. k x i ). (3) i=
9 7 In the first stage at date 0, firm i, anticipating that the spot market equilibrium is given by equations (2) and (3) in the second stage, chooses a market order, x i, so as to maximize its expected date profits from the spot and futures markets: max x i { E (k + ) 2 ( a + θ + x i + )( x h a + θ kx i + ) x h h i h i [ + k + ( a + θ + x i + ) ] x h p f x i θ, x i }. (4) h i The optimal trading strategy is a function, X i, such that x i = X i (θ). Similarly, informed trader j, anticipating that the spot market price at date is given by equation (3) in the second stage, chooses a market order, y j, so as to maximize his or her expected date profits: max y j {[ E k + ( a + θ + k ) x i i= p f ] y j θ, y j }. (5) The optimal trading strategy is a function, Y j, such that y j = Y j (θ). 0 The market maker, anticipating that the spot market price at date is given by equation (3) in the second stage, observes the aggregate order flow, z = k i= x i + nj= y j + u, but not x i, y j, or u separately. She sets a single futures price, p f, at which she will execute z. Conditioned on observing z, the market maker earns zero expected date profits so that [ p f = E k + ( a + θ + Her pricing rule is a function, F, such that p f = F (z). k i= ) ] x i z. (6) As is commonly assumed, we consider only the set of symmetric, linear, rational expectations equilibria. That is, X i (θ) = α + βθ, Y j (θ) = γ + δθ, and F (z) = µ + λz. 0 Both the producing firms and the informed traders determine their optimal trading strategies fully anticipating the effects of their own market orders on the pricing rule adopted by the market maker.
10 Proposition. If (k 2 + )(nk + 2k + n) σ θ > n(k 2 k + ) k +, then there exists a unique symmetric, linear, rational expectations equilibrium in which Xi O a(k + n + )(k ) (θ) = (k 2 + )(nk + 2k + n) σ θ + k(k 2 + ) nk n 2 θ, (k 2 + )(nk + 2k + n) σ θ (7) Yj O k 2 + (θ) = θ, (8) (k 2 + )(nk + 2k + n) σ θ F O (z) = a(k2 + )[ (k 2 + )(nk + 2k + n) σ θ k(nk n 2)] + (k + ) 2 [k 3 + k + (k 2 + )(nk + 2k + n) σ θ p O (θ) = { [ a k + [ +θ qi O (θ) = { [ a + k + [ +θ + (k 2 + )(nk + 2k + n) σ θ ] k(nk n 2) (k + ) 2 (k + n + ) k(k + n + )(k ) (k 2 + )(nk + 2k + n) σ θ k(nk n 2) (k 2 + )(nk + 2k + n) σ θ (k + n + )(k ) (k 2 + )(nk + 2k + n) σ θ (nk n 2) (k 2 + )(nk + 2k + n) σ θ where the superscript O denotes the observable case. 8 z, (9) + k(k 2 + ) ]}, (0) + k(k 2 + ) ]}, () ] ] Proof. See the appendix. Since the producing firms are risk neutral, they participate in the futures market not for the usual hedging purposes. Instead, futures trading serves two other purposes. First, the producing firms try to exploit their private information about the demand condition for speculative profits from their futures positions. Second, the producing firms use futures transactions as a commitment device to attain advantageous positions in the spot market. This is evident from equation (2) that by taking
11 9 a short futures position firm i credibly commits to a higher level of output which would induce the rival firms to cut their production. The speculative goal and the commitment goal of futures trading may be inconsistent with each other. In the futures market, the producing firms compete against the informed traders who also have private information about θ. What distinguishes the producing firms from the informed traders is that market orders submitted by the former affect not only the transaction price in the futures market, F (z), but also the prevailing spot price at which all of the futures contracts will be settled. From Proposition, as the demand condition in the spot market improves, i.e., as θ increases, the producing firms are more likely to take short positions, which are exactly opposite to those taken by the informed traders. Firm i s expected trading profits are given by { [ E Xi O (θ) k + ( a + θ + k h= ) ]} Xh O (θ) F O (z) σ 2 = u(n + 2 nk) [ (k (k + + n)(k + ) 2 (nk + 2k + n) 2 + )(nk + 2k + n) σ ] θ k(nk n 2). Since (k 2 + )(nk + 2k + n) σ θ > n(k 2 k + ) k +, it is easily shown that the term inside the squared brackets on the righthand side of the above equation is strictly positive. As long as nk n 2 > 0, which holds if n > 2 and k > 2, firm i trades in the futures market expecting to lose money. In other words, unless the number of participants in the futures market is very small, the producing firms expect to lose from futures trading, despite of having private information about θ. Informed trader j expects to earn the following trading profits: { E Y O j [ (θ) k + ( a + θ + k h= ) ]} Xh O (θ) F O (z) This feature of the model is very different from typical market microstructure models where market orders submitted by traders do not affect the fundamental payoff of the security. See Kyle (985) and Admati and Pfleiderer (988) for details on market microstructure models with strategic informed traders.
12 = σu(k ) [ (k (k + + n)(k + ) 2 (nk + 2k + n) 2 + )(nk + 2k + n) σ ] θ k(nk n 2), which is strictly positive and decreasing in n. Put differently, as the competition among the informed traders increases, each informed trader s expected trading profits would be reduced. Since the market maker sets the price schedule such that she expects to break even, the expected trading losses of the liquidity traders always equal the combined trading profits earned by the producing firms and the informed traders, and are given by λσ 2 u = σu 2 [ (k (k + ) 2 (k + n + ) 2 + )(nk + 2k + n) σ ] θ k(nk n 2). 0 One important feature of this model is the possibility of nonexistence of linear rational expectations equilibria in the futures market. To ensure the sufficient condition for the existence of linear rational expectations equilibria as given in Proposition, the number of the informed traders and the producing firms should not be too large. 2 The intuition is as follows. Both the informed traders and the producing firms are strategic traders in the futures market. They choose their optimal trading strategies in a way that fully takes into the account all plausible consequences on the futures price to be set by the market maker. We can find from Xi O (θ) and Yj O (θ) as given in Proposition that the informed traders and the producing firms opt for opposition positions in the futures market. Suppose, for instance, that the number of the producing firms increases. Then, the total size of positions taken by them becomes larger, which provides the informed traders chances to increase their positions 2 The sufficient condition for the existence of linear rational expectations equilibria is that (k2 + )(nk + 2k + n) σ θ n(k 2 k + ) + k > 0. For a fixed σ θ, the lefthand side of the above inequality decreases in n and k. Thus, for sufficiently large n + k, there do exist linear rational expectations equilibria in the futures market.
13 without too much impact on the futures price. Therefore, increases in the number of the informed traders and the producing firms have escalating effects in that both groups of traders take ever increasing futures positions, which would possibly lead to the collapse of the futures market The unobservable case In this section, we consider the unobservable case where the futures positions of the producing firms remain private information at date prior to the firms making their production decisions. Given the futures price, p f, a set of conjectured futures positions, and the realization of demand parameter, θ, firm i conjectures that firm h s output level is given by q h = ( a + θ kx c h + ) x c l, (2) k + l h from the analysis in the previous section. Based on this conjecture, firm i s optimal output is given by q i = 2 ( a + θ ) q h x i h i = ( a + θ k + k + 2 x i + h i x c h k ) x c i, (3) 2 where the second equality follows from equation (2). From firm i s point of view, the spot market price of the commodity is given by p = ( a + θ + k + k + 2 x i + h i x c h k ) x c i, (4) 2 3 Allaz (992), Allaz and Vila (993), and Hughes and Kao (997) do not have this problem of nonexistence of linear rational expectations equilibria since they consider competitive futures markets populated with nonstrategic traders.
14 2 where the equality follows from equations (2) and (3). In the first stage at date 0, each of the producing firms, anticipating that the spot market equilibrium is given by equations (3) and (4) in the second stage, chooses a market order so as to maximize its expected date profits from the spot and futures markets. Likewise, each of the informed traders, anticipating that the spot market price at date is given by equation (4) in the second stage, chooses a market order so as to maximize his or her expected date profits. The market maker, anticipating that the spot market price at date is given by equation (4) in the second stage, observes the aggregate order flow and sets a single futures price to balance supply and demand. The following proposition characterizes the unique symmetric, linear, rational expectations equilibrium. Proposition 2. If 4 n + k σ θ > (k ) 2 + n(k + ), then there exists a unique symmetric, linear, rational expectations equilibrium in which X U i (θ) = Y U i (θ) = θ n + k σ θ, (5) F U (z) = a k + + ( n + k σ θ + k)z (k + )(n + k + ), (6) p U (θ) = [ ( )] k a + θ + k + n + k σ θ, (7) qi U (θ) = [ ( )] a + θ k + n + k σ θ. (8) where the superscript U denotes the observable case. Proof. See the appendix.
15 3 The results of the unobservable case are qualitatively different from those of the observable case. In the former case, we are essentially back to the Kyle (985) and Admati and Pfleiderer (988) framework with n + k informed traders. Two observations are in order. First, both the informed traders and the producing firms opt for the same futures trading strategies. Second, as the demand condition in the spot market improves, i.e., as θ becomes larger, the producing firms are more likely to adopt larger long positions in the futures market. When the futures positions taken by the producing firms are observable, they can be used as a credible commitment device of being engaged in more aggressive output decisions: As a firm takes a short futures position observed by other firms, it credibly signals that the firm would produce a higher level of output, which, in Cournot competition, leads other firms to cut their production. When a producing firm decides its trading strategy in the futures market, it takes into account this strategic effect on other firms output decisions. However, if the producing firms cannot observe other firm s futures positions, this strategic effect of futures trading disappears. In case of unobservable futures positions, the firms simultaneously determine their output levels based on their conjectured levels of other firms futures positions. Even if a firm deviates from the futures position conjectured by other firms, it cannot affect other firms production decisions since it is not observable, thereby making the strategic effect of futures positions in the spot market no longer exist. 4 As shown in Proposition 2, as the number of the informed traders and the produc 4 Denote firm i s expected profits as Π i = E{[a + θ k h q h q i (x i )]q i (x i ) + [a + θ k h q h q i (x i ) p f ]x i }. Since other firms output is not affected by x i in case of unobservable futures trading, = Πi q i q i x i + Πi x i only firm i s output is affected by x i. Thanks to the envelope theorem, dπi dx i = Πi x i. This is essentially equivalent to the case where the producing firms determine their futures trading strategies without any consideration of the effects on the spot market.
16 4 ing firms increases, it is more likely that there will be no linear rational expectations equilibria in the futures market. This is due to the producing firms ability to affect through their futures trading not only the transaction price of the futures market but also the spot price at which the futures contracts will be settled. As there are more informed traders and producing firms competing in the futures market, the futures price becomes less sensitive to the aggregate order flow, which consequently leads the producing firms to take more aggressive trading strategies, particularly due to their ability to affect the spot price, thereby making the breakdown of the futures market more plausible. 5 Expected trading profits earned by each informed trader or firm are given by ( n + k σ θ + k)σ 2 u (k + )(n + k + )(n + k). Unlike in the observable case, the producing firms expect to earn positive profits from futures trading. The expected consumers surplus is given by 2 E{[a + θ p(θ)]kq i(θ)}. Let CS O and CS U be the expected consumers surplus in the observable and unobservable case, respectively. Then, we have CS O CS U equal to k 2 2(k + ) 2 {E[XO i (θ) 2 ] 2E[X O i (θ)(a + θ)] E[X U i (θ) 2 ] + 2E[X U i (θ)(a + θ)]}. From Proposition, we have E[X O i (θ) 2 ] 2E[X O i (θ)(a + θ)] > 0. 5 The equilibrium λ set by the market maker equates the expected trading losses incurred by the liquidity traders with the combined expected trading profits earned by the informed traders and the producing firms. As there are more informed traders and producing firms engaging in futures trading, their competition results in smaller total profits earned by them, thereby lowering the equilibrium λ. As the detailed analysis in the appendix shows, if λ is sufficiently small, then the producing firms take infinitely large positions since their orders substantially improve the spot price with relatively little impact on the futures price.
17 5 From Proposition 2, we have ( )] E[Xi U (θ) 2 ] 2E[Xi U (θ)(a + θ)] = n + k σ θ n + k σ θ 2 σθ 2 < 0. Thus, we have CS O CS U > 0. The expected producers surplus is given by E[kp(θ)q i (θ)]. Let P S O and P S U be the expected consumers surplus in the observable and unobservable case, respectively. Then, we have P S O P S U equal to k (k + ) 2 { ke[xo i (θ) 2 ]+(k )E[X O i (θ)(a+θ)]+ke[x U i (θ) 2 ] (k )E[X U i (θ)(a+θ)]}. Using the above results, we have P S O P S U < 0. Thus, we establish the following proposition. Proposition 3. Consumers are better off while the producing firms are worse off should the firms make their futures positions observable than unobservable. The possibility of futures trading prior to the production stage allows the producing firms to adopt futures trading as a commitment device of taking more aggressive output decisions. This leads to the product market equilibrium in which more output is produced at a lower price. Welfare implication of making the producing firms futures positions observable is clear: consumers are better off while the producing firms are worse off. Now we move on to the issue of how informative the futures price is about the spot price. Proposition 4. Var O (p p f ) Var O (p) = k 2 + (k + )(n + k + ) > VarU (p p f ) Var U (p) = n + k +.
18 6 Proof. See the appendix. As we can see from this proposition, in the observable case, as the number of the informed traders increases, the futures price becomes more informative in that the variance of p conditioned on the futures price goes down. This concurs with the standard result of market microstructure models with more private information owned informed traders reflected in the transaction price as the number of informed traders increases and thereby they submit larger trading orders to the market maker. This is not necessarily true for the case of increasing number of the producing firms. Since the producing firms futures trading not only affects the future price but also the spot price at which futures contracts will be settled, increasing number of the producing firms weakens the informativeness of the futures price. 6 Contrary to the observable case, in the unobservable case, as the number of the informed traders and the producing firms increases, the futures price becomes more informative in that the variance of p conditioned on the futures price goes down since both the informed traders and the producing firms take the same trading strategies in the futures market. 5. Conclusion In this paper, we have analyzed the incentives for firms to engage in futures trading in the context of an oligopolistic product market and strategic trading in a futures market. If the producing firms are risk neutral and have perfect information about 6 In the context of standard market microstructure models, greater number of the producing firms essentially increases the variance of the fundamental payoff of the futures contracts, and thereby the informativeness of the futures price deteriorates.
19 7 product market conditions, risk reduction cannot be an incentive for their participation in the futures market any more. There are two reasons that prompt the producing firms to trade futures contracts. First, the producing firms can use futures trading strategically to improve their positions in the product market competition. Second, they participate in the futures market as informationbased traders to gain trading profits. As informationbased traders, the producing firms not only have access to the private information on the spot price at which futures contracts will be settled, but also they can directly affect it since their product market decisions determine the spot price in the product market. Depending on whether the futures positions taken by the producing firm are observable or not, we have qualitatively different results. If the futures positions taken by the producing firms are observable as they make their product market decisions, their motivation of adopting futures trading as a credible commitment device to take more aggressive product market strategies outweighs the incentive to earn trading profits from informationbased trading. In fact, the producing firms expect to lose money from futures trading although they possess utmost advantage against other participants in the futures market. However, if the futures transactions of the producing firms are not observable, the effectiveness of futures trading as a commitment device in the product market competition disappears, and they trade in the futures market just to earn trading profits. We can draw a number of interesting policy implications from this paper. First, as firms are allowed to be trading in futures markets prior to making product market decisions, disclosure policies on their futures positions result in vastly different welfare effects. If firms are required to publicly disclose their futures positions, thereby every participant in the market are able to observe them, then as the analysis of this papers
20 8 shows, consumers are better off while producing firms are worse off. No disclosure requirement on the firms futures positions is advantageous to producing firms but disadvantageous to consumers. 7 Second issue is on the stability of futures markets. Analysis in this paper demonstrates that if the futures price is set by a riskneutral market maker who earns zero expected profits, then future market equilibria may not exist when there is a large number of informationbased traders. This is due to the presence of producing firms in futures markets who can affect both futures prices and spot prices at which futures contracts will be settled. The assumption of a riskneutral market maker earning zero expected profits is valid if sufficiently large number of market makers are competing to facilitate futures trading. 8 In this case, restricting the competition among market makers are likely to ensure the existence of futures market equilibria even in the presence of large number of informationbased traders. 9 This paper assumes that all market participants including producing firms are risk neutral. Extending the current model to the case of risk aversion of producing firms can reveal different motivations for futures trading including risk reduction, which is left for further research. 7 United States Statements of Financial Accounting Standards (SFAS) and International Accounting Standards Committee (IASC) do not have specific clauses on the disclosure of futures positions. For more details, see Hughes and Kao (998). 8 Low entry barrier into market making of futures markets is enough to squeeze incumbent market makers profits to a very low level. 9 In the context of the model presented in this paper, sufficiently large λ prevents the producing firms from taking infinite futures positions, thereby breaking down the futures market equilibrium. If the market maker sets the price schedule with greater λ than the ones given in Propositions and 2, then she is able to earn positive expected profits, and the expected losses incurred by the liquidity traders will increase although the futures market equilibrium might be restored. One solution for the welfare losses of the liquidity traders is to charge fixed entry fees to market makers to make up increased trading losses of the liquidity traders.
21 9 Appendix Proof of Proposition. Suppose that firm i conjectures that firm j will submit a market order of α c + β c θ, that each informed trader will submit a market order of γ + δθ, and that the market maker uses a linear pricing rule of µ + λz. The expected date profits of firm i are given by (k + ) [a + θ + x 2 i + (k )(α c + β c θ)][a + θ kx i + (k )(α c + β c θ)] { } + k + [a + θ + x i + (k )(α c + β c θ)] µ λ[x i + (k )(α c + β c θ) + n(γ + δθ)] x i. Solving the firstorder condition yields x i = 2(a + θ) + (k )(2 λ(k + )2 )(α c + β c θ) µ(k + ) 2 λn(k + ) 2 (γ + δθ). 2(λ(k + ) 2 ) The secondorder condition requires that λ > /(k + ) 2. Let x i = α + βθ. We can solve for the symmetric Nash equilibrium by setting α c = α and β c = β: α = 2a µ(k + )2 λnγ(k + ) 2, β = (k + ) 3 λ 2k 2 λnδ(k + )2 (k + ) 3 λ 2k. (A.) Suppose that informed trader j conjectures that each of the other informed traders will submit a market order of γ c + δ c θ, that each of the firms will submit a market order of α + βθ, and that the market maker uses a linear pricing rule of µ + λz. The expected date profits of informed trader j are given by { } k + [a + θ + k(α + βθ)] µ λ[k(α + βθ) + y j + (n )(γ c + δ c θ)] y j. Solving the firstorder condition yields y j = a + θ µ(k + ) + k(α + βθ)( (k + )λ) λ(γc + δ c θ)(n )(k + ). 2(k + )λ
22 20 The secondorder condition requires that λ > 0. Let y k = γ + δθ. We can solve for the symmetric Nash equilibrium by setting γ c = γ and δ c = δ: γ = a µ(k + ) + kα( (k + )λ), δ = (k + )λ(n + ) + kβ( (k + )λ). (A.2) (k + )λ(n + ) Suppose that the market maker conjectures that each of the two firms will submit a market order of α + βθ, and that each informed trader will submit a market order of γ + δθ. The aggregate order flow, z = k(α + βθ) + n(γ + δθ) + u, is normally distributed. By the projection theorem, we have F (z) = { } E k + [a + θ + k(α + βθ)] z = k(α + βθ) + n(γ + δθ) + u = ( + kβ)(kβ + nδ)σθ 2 (a + kα) + k + (k + )[(kβ + nδ) 2 σθ 2 + (z kα nγ). σ2 u] Let p f = µ + λz. Thus, we have µ = k + [a+k( (k+)λ)α (k+)nλγ], λ = ( + kβ)(kβ + nδ)σ 2 θ (k + )[(kβ + nδ) 2 σ 2 θ + σ2 u]. (A.3) If we can solve for a unique solution of equations (A.), (A.2), and (A.3), we are done. yields Substituting δ in equation (A.2) into β in equation (A.) and rearranging terms nk n 2 β = (k + ) 2 (n + k + )λ + k(nk n 2). (A.4) Substituting equation (A.4) into δ in equation (A.2) and rearranging terms yields δ = k 2 + (k + ) 2 (n + k + )λ + k(nk n 2). (A.5) Substituting equations (A.4) and (A.5) into λ in equation (A.3) and rearranging terms yields λ = (k 2 + )(nk + 2k + n) σ θ k(nk n 2) (k + + n)(k + ) 2. (A.6)
23 2 Since λ is required to exceed /(k + ) 2, we need (k 2 + )(nk + 2k + n) σ θ > n(k 2 k + ) k + Substituting equation (A.6) into equations (A.4) and (A.5) yields nk n 2 β =, δ = (k 2 + )(nk + 2k + n) σ θ k 2 +. (k 2 + )(nk + 2k + n) σ θ Substituting µ in equation (A.3) into γ in equation (A.2) yields γ = n n + γ, which implies that γ = 0. Substituting γ = 0 into α in equation (A.) and µ in equation (A.3) yields α = (k + ) 3 λ 2k (2a (k + )2 µ), µ = [a + k( (k + )λ)α]. k + Solving the above equation for α and µ and using equation (A.6) yields a(k + n + )(k ) α =, k(k 2 + ) + (k 2 + )(nk + 2k + n) σ θ µ = a(k2 + )[ (k 2 + )(nk + 2k + n) σ θ nk 2 + nk + 2k]. (k + ) 2 [k 3 + k + (k 2 + )(nk + 2k + n) σ θ ] This completes our proof. Proof of Proposition 2. Suppose each informed trader will submit a market order of γ + δθ, and that the market maker uses a linear pricing rule of µ + λz. The expected date profits of firm i are given by in symmetric rational expectation equilibrium. + (k + ) 2 { k + [ a + θ + k + [ a + θ + k + 2 x i + k 2 2 x i + k 2 ][ (α c + β c θ) a + θ k + ] (α c + β c θ) 2 x i + k 2 ] (α c + β c θ) µ λ[x i + (k )(α c + β c θ) + n(γ + δθ)] } x i.
24 22 Solving the firstorder condition yields x i = (k + )(4λ ) [ 2(a + θ) + (k )( 2(k + )λ)(α c + β c θ) 2(k + )µ 2(k + )nλ(γ + δθ)]. The secondorder condition requires that λ > /4. Let x i = α + βθ. We can solve for the symmetric Nash equilibrium by setting α c = α and β c = β: α = (k + ) 2 λ k (a (k + )µ (k + )nλγ), β = ( (k + )nλδ). k + ) 2 λ k (A.7) Since the unobservable case is observationally equivalent to the observable case for the n informed traders and the market maker, we have γ and δ given in equation (A.2) and µ and λ given in equation (A.3). If we can solve for a unique solution of equations (A.7), (A.2), and (A.3), we have following solutions. yields Substituting δ in equation (A.2) into β in equation (A.7) and rearranging terms β = (k + )(n + k + )λ k. (A.8) Substituting equation (A.8) into δ in equation (A.2) and rearranging terms yields δ = (k + )(n + k + )λ k = β. (A.9) Substituting equations (A.8) and (A.9) into λ in equation (A.3) and rearranging terms yields λ = n + kσθ / + k (k + )(n + k + ). (A.0) Since λ is required to exceed /4, we need 4 n + kσ θ / > (k ) 2 + n(k + ). Substituting equation (A.0) into equations (A.8) and (A.9) yields β = δ = n + kσθ /.
25 23 Substituting γ = 0 into α in equation (A.7) and µ in equation (A.3), and solving the above equation for α and µ yields α = 0, µ = a k +. This completes our proof. Proof of Proposition 4. From the conditional variance of normally distributed random variables, we have V ar(p p f ) = V ar(p)( Cov(p, pf ) 2 V ar(p)v ar(p f ) ). From the proof of Proposition, we have λ = Cov(p, kβθ + nδθ + u) V ar(kβθ + nδθ + u) and following equalities can be obtained from equations (A.4) and (A.5). Cov(p, p f ) 2 V ar(p)v ar(p f ) Cov( (θ + kβθ), kβθ + nδθ + u)2 k+ = V ar( (θ + kβθ))v ar(kβθ + nδθ + u) k+ = λ Cov( (θ + kβθ), kβθ + nδθ + u) k+ V ar( (θ + kβθ)) k+ = λ Cov( λ(k+)(n+k+) θ, (nk+2k+n) θ + u) (k+) 2 (n+k+)λ+k(nk n 2) (k+) 2 (n+k+)λ+k(nk n 2) λ(k+)(n+k+) V ar( = nk + n + 2k (k + )(n + k + ). (k+) 2 (n+k+)λ+k(nk n 2) θ) We follow the similar steps to show that Cov(p,p f ) 2 = Cov(θ,pf ) 2. V ar(p)v ar(p f ) V ar(θ)v ar(p f ) By following the similar steps as above, and from equations (A.8) and (A.9), we derive the result of the unobservable case.
26 24 References Admati, A.R. and P. Pfleiderer, 988, A theory of intraday trading patterns: Volume and price variability, Review of Financial Studies, Allaz, B., 992, Oligopoly, uncertainty and strategic forward transactions, International Journal of Industrial Organization 0, Allaz, B. and J.L. Vila, 993, Cournot competition, forward markets and efficiency, Journal of Economic Theory 59, 6. Broll, U. and K.P. Wong, 2002, Imperfect forward markets and hedging, Economic Notes 3, forthcoming. Hughes, J.S. and J.L. Kao, 997, Strategic forward contracting and observability, International Journal of Industrial Organization 6, Hughes, J.S. and J.L. Kao, and M. Wiiliams 2002, Public Disclosure of Forward Contracts and Revelation of Proprietary Information, Working Paper. Kyle, A.S., 985, Continuous auctions and insider trading, Econometrica 53,
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