Abstract
This chapter presents an evolutionary model of a futures market to justify the eventual occurrence of an informationally efficient equilibrium. While the literature usually justifies informational efficiency in the context of rationality, here, in this dynamic futures market, traders do not maximize their profits or utilities nor do they form rational expectation about spot prices. Instead, they are preprogramed with some predetermined behavioral traits (such as trading types (buyer or seller), traders’ inherent abilities to predict the spot price). With the markets serving as a selection process of information, it can be shown that the proportion of time that the futures price equal to the spot price converges to one with probability one.
This chapter is based on my article published in The Review of Financial Studies 11(3): 647–674, 1998.
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Notes
- 1.
In the traditional rationality models agents make choices to maximize certain objectives. Here, in this chapter, the evolutionary approach is deployed for the purpose of abandoning the rationality of making choices. Instead the choice variables are exogenously prespecified.
- 2.
Analogous to the biological term of genotype, the economic agent’s type is characterized by his or her choice variable. The choice variable could be interpreted as arising from an agent maximizing his or her own objective within his or her resource constraints, information constraints and his or her ability constraints.
- 3.
In Figlewski (1982), informational efficiency is obtained in the long run only in the special case when all traders have independent information. In contrast, this is not a required assumption in the evolutionary model of this chapter.
- 4.
The justification for this assumption is as follows. For each of a large number of independent producers the output level is randomly determined at the beginning of each time period. Since the commodity is nonstorable, the total output of all producers must be delivered to the spot market at the end of each time period and sold at a market clearing spot price. Since the individual output levels of producers are determined in the beginning of each time period, the industry’s aggregate output is also determined at the beginning of each time period. Given any exogenously specified continuous demand schedule in the spot market, the spot price is determined at the beginning of each time period.
- 5.
A more elaborate production model, which characterizes the costs of individual producers, could be specified. This adds very little to the issues examined in this chapter, which focuses on the effect of various speculators’ behavior on the futures market.
- 6.
The producers honor their futures contracts sold at the beginning of each time period, by transferring, at the end of the time period, to the buyers of the futures contracts the corresponding revenue received in the spot market and in return the producers receive from the buyers the value of these same contracts at the futures price previously agreed upon. The short sellers deliver to the buyers their short sales valued at the spot price and receive from the buyers the value of the same contracts at the previously agreed upon futures price.
- 7.
To prevent unrealistic negative predictions, the range of predictions sometimes may have to be truncated if p s + ν s t < d. More precisely, \({b}_{s}^{t} =\max [{p}_{s} + {\nu }_{s}^{t},d]\).
- 8.
This reflects the impossibility of speculators learning from other traders and learning from their past experiences.
- 9.
One can interpret the b s t as a signal that speculator t receives at time s. The prediction error ν s t is the noise that speculator t brings into the market at time s. As long as b s t is not equal to p s speculator t is acting upon noise. Of course, given the differences in the distribution of speculators’ prediction errors, different speculators have different probabilities of acting upon noise. In this paper no assumptions are made with respect to the type of the distribution of the prediction error. (This stands in contrast to most papers including recent papers by Kyle and Wang (1997) and Fischer and Verrecchia (1997).)
- 10.
Because of the discreteness of prices, when p s f < b s t a buyer may not be able to spend all of his or her speculative wealth (f t V s − 1 t) if the supply of futures contracts lies between \(\frac{{f}_{t}{V }_{s-1}^{t}} {{p}_{s}^{f}+d}\) and \(\frac{{f}_{t}{V }_{s-1}^{t}} {{p}_{s}^{f}}\). Thus, a step-like demand curve is used for any buyer.
- 11.
Ensuring no defaults can also be accomplished if the spot price is constrained such that it cannot exceed twice that of the futures price. This latter assumption is similar to that used by Feiger (1978). Either assumption leads to the same conclusions.
- 12.
More precisely, consider \({q}_{s}^{t} = -k\frac{{f}_{t}{V }_{s-1}^{t}} {{p}_{s}^{f}},\) (k > 0) (1′) where \({p}_{s} \leq \overline{m}d\) (2′) and p s f ≥ d (3′). To ensure that short sellers honor their transactions it is assumed that the speculative wealth \({f}_{t}{V }_{s}^{t} = ({p}_{s} - {p}_{s}^{f}){q}_{s}^{t} + {f}_{t}{V }_{s-1}^{t} \geq 0\) (4′). Therefore, with equations (1′), (2′), and (3′), equation (4′) is ensured when \(k \leq \frac{1} {\overline{m}-1}\). Suppose this constraint is not imposed for short sellers and suppose that k is set equal to one. Then in the extreme case of p s f = d and \({p}_{s} = \overline{m}d\) it would follow that \({f}_{t}{V }_{s}^{t} = ({p}_{s} - {p}_{s}^{f}){q}_{s}^{t} + {f}_{t}{V }_{s-1}^{t} < 0\) for \(\overline{m} > 2\). This implies that the short seller is not able to honor his or her contracts.
- 13.
For similar reasons described in footnote 10, a step-like demand curve is used for any seller.
- 14.
The precise way that the remaining supply is allocated has no effect on the results of the chapter.
- 15.
Although the model of this chapter is formulated in a particular manner, the conclusions do not rely on the range of trading types. The proof can be modified to provide the same conclusions if all speculators are buying contracts in some time periods and in other time periods selling contracts (see Luo 1995b). Furthermore, the results of the model would also hold with continuous rather than discrete futures and spot prices.
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Luo, G.Y. (2012). Evolution and Informationally Efficient Equilibrium in a Commodity Futures Market. In: Evolutionary Foundations of Equilibria in Irrational Markets. Studies in Economic Theory, vol 28. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-0712-6_4
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