, (1)V.Popkov, Å.Shipitsyn, D.Berg
International A.Bodganov Institute,
4 Chebyshev St., Ekaterinburg, 620062, Russia
High level of transparency, liquidity, volatility and interconnection was achieved by modern financial markets (both stock and currency ones), basically due to PCs and Internet; besides, the volume of tenders on the markets considerably increased. This situation allows to consider the markets as global dynamic networks.
A structure of the financial market dynamic network, given on fig. 1a, proves the mentioned market to be a typical example of a dual system (i.e. the system, consisting of two different and indissolubly bound parts, mutually supplementing and determining one another) [1] , for the characteristic feature of duality is four connected in pairs elements, one being considerably different from others [1] (in the analyzed case the role of the special element seems to be taken by "universal equivalent” – money, as far as it is the growth of its amount, that is the final goal of every subject of the financial market).
In accord with [1], the theory of duality represents one of possible directions for the creative development of Tektology (Universal organizational science) [2], whose basic statement is the thesis about unity of organizational methods of world cognition, contending, in particular, that transferring methods of analysis from one scientific field to some other often brings to highly positive results, more than that it may even cause a break in cognition and transformation of the world [2]. Besides, fig. 1 proves a crystalline solid state to have just the same structure of its dynamic network as the financial market does (a typical economic object). Similar are the functioning mechanisms of both networks mentioned above: electrons move within a crystalline solid state by skipping from one atom to another, while money and goods move within the financial market by passing from one buyer or seller to some other. Thus, we can assert, that dynamic networks of the financial market and a crystalline solid state are isomorphic to each other.
The circumstances mentioned above point to the opportunity to efficiently study financial markets with the help of thoroughly developed and successfully applied methods of analyzing crystalline solid states. From the standpoint of Tektology [2] this means practical application of the principle of unity of the world cognition organizational methods (by transferring those methods from physics to economy).
We should note the fact, that the usage of physical methods in economic analyses often proves to be highly fruitful. This statement can be well illustrated, for example, by the work [3], describing a very successful attempt to study the financial market with the help of Ising model, one of the most popular models of solid state physics.
Let’s also mark the fact, that crystalline lattice of a solid state (the aggregate of subjects of the financial market) represents a static network (characterized only with its structure, while addition of electron gas (the aggregate of objects of the financial market) "excites” this network and turns it from static to dynamic one (characterized with the process, continuously proceeding in its structure). A similar classification was given in the works [4, 5], where the static network was called "dead” (or "not excited”), while the dynamic one – "alive” (or "excited”). The change from the first to the second (i.e. "reanimation” of the "dead” network) takes place due to engaging electromagnetic field, which automatically arises in our problem as well, if we include electron gas into consideration (as far as electrons possess electric charge continuously moving on a crystalline lattice, and these are moving electric charges that create electromagnetic field).
Basic fundamental theoretical models of solid state physics are t – J model [6] (representing a special case of Hubbard model [7] ) and Anderson model [8]. Recently there was taken a rather successful attempt to unite them and create a new synthetic model, called t – J – A model [9]. It supposes, that movement of electrons from one atom to another occurs by leaps (i.e. at any fixed moment of time all electrons in the considered crystalline solid state are placed on atoms, not in interatomic space), each atom having only one energetic level (i.e. at any fixed atom there can take place not more than one electron at the same time). Therefore, in the frame of t – J – A model three electron conditions of an atom are possible:
, (1)
they mean correspondingly absence of electrons and presence of one electron with "up” or "down” spin (in the first case the considered atom has zero spin, in the second – "down” spin, and in the third – "up” spin).
Proceeding from the mentioned isomorphism of the financial market and a crystalline solid state dynamic networks, electrons with "up” and "down” spin can be compared correspondingly to money and goods (goods are shares and currency), while atoms with "up” and "down” spin – to buyers and sellers. This (1) means, that an atom has "up” ("down”) spin in case there is an electron with "up” ("down”) spin on it. Taking into consideration the above given comparison of physical notions with economic ones this statement means, that the buyer is the subject of the financial market, having money, while the seller is the one, having goods. Possession by one subject with money and goods at the same time is not permitted. This economic restriction follows from the applied physical model (forbidding simultaneous placement of two and more electrons on one atom) and allows to simplify the initial task in order to facilitate the further solution (promoting separation of buyers and sellers). We may say, that the trade on the considered financial market is conducted by fixed lots (documentary lots and money ones, corresponding to their cost), each subject of the market having not more than one such lot at any moment of time.
We should say, that the first of the mentioned in (1) elements – atom with no electrons (i.e. the subject of the financial market with neither money, nor goods) can be considered a static element of the dynamic network (or a passive participant of the market, while the other two – atoms, having one electron with "up” or "down” spin (i.e. the subjects of the financial market with either money or goods being buyers and sellers correspondingly) can be considered dynamic elements of the network (or active participants of the market). Increase in number of electrons in the considered crystalline solid state (influx of money and goods to the financial market) results in "activation” of passive participants, i.e. decrease in the number of static elements of the dynamic network.
The above mentioned t – J – A model can be characterized with three basic physical parameters, each can be given economic interpretation. The first parameter – exchange energy J is a characteristic of the exchange process of electrons with "up” and "down” spin between two atoms (i.e. simultaneous transition of electron with "down” spin from the first atom to the second, and electron with "up” spin from the second atom to the first, which results in the change of the first atom’s spin from "down” to "up”, and the second’s one – from "up” to "down”); the higher is the value J, the easier and quicker such an exchange goes on. In economic language this process corresponds to the exchange of money and goods between a buyer and a seller, therefore, J parameter can be interpreted as a degree of easiness and quickness of sale and purchase of goods, i.e. liquidity.
The second parameter of t – J – A model – width W of the
energy dispersedness (dispersion -?) zone (formed by all various electronic
energetic levels, existing on atoms of the considered solid state) is a characteristic
of the disorder in a crystalline lattice. Let’s note, that an electronic energetic
level of an atom represents a summary energy of the atom in its two conditions
– 
and 
(i.e. electrons with "up” and ‘down” spin, placed on it in turns), while
W is equal to difference between maximum (the highest) and minimum
(the lowest) electron energetic levels of atoms of the considered solid state.
If we now compare a physical notion "energy” and an economic category "cost”,
electron energetic level of an atom could be interpreted as a summary cost of
money and goods, which can have various meanings only due to fluctuations of
their price. Thus, W parameter in fact represents the difference
between minimum and maximum prices of goods, therefore, this parameter can be
interpreted as a changeability degree of the market price of goods, i.e. volatility.
The third parameter of t – J –A model – concentration of electrons n
(varying from 0 to 1) represents relation of the number of electrons
in a solid state ( 
) to the number
of atoms in it ( 
) :
, (2)
where
, (3)
while 
and 
– are correspondingly the number of electrons with "down” and "up”
spin. From the economic standpoint n parameter can be given a
dual mutually supplementing interpretation.
On the one hand, the value n represents the relation of cumulative amount of the financial market resources (i.e. cumulative amount of money and goods, circulating on it) to the number of all participants of the market (both active and passive ones), which allow to consider it as concentration of resources. On the other hand, as far as electrons within t – J – A model can be placed only on atoms (not more than one electron on each atom), the value n represents concentration of atoms, engaged by electrons i.e. relation of the number of atoms, engaged by electrons, to the full number of atoms in the considered solid state. Thus, in economic language n parameter can be called a concentration of active participants of the market (representing the relation of the number of the market active participants to the full number of the market participants), or a dynamic factor (factor of wholeness) of the financial market network (showing which part of elements of the dynamic network is engaged in interaction with each other).
In the frame of t – J – A model instead of electron concentration n
there can be used a conjugated magnitude – hole concentration 
, connected with 
through elementary
relation
(4)
and, in particular, represents concentration of atoms not engaged by electrons
(i.e. relation of the number of atoms not engaged by electrons to the full number
of atoms in the considered solid state). From the economic standpoint
value is a concentration of passive participants of the market (representing
the relation of the number of passive market participants to the full number
of market participants), or a static factor (factor of dissociation) of the
financial market dynamic network (showing which part of elements in the dynamic
network does not interact with the rest of the network). In other words,
parameter can be interpreted as a degree of the market dissociation (disconnectedness)
t – J – A model allows to describe three types of various magnetic orderings of a crystalline solid state – antiferromagnetism ( A ) , ferromagnetism ( F ) and saturated ferromagnetism ( SF ) [9 – 14] (fig. 2). An antiferromagnetic structure is characterized by a strict alternation of atoms with "up” and "down” spin (that’s why all nearest neighbours of every atom have spin, opposite to it), a ferromagnetic one by the prevalence of atoms with "up” spin (over atoms with "down” spin), while a saturated ferromagnetic one by the presence of only atoms with "up” spin (and absolute absense of atoms with "down” spin). Antiferromagnetic, ferromagnetic and saturated ferromagnetic structures are described with the following relations correspondingly:
, (5)
(6)
è
, (7)
where 
and 
is the number of atoms with "up” and "down” spin.
The following economic interpretations can be given for the three mentioned magnetic structures. Antiferromagnetic represents a stable (balanced) market, with the number of buyers equal to the number of sellers (according to fig. 2a each buyer being surrounded by sellers, and each seller by buyers; this fact increases efficiency of activities of all financial market subjects). Ferromagnetic means an unstable (not balanced) market, where the number of buyers is less than the number of sellers. Finally, saturated ferromagnetic in fact represents a complete absence of a real market, for in this case all active participants of the market are sellers, and there are no buyers. Such a situation can be looked upon as a global crisis, when the market completely looses liquidity, and sellers can not market goods for buyers have no money.
Thus, there has been found a detailed one-to-one correspondence between a crystalline solid state (described by t – J – A model) and the financial market. The results of economic interpretation of physical notions, we have carried out, are given in table 1 .
Table 1 allows to give an applicable to the financial market economic interpretation of physical results, received when describing a crystalline solid state in the frame of t – J – A model. Fig. 3 presents a magnetic phase diagram of the very model, constructed in [9 – 14]. In the left upper half of the diagram (including axis of ordinates) there is shown an antiferromagnetic phase, in the right lower half a ferromagnetic one, the phase of saturated ferromagnetism exists only on the abscissa axis. Appearance of disorder results in the growth of antiferromagnetic field and decrease of a ferromagnetic one, i.e. disorder strengthens antiferromagnetism and suppresses ferromagnetism [9]. From the economic point of view fig. 3 represents a phase diagram of the financial market condition, which allows to make the following conclusions.
In case of absolute absence of liquidity ( J=0 ) a real financial market is also absent in fact ( as far as all active participants of the market are sellers, there are no buyers). This fact seems to be natural, for the financial market has an extremely speculative character (i.e. all purchases are made only with the view for the future sales), as far as none of the participants of the market would buy goods, he will not be able to sell later (absence of liquidity means impossibility to sell the bought goods); and at the same time everybody having non-liquid goods would strive to get rid of it (by selling it).
Liquidity ( J>0 ) results at once in appearance of a real market. In case of low liquidity (low positive values of J parameter) this market is unstable (not balanced), however, growth of liquidity (as values of J parameter increase) turns it into a stable (balanced) one. We should note, that the higher is dissociation of the market (and the lower is concentration of its resources n ) , the higher liquidity (i.e. the higher value of J parameter) is required to pass from an unstable market to a stable one.
Everything said in two above paragraphs can be shortly formulated as follows: gradual decrease of liquidity (from high positive values to zero) results first in destabilization of the financial market, and then – to its complete disappearance (the higher is the market dissociation and the lower is the concentration of its resources, the less stable against liquidity decrease is the stable condition of the market).
Let’s also mention the fact, that growth of the market dissociation
(and reduction of its resource concentration
) turns a stable (balanced) market into an unstable (not balanced) one, i.e.
at any liquidity value J there exists a critical value of dissociation
, excess of which ( 
) brings to destabilization of the market, the value
increasing with growth of J (therefore, a high-liquid market is
more stable against growth of dissociation and reduction of resource concentration,
as compared to a low-liquid one).
This situation reminds the percolation phenomenon (i.e. leaking, or casual
distribution of fluid on a crystalline lattice of the environment), analyzed
in the fractal theory [15]. The analysis of this phenomenon proves, that growth
of the number of breakup connections between junctions of the lattice results
in the fact, that leaking of fluid through the environment stops, the quantitative
characteristics of the process being a critic concentration of breakup connections
[15] , named "threshold of leaking”,
and which is actually analogous to the value
already considered by us.
Besides, we may see, that appearance of volatility ( W>0 ) results in growth of a stable (balanced) market field and reduction of an unstable (not balanced) market field. Therefore, volatility favours stabilization of the financial market. This conclusion seems rather regular, for volatility characterizes price fluctuations, while the financial market (having a speculative character) attracts its all participants (subjects) with expectation of the prices variation.
From the other hand [16] showed, that only a small disorder favours strengthening
of antiferromagnetism (small positive values of W parameter) ;
if there takes place the growth of disorder (as the values of W
parameter grow) extension of the antiferromagnetic field on the phase diagram
is replaced with its compression, while at infinitely big disorder ( 
) the antiferromagnetic field simply disappears i.e. antiferromagnetism is completely
supresse). Translating everything said above from the physical language to the
economic one, we understand, that only a not too high (a reasonable) volatility
is for the good of the financial market; if price fluctuations reach giant scales,
it results in complete destruction of the market (as far as in such situation
sharp growth of risks and huge financial losses of subjects of the market result
in mass outflow of money and participants of trade from the market).
Finally, fig. 3 demonstrates, that appearance of volatility ( 
) provides stabilization of the market only if there is final liquidity ( 
) ; in case, when the latter is absent ( J=0 ) no volatility values
W ( W>0 ) would allow to create a real financial
market. At the same time, appearance of any final liquidity ( J>0
) brings at once the appearance of a real market (stable or unstable one – depending
on the value of its dissociation degree )
even if there is no volatility ( W=0) . So, liquidity is a more
important factor of the financial market destabilization, as compared to volatility.
So, we have completely analyzed an economic sense of the magnetic phase diagram of t – J – A model. Another important and interesting physical result, received in the frame of the given model, is the concentration dependence of Neel and Curie temperatures * [9, 10], shown on fig. 4 .
Comparison of Neel 
and Curie 
temperatures correspondingly to volumes of demand and supply of goods on the
market, and concentration of holes to
the price of these goods, brings to complete qualitative coincidence fig. 4á
with the result of the work [17] shown on fig. 5], which is called "the
bible of modern neoclassic economic theory” [18], and its author Leon Walras
is "one of the greatest economists” [19]. The mentioned coincidence of
physical results (received by means of mathematical methods) with economic conclusions
would seem not casual and not surprising at all, if we take into consideration
the following statement by L.Walras: "A pure political economy, or the
theory of exchange of goods and barter cost, is a physical and mathematical
science” [17].
From the economic standpoint fig. 4b ( as well as fig. 5 ) shows, that at the
zero price of goods ( 
) there is
no supply ( 
) , while demand is only
determined by a complete (absolute) need in the very goods and is a final value
( 
) . Under the growing price ( 
) demand gradually goes down to zero ( 
) , while supply, becomes different from zero ( 
) only in case of some fixed final price (high enough for the sale of goods
to remain profitable) and then monotonously goes up. Abscissa of intersection
point of supply and demand curves (or, in other words, curves of sales and purchases)
determines an equilibrium (correct) price of goods [17].
However, fig. 4 in fact contains more information, than fig. 5, i.e. economic results of our financial market analysis (based on physical results of crystalline solid state study in the frame of t – J – A model) supplement and expand conclusions of the work [17], generalizing them in case of variable liquidity J, thus, allowing to analyze the influence of liquidity variations on demand, supply, and equilibrium price.
Fig. 4à proves, that in case of complete absence of liquidity ( J=0
) demand for the goods is also absent (
) , while supply differs from zero (
) at any (even any small) positive price (
) ; equilibrium price, in this case, being peer to zero point. This conclusion
seems to be true, for liquidity characterizes simplicity and rapidity of sale,
while absence of liquidity means, that this kind of sale is impossible. As far
as any financial market has a speculative character, there is nobody wishing
to buy goods in case, if posterior sale is impossible; the owners of absolutely
non-liquid goods are, obviously, ready to sell them at any price, different
from zero (if only not for free).
Fig. 4b proves, that if liquidity (
) appears in the field of small prices
there at once appears demand different from zero ( 
) , while supply becomes to differ from zero (
) only if a positive value of price is not too small 
(the
higher is the price, the higher is liquidity J) ; an equilibrium
price under final liquidity becomes different from zero. In case of further
growth of liquidity demand increases, while supply falls down (this statement
turns to be true for all prices); in this case supply and demand curves are,
in fact, dislodged to the right on an abscissa axis, which results in the fact
that an intersection point of these curves moves in the same direction, i.e.
growth of liquidity results in growth of equilibrium price. All these results
well correspond to the speculative character of the financial market.
Thus, making use of isomorphism of the financial market and a crystalline solid state dynamic networks, we have successfully applied theoretical methods of solid state physics to analyze the dynamics of the financial markets (both stock and currency ones). Results, we have received, prove the high efficiency of transiting methods from physics to economy, and allow to hope for successful continuation of the started activity.
|
Designation |
Physical sense |
Economical interpretation |
|
|
Electrons with "down” spin |
Money |
|
|
Electrons with "up” spin |
Goods (shares or currency) |
|
|
Atoms with "down” spin |
Buyers |
|
|
Atoms with "up” spin |
Sellers |
|
J |
exchange |
Liquidity |
|
W |
Disorder |
Volatility |
|
|
Concentration of holes |
Dissociation of market |
|
n |
Concentration of electrons |
Concentration of market resources |
|
A |
Antiferromagnetic |
Stable (balanced) real market |
|
F |
ferromagnetic |
Non-stable (not balanced) real market |
|
SF |
Saturated ferromagnetic |
Absence of real market |
Fig.1 . Isomorphic structures of dynamic networks:
à – the financial market (economic object),
á – a crystalline solid state (physical object).
Buyers and sellers an exchange places due to the speculative character of the financial market (i.e. making purchases only with a view to further sales, and selling only for the future purchases), as for atoms with "down” and "up” spin – due to the possibility of mutual exchange of electrons with "down” and "up” spin, placed on them (considering the fact, that the spin of an atom is strictly determined by the spin of an electron, placed on it).
"Up” and "down” spin is a direction of the spin’s vector correspondingly along and against external magnetic field.
Fig. 2 . An outlined idea of antiferromagnetic ( à ) , ferromagnetic ( á ) and saturated ferromagnetic ( â ) magnetic orderings of solid state atoms
(the example of two-dimensional square lattice of 
size
in case of n=1) .
Fig. 3 . A qualitative aspect of the magnetic phase diagram of t – J – A model (or the diagram of the financial market condition)
on the plane of parameters 
if W=0
(full line) and W>0 (dashed line).
Fig. 4 . Dependence of Neel ( )
and Curie ( ) temperatures
On the concentration of holes if
W=0
(or dependence of demand and supply on the price,
i.e. curves of purchases and sales):
à – J=0,
b – J>0:
(full lines) and
(dashed lines),
; 
and 
– being equilibrium prices correspondingly
if and
.
Fig. 5 . Dependence of the demand ( 
) and supply ( 
) volumes in the market
on the price (
) of these goods (curves
of sales and purchases);
– equilibrium price (i.e. the price
of real exchange of goods on money between sellers and buyers).
Fig.1 . Isomorphic structures of dynamic networks:
à – the financial market (economic object),
á – a crystalline solid state (physical object).
Buyers and sellers an exchange places due to the speculative character of the financial market (i.e. making purchases only with a view to further sales, and selling only for the future purchases), as for atoms with "down” and "up” spin – due to the possibility of mutual exchange of electrons with "down” and "up” spin, placed on them (considering the fact, that the spin of an atom is strictly determined by the spin of an electron, placed on it).
"Up” and "down” spin is a direction of the spin’s vector correspondingly along and against external magnetic field.
Fig. 2 . An outlined idea of antiferromagnetic ( à ) , ferromagnetic
( á ) and saturated ferromagnetic ( â ) magnetic orderings of solid state atoms
(the example of two-dimensional square lattice of size
in case of n=1) .
Fig. 3 . A qualitative aspect of the magnetic phase diagram
of t – J – A model (or the diagram of the financial market condition)
on the plane of parameters if W=0
(full line) and W>0 (dashed line).
Fig. 4 . Dependence of Neel (
) and Curie ( ) temperatures on the
concentration of holes if W=0
(or dependence of demand and supply on the price, i.e. curves of purchases
and sales):
(full lines) and (dashed lines), ;
and – being equilibrium prices
correspondingly if and
.Fig. 5 . Dependence of the demand (
) and supply ( ) volumes in the market
on the price () of these goods (curves
of sales and purchases);
– equilibrium price (i.e. the price
of real exchange of goods on money between sellers and buyers).