Abstract
In previous chapters we have seen the advent of a new society, with the growing complexity of the market and technological innovations. We demonstrated the move away from the Gaussian world towards that of heavy tail distribution. To fully understand the latter distribution, we need to recognize a generalized central limit theorem, where the Gaussian distribution is simply a special case. We are therefore able to investigate several stable distributions and expand our idea of price equilibrium as a stable distribution. We will also discuss the central issue of how the market economy can generate complex dynamics. We use the trader dynamics system designed by Philip Maymin, and recreate his simulation dynamics for an automaton. This explicitly takes into account traders’ decisions in combined multiple layers: actions, mind states, memories of the trader and the market. These factors are normally regarded as influential components that govern price movements. In financial transactions, we dispense with psychological effects at individual and aggregate levels. These may no longer be secondary effects.
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- 1.
A more sophisticated version of this is the auto-regressive integral moving average model, ARIMA(p, q, r). MA(1) is an approximate function of AR(∞).
- 2.
Also see Fama (1965).
- 3.
The British physician and natural scientist Francis Galton (1822–1911), who first tried to measure intelligence, stressed the law of the large number: “I hardly know anything other than this whose imagination is influenced in such a way like the marvelous form of cosmic order, which is expressed by the law of the error frequency. It prevails with bright serenity, and complete self-denial in the middle of the wildest confusion. The greater the rabble, the greater the obvious anarchy. Its influence is surely the highest law of senselessness”.
- 4.
- 5.
- 6.
See Cook (2010).
- 7.
See Appendix for “ An elementary derivation of the one-dimensional central limit theorem from the random walk”.
- 8.
ø(1) represents a function tending to 0.
- 9.
- 10.
- 11.
The power law differs from the power distribution. The probability density for values in a power distribution is proportional to x a−1 for 0 < x ≤ 1∕k and zero otherwise.
- 12.
A linear combination of independent identically distributed stable random variables is also stable.
- 13.
As already noted, the generalized CLT still holds.
- 14.
DFR indicates that the ability to make more money might increase with one’s income (Singh and Maddala 2008, p. 28).
- 15.
c is eliminated. If x = 0, F(0) = 0. It follows that \(c = b^{1/a}\).
- 16.
It is clear that F → 1 as x → ∞.
- 17.
- 18.
The number of states may be more than two.
- 19.
Note in the Snapshots how sometimes volatility starts small and then increases, and sometimes it starts large and decreases. A price series that appears stable can suddenly collapse, and after a collapse can remain low, or can come back up. Because the investor looks back a fixed number of days, the price series is guaranteed to eventually cycle. However, this cycle length can be quite long. For example, a 15-business-day look-back, absent any external changes, will cycle in about 130 years.
- 20.
Ticks are approximately of the order of seconds.
- 21.
The strength of a buy or sell is an exponent of the base b. For example, when b = 3 and k = 4, the four possible actions are \(-3,-1, 1\), and 3, meaning sell three times more than normal, sell normally, buy normally, or buy three times more than normal. If k = 6, then two additional actions are −9 and 9. According to Maymin, “[i]f a normal buy would increase the market price by one percent, a nine times more fervent order would increase the market price by nine percent” (Maymin 2011a, p. 4).
- 22.
In the Tokyo Stock Exchange, as set out in Chap. 4, the morning or afternoon session under the double auction rule will start after a batch auction on the preceding 30 min of orders.
- 23.
In HFT, as set out in Chap. 4, a cross-operation is feasible if it can happen more quickly than a trading time unit.
- 24.
According to Wikipedia, “The term - a reference to the manipulation of a simple hand puppet made from a sock - originally referred to a false identity assumed by a member of an internet community who spoke to, or about, himself while pretending to be another person”.
- 25.
The U-Mart experiment is helpful in understanding how various random behaviors could intermediate a coordination of contracts in dealings.
- 26.
The minimal model of simulating prices of financial securities using an iterated finite automaton, where length of generated series = initial condition.
- 27.
k = 15 and w = 15 means the maximal cycle length is longer than the age of the universe in seconds.
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Aruka, Y. (2015). The Complexities Generated by the Movement of the Market Economy. In: Evolutionary Foundations of Economic Science. Evolutionary Economics and Social Complexity Science, vol 1. Springer, Tokyo. https://doi.org/10.1007/978-4-431-54844-7_6
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