EUROPHYSICS CONFERENCE "APPLICATION OF PHYSICS TO FINANCIAL
ANALYSIS – 3”, LONDON, DECEMBER, 2001
PRESENTATION
Financial Market Analysis in the Frame of t – J Model
of Solid State Physics
E. V. Shipitsyn , V. V. Popkov , D. B. Berg
International A.Bogdanov Institute,
4 Chebyshev St., Ekaterinburg 620062, Russia
1. Economic interpretation of physical notions, guided
by financial market analysis
Application of physical methods in economical analysis often
proves to be highly fruitful. For example, book [1] can perfectly illustrate
this very statement. The book describes a rather successful attempt to analyze
financial market by means of Ising model, one of the most popular models of
solid state physics. The analysis we carried out in the previous work [2] revealed
isomorphism of dynamical networks of the financial market (either stock or currency
one) and crystalline solid state. This fact allows to efficiently study financial
markets with the help of already worked out and applied methods of solid state
physics.
One of the basic fundamental theoretical models of solid state
physics is t – J model [3], representing a particular case of a well-known
Hubbard model [4]. t – J model describes exchange processes of electrons
with different spin between two arbitrary atoms (each atom having not more than
one electron). Spin of an electron can have either "up” or "down”
meaning, while spin of an atom coincides with spin of the electron placed on
it. Electrons with "down” and "up” spin can be refereed to money and
goods (goods are shares and currency), while atoms with "down” and "up”
spin – to buyers and sellers correspondingly [2].
t – J model is characterized with two physical parameters,
each being able to have economical interpretation. The first parameter – exchange
energy 
– 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 , 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,
parameter can be interpreted as a degree of easiness and quickness of sale and
purchase of goods, i.e. market liquidity (which can be quantitatively determined
as, for instance, volume of tenders in the unit of time).
The second parameter of t – J model – electron concentration
– represents relation of the number
of electrons in a solid state to the number of atoms in it. From the economic
point of view value can be considered
as the relation of the total cost of all goods circulating in the market (shares
or currency) to the total amount of these goods, i.e. as the market price of
the goods.
In the frame of t – J model there can be described three
types magnetically ordered phases (characterized with various value of magnetization
) – paramagnetism ( 
) , saturated ferromagnetism ( 
)
and non-saturated ferromagnetism ( 
) . Relative numbers of sellers 
and
buyers 
on the market (i.e. their
total shares:
and 
, (1)
where 
and 
– absolute numbers of sellers and buyers) can be compared correspondingly to
relative magnetization 
and its conjugated
value 
(relative "non-magnetization
”):
and 
(2)
Non-saturated ferromagnetism ( 
and 
) then would correspond with
a stable market (a market with both sellers and buyers), while saturated ferromagnetism
( 
and 
) and paramagnetism ( 
and 
) – with two opposite types of a non-stable market: "not buying” market
(a market with sellers, but with no buyers) and "not selling” market (a
market with buyers, but with no sellers), correspondingly.
Thus, there has been found a detailed one-to-one correspondence
between a crystalline solid state (described by t – J model) and the
financial market. The results of economic interpretation of physical notions,
we have carried out, are given in table 1.
2. Phase diagram of financial market state
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 model. Fig. 1 presents a magnetic phase
diagram of the very model, constructed in our work [5]. From the economic point
of view Fig. 1 represents a phase diagram of the financial market condition,
which allows to make the following conclusions.
In case of absolute absence of liquidity ( 
) at any price of goods different from zero ( 
) the financial market is a "not buying” one (i.e. all participants of
the market are sellers, and there are no buyers). This fact seems to be very
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 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).
Appearance of liquidity ( 
) means that there appears a possibility to sell the bought goods. Growth of
liquidity at any fixed non-zero price of goods (for example, at 
) results first in transformation of the "not buying” unstable market into
the stable market (which means appearance of buyers), and then – in transformation
of the stable market into the "not selling” unstable market (which means
disappearance of sellers). It should be noted, that the higher is the price,
the higher liquidity is required to carry out both mentioned phase transitions.
In case of zero price of goods ( 
) at any liquidity different form zero (
) the financial market becomes a "not selling” one (which means that all
participants of the market are buyers, and there are no sellers). Growth of
the price of goods ( ) at any fixed
non-zero liquidity (for example, at 
) results first in transformation of the "not selling” unstable market
into the stable market (which means appearance of sellers), and then – in transformation
of the stable market into the "not buying” unstable market (which means
disappearance of buyers), the first of the mentioned phase transition taking
place at 
, while the second – at
( 
) ; critical points 
and 
represent those values of price ,
corresponding with liquidity , which
stipulate appearance (or disappearance) of sellers and buyers correspondingly.
It should be noted, that the higher is liquidity (i.e. the higher is value 
) , the higher price is required both for appearance of sellers and disappearance
of buyers (i.e. the higher are values
and ) .
3. Supply, demand and equilibrium price on the financial market
In our work [6] there has been established the following
connection between magnetization
, electron concentration and exchange
energy :
, (3)
where
, (4)
, (5)
, 
,
, (6)
If we take into consideration expressions (3) – (6) the equations
(2) will represent functional dependencies of relative numbers of sellers
and buyers on the financial market
upon the market price of goods and
market liquidity . A graphical picture
of these dependencies is given on Fig. 2 , where the following can be concluded.
At a low price of goods ( 
) all
participants of the market are buyers, and there are no sellers (the market
is unstable – a "not selling” one). As the price grows ( 
) in addition to buyers in the market there appear sellers (the market comes
to be stable). The further growth of the price would cause decrease in the number
of buyers and increase in the number of sellers. If the price keeps on growing
( 
) buyers completely disappear,
therefore these are only sellers, who remains in the market (the market becomes
unstable again – "not buying” this time).
You might note, that Fig. 2 completely qualitatively corresponds
with the result of the work [7], shown on Fig. 3 , which is called "the
Bible of the modern neoclassical economic theory” [8], while its author Leon
Walras is called ”one of the greatest economists” [9]. Fig. 3 demonstrates,
that at zero price of goods ( 
)
there is supply ( 
) , while demand
is solely determined by absolute need in these goodsà and is the final value
( 
) . If the price grows ( 
) demand gradually goes down to zero ( 
) , while supply comes to be different from zero ( 
) only at some fixed final price (big enough for the sale to be profitable)
and then monotonously grows. Abscissa of intersection point of supply and demand
curves (in other words, curves of purchases and sales) determines an equilibrium
(fair) price of the goods 
[7].
As we’ve already mentioned t – J model assumes that
each atom should have not more than one electron. This very condition means,
that the trade on the financial market we study within the frame of t – J
model is carried out by means of fixed lots (documentary lots and money ones,
adequate to their cost), and each subject of this market (either a seller or
a buyer) can at any moment of time have not more than one such lot (documentary
or money one, correspondingly). The latter statement proves the fact, that within
the frame of our (model) analysis of the financial market the requirement 
is equivalent to the requirement 
, which means, that critical price 
we introduced in this work coincides with equilibrium price
figured in the work [7]:
(7)
Equality (7) allows us to find equilibrium price
from the equation , which with consideration
of (2) looks as follows
(8)
At any fixed liquidity
numerical solution 
of the equation
(8) with the consideration of (3) – (7) makes an equilibrium price
. A diagram of dependency of equilibrium (fair) price on market liquidity in
given on Fig. 4 , and makes evident the fact, that growth of liquidity causes
monotonous growth of equilibrium price.
4. Conclusions
Basing on isomorphism of financial market dynamic networks
and a crystalline solid state, we established in the previous work [2], we have
successfully applied theoretical methods of solid state physics (in the frame
of t – J model) to analyze financial markets (both stock and currency
ones), namely:
- Constructed a phase diagram of the financial market condition,
- Established functional dependencies of relative numbers of sellers and buyers
on the market upon the market liquidity and the market price of goods (shares
or currency),
- Constructed a diagram of dependency of equilibrium (fair) price upon the
market liquidity.
The received results prove the high efficiency of transition
of methods from physics to economy we’ve applied, and allow to hope for successful
continuation of the activity we have begun.
The work has been carried out with partial support of RGNF
Grant ¹ 01-02-00114a.
REFERENCES
- Peters E.E. Chaos and Order in the Capital Markets. New York: John Wiley
and Sons, 1996.
- Popkov V.V., Shipitsyn E.V., Berg D.B. // Europhysics Conference Abstracts,
2001, V. 25F, P. 79.
- Hirsch J.E. // Phys. Rev. Lett., 1985, V. 54, N. 12, P. 1317.
- Hubbard J. // Proc. Roy. Soc., 1963, V. A276, N. 1365, P. 238.
- Izyumov Yu.A., Letfulov B.M., Shipitsyn E.V. // Fiz. Met. Metalloved., 1991,
N. 7, P. 90.
- Letfulov B.M., Shipitsyn E.V. // Fiz. Met. Metalloved., 1991, N. 10, P.
61.
- Walras L. Les elements d’economie politique pure. Paris: Economica, 1900.
(Walras L. Elements of Pure Economics. Irwin: W. Jaffe, 1954.)
- Negishi T. History of Economic Theory. Amsterdam: North-Holland, 1988.
- Schumpeter J.A. History of Economic Analysis. Oxford University Press, 1954.
Table 1 . Table of correspondence of physical and economic
notions.
| Designation |
Physical sense |
Economical interpretation |
| |
Exchange energy |
Market liquidity |
| |
Electron concentration |
Market price |
| |
Relative magnetization |
Relative number of sellers on the market
|
| |
Relative "non-magnetization” |
Relative number of buyers on the market 
|
| NF |
Non-saturated ferromagnetism |
Stable market (a market with both sellers and buyers) |
| SF |
Saturated ferromagnetism |
"Not buying ” unstable market (a market with sellers,
but with no buyers) |
| P |
Paramagnetism |
"Not selling” unstable market (a market with buyers,
but with no sellers) |
COMMENTS TO FIGURES
Fig. 1 . Magnetic phase diagram of t – J model (of a
phase diagram of the financial market state) on the plane of parameters 
.
Fig. 2 . Dependency of relative numbers of buyers
and sellers on the market price at
arbitrary fixed liquidity :
– "not selling”
unstable market ( P ) ,
– stable market
( NF ) ,
– "not buying”
unstable market ( SF ) ;
– price of coincidence
for the numbers of sellers and buyers,
determined from the condition
and corresponding with maximum stable market ( 
) .
Fig. 3 . Dependency of demand volume 
and supply volume 
of goods
on the market upon the price
of
these goods (curves of purchases and sales);
– equilibrium
price (the coincidence price of supply and demand,
i.e. the real price of exchange goods on money),
determined from the condition
[7].
Fig. 4 . Dependency of equilibrium price
on market liquidity .
