*Flash crash*

Version 2.0

Avalanche-like processes are common in living systems
stuffed with biofeedback. Two characteristic examples are hippocampal burst and
the collapse of a financial market. Although happening on different levels of
organization, they may be related because the hippocampus is an organ in the
brain which participates in decision making. Both are based on positive
feedback and probably may be described by similar differential equations. Let's
explore the second case in details. The choice is quite obvious out of
pragmatic considerations. On one hand, this phenomenon is important from
different points. On the other - a lot of data was gathered about its
mechanisms. Detailed explanation makes it possible to create a working model
with variable parameters. Changing these parameters, one is able to explore
different situations that may happen in reality. Later, the same model may be
applied to different cases.

Free market is a self-regulating system consisting of
many small independent traders. Each one buys or sells in search of profit.
When bears sell, prices drop. When bulls buy, they rise. Biofeedback may be
considered on the macro-level, but it is explained by individual psychology and
statistics of decisions. Obviously, there is stabilizing negative feedback.
When the price deflects from the current average and drifts towards a border of
the Bollinger band, more and more traders will want to return it back. With
linear approximation, we will even be able to draw parallels to mechanics and
receive the output process in the form of harmonic oscillations.

They will show extinction and require some stimulus to
start.

Obviously, nothing here is looking like exploding
developments. Such processes require self-enhancement rather than self-correction.
That is, when prices move up, away from the average, traders should feel
incentive to buy. We can easily find confirmation on real charts.

Harmonic oscillations have rounded tops. Instead,
price dynamics is full of fragments with sharp, narrow summits. While harmonic
oscillations are similar to electric motors with smooth rotation, market
dynamics is like a combustion engine. This may be explained by the logic of
individual traders. There are 2 strategies to open a position. 1. Trade a
reversal. 2. Join a rally. In the first case, bulls enter trading when bears
close their positions. Both produce the same effect of negative feedback. The
second case is principally different. When traders notice accelerating
movement, they join and thus further accelerate it. The result will be
exponential growth or fall.

At the beginning of this curve, you see the sharp
bottom of the previous crash followed by the gradual restoration to the normal
level. Then, the next exponential downfall begins.

When will it stop? Obviously, when the first strategy
becomes prevalent. How is it possible if both forces grow linearly with the distance
from the average? Each line is determined by 2 parameters. We can create a
model with them, then experimentally and theoretically try different
combinations. Avalanche processes may unfold in both directions. Below you see
a mechanism that stops an exponential growth.

This combination ensures that purchasing starts earlier than selling, but the slope of the selling line is larger so the move will be eventually reversed. X - distance of current price from moving average. Y - average speed of buying (red) and selling (blue).

**Supplement**

**How to generate such curves using WinNB.**

Harmonic oscillations are produced by the classical
differential equation

d^2x/dt^2 =
-k * x, k > 0

The meaning
is that the returning force is proportional to the displacement from the
central point. For the purpose of numerical integration this will look like:

value will
be @X1 units if

value is @X units; difference is @DX
units;

@DX1 = @DX - @K * @X;

@X1 = @X + @DX1.

Now let’s add extinction

d^2x/dt^2 =
-c * dx/dt -k * x, c > 0, k > 0

and

value will
be @X1 units if

value is @X units; difference is @DX
units;

@DX1 = @DX - @C * @DX - @K * @X;

@X1 = @X + @DX1.

In order to implement self-supporting positive feedback,
leave only the first derivative and change the sign

dx/dt = k *
x, k > 0

value will
be @X1 units if

value is @X units; difference is @DX
units;

@DX1 = @DX + @K * @X;

@X1 = @X + @DX1.

Copyright (c)
I. Volkov, April 23, 2020 - April 27, 2020