Nov

28

In the office we were talking about the repeated action of the S&P’s move to a certain level, and then it’s falling back from this level, that occurs on a day like today. This repeats until the potential energy of the market is converted to kinetic energy, and the market rises higher. We were looking for analogies for this, such as power lifting where you bounce the weight before extending it to maximum lift, or pole vaulting where you can take up to three tries to get over the bar. In the process of this we were also considering the energy transfer involved in making a child’s swing set go higher with each swing. The following brief explanation was found but I would be interested in any ideas people have on a proper model for the back and forth; the trying to get there but failing, that happens so often in the markets.

Each time the swing moves forward and then returns to its starting position counts as one cycle. Using a stop watch determine the length of time a swing needs to complete say 20 cycles. Divide 20 cycles by the time and you have the swings frequency in cycles per second or Hertz (Hz).

Since a swing is basically a pendulum it’s possible to calculate its resonant or natural frequency using pendulum equations as follows:

Note that the natural frequency of the swing is not influenced by the mass of the person in it. In other words’ it makes no difference whether a swing has a large adult or a small child in it. It will have the about the same natural frequency. Slight differences can be caused by slightly different locations of the person’s center of mass. This is located about two inches below the navel. When people are sitting the center of mass is in about the same place relative to the seat of the swing regardless of whether the person is an adult or a child.

If a forcing function is applied to a swing at the natural frequency of the swing it will resonate. The amplitude of the swing will increase during each back and forth cycle. The forcing function can be provided by a second person pushing on the swing. In this case even a small child can make a large adult swing by pushing in sync with the swing’s back and forth cycle. The forcing function can also be provided by the person in the swing. In this case the person in the swing shifts her center of mass very slightly by changing the position of her legs or torso. This creates a slight pushing force which makes the swing go higher and higher. It takes a very small force but it has to be timed perfectly.

The big question is what keeps the swing from flying apart or spinning over the top of the swing’s frame and subsequently killing its rider? After all, if it is a resonating system then it should be very dangerous to keep applying force in time with the swing’s frequency. The answer is fairly simple. The equation given above is only good for small angles. When the swing goes beyond a certain height it is no longer possible for the person in it to apply the necessary small force in sync with the natural frequency because the natural frequency changes. In other words the motion of the system is naturally limited.

Jim Sogi offers:

The apparent back and forth motion around the round number is a chart artifact, and as with so many chart artifacts is an illusion. The motion is in three dimensions and only appears on the chart in two. The model is a tether ball, like at summer camp. It has circular momentum from whacking it, and tightens, then rebounds off and unwinds. The angle of the wind depends on the angle of the whack. Circular math a’la Newton might work.

The other model is a guitar string. It has harmonics and standing waves along its length as the axis of vibration meet along the string, similar to price action harmonics. The higher harmonics are recreated in the higher and lower price levels.

Gary Rogan comments:

I also view the market gyrations as something similar to a swing, except it’s nothing like a physical, earthly swing because there are two forces involved, and one of them is “unusual” for a physical-world system. In the physical world, there is only gravity (other than a small amount of friction) involved in the dynamics of a swing that results in a simple differential equation describing the motion for small deviations. I see two basic “forces” involved in market motion: “momentum” and “value pricing”. Positive momentum is the force that causes people to buy when the market is moving up (buying interest proportional to market velocity), negative momentum is the force that causes people to sell when the market is moving down. Thus momentum is a force proportional to velocity, sort of like inverse friction that doesn’t exist in the real world. Value pricing is what causes people to buy when prices are “too low” and sell when they are “too high”.

Of course all of this exists in the environment of slow upward drift and real-world-like friction of various trading costs as well as news events and money-supply formations that are not completely dependent on the immediate market dynamics. The relative amplitudes of the two forces also change with time.

Normally the two forces are balanced enough to keep the market gyrating around some sort of a temporary equilibrium that itself is slowly drifting. However, when the momentum force gets too high (as in 2000) it will break the swing.

Jeff Sasmor adds:

Another thing to consider is inertia. There is a nice article on this in Wikipedia and other sources.

The principle of inertia is one of the fundamental laws of classical physics which are used to describe the motion of matter and how it is affected by applied forces. Inertia is the property of an object to resist changes in velocity unless acted upon by an outside force. Inertia is dependent upon the mass and shape of the object. The concept of inertia is today most commonly defined using Sir Isaac Newton’s First Law of Motion, which states:

Every body perseveres in its state of being at rest or of moving uniformly straight ahead, except insofar as it is compelled to change its state by forces impressed. [Cohen & Whitman 1999 translation]

Perhaps this explains the recent upwards moves in stocks in spite of multiple discouraging memes. Humans have a lot of inertia, we’ve probably programmed a lot of it into the machines that do a lot of the trading these days.

It’s odd that this came up today, I was mulling the concept last night before falling asleep. Interesting questions that came up are:

It is a system with a lot of inputs and time-varying coefficients. Maybe it’s a reverb chamber?

David Wren-Hardin mentions:

Swings and oscillations are found throughout nature where systems on different time courses interact with each other. One obvious relationship is the classic predator-prey population dynamic. As prey animals increase in number, predator numbers rise on a lagging basis. A peak in prey animals is followed by a crash as they consume their resources, dragging the numbers of predators with them. One can cast value investors in the role of rabbits, with their steady grazing on low-calorie fare, and the momentum investor in the role of the coyote, waiting for concentrated packets of dense nutrients. Or one could place the casual investor in the role of rabbit, and the average financial professional in the role of coyote, but I’ll refrain from that comparison so not to risk defaming the coyote.

Animals also use oscillations to find out information about their environment, much like the technical analyst or trading-surfer surveying their charts. The weakly electric fish, Eigenmannia, emits an electric signal as a sort of radar to find objects in its surroundings. The problem arises when another Eigenmannia is nearby, sending out a signal at a frequency near the first fish’s signal. This results in a “beat” frequency equal to the difference of the frequency of the two signals, composed of amplitude and phase modulations. Much like the market, when the agendas of different market participants collide, the result is confusion and little information for anyone. The fish responds by moving the frequency of its signal away from the other, a process known as the Jamming Avoidance Response. The fish doesn’t know if it is higher or lower, and has to solve the problem based on how receptors spaced over its body are receiving the phase information of the two signals. In essence, each receptor “votes” on whether it perceives the signal to be leading the other, i.e., it’s at a higher frequency, or lagging, i.e., a lower frequency. Any one neuron may be wrong, but in the aggregate, the animal arrives at the correct conclusion. In classic research, the late Walter Heiligenberg termed this organization a “neuronal democracy”.

As traders, individual neurons awash in the market’s oscillations, we are faced with the same problem. Are we leading? Are we lagging? It may come as little comfort that the market will eventually get it right, even if we are wrong.

GM Nigel Davies offers:

In chess this would be quite a typical scenario. Often when you inflict some kind of permanent damage (structural or material), there is a temporary release of energy from the other side’s pieces. The ‘trick’ is to balance the gains against the likely reaction, and this is also necessary. To improve a position you often have to allow some temporary (hopefully) counter play, kind of like a wrestler letting go of an opponent temporarily so as to get a better grip.

Dr. Michael Cook adds:

Gary comments that market gyrations are “nothing like a physical, earthly swing” because there are two forces involved. How about the case of a damped oscillation, which has physical analogues? Using this analogy, momentum investors are “damped” by the “restoring force” supplied by value investors.

And what happened in the bubble was the disappearance of effective value investors, which led to an un-damped oscillation, which, when driven at the appropriate frequency, leads to wider and wider oscillations which no physical — or financial — system can sustain.

The collapse of the Tacoma Narrow Bridge is the canonical example, and here is an illustration of the math behind the phenomenon.

Rick Foust contributes:

Imagine a ball rolling down a slight incline that has a crown in the middle and rails on the sides, similar to a highway with guard rails. The ball seeks the nearest rail, bounces repeatedly and eventually stays on the rail as it continuous forward.

Now imagine that the roadway has an irregular surface and rough rails. The ball will once again seek a rail. But this time, it will do so in a careening fashion that depends on the roadway surface. As it encounters a rail, it will briefly run down the rail, bouncing as it goes, until it eventually hits a point of roughness large enough to kick it to the other side. The amount of roughness required to cause a change in state depends on the slope of the underlying surface.

In the market, the rails are accumulations of large and small limit orders. Rail roughness is created by variations in order size and position. The roadway surface is formed by underlying market orders that create a natural drift. The roadway surface may undulate in a rhythmic fashion, similar to the Tacoma bridge, if market participant psychology is undecided. Or it may consistently lean in one direction if there is a prevailing sentiment.

At some point, limit orders at one rail or the other are exhausted, pulled or merely absent. At that point, the ball is free to discover the location of other rails. Stops are now run, creating new market orders. New participants are drawn in. If the new rails encountered are small and scattered, the ball will plow through them and may even gain momentum until it eventually encounters a rail large enough to stop it. Until this rail is reached, the underlying roadway slope will likely increase as sentiment is self-reinforced.


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