Most of parameters used to describe states and dynamics of financial market depend on proportions of the appropriate variables rather than on their actual values. Therefore, projective geometry seems to be the correct language to describe the theater of financial activities. We suppose that the object of interest of agents, called here baskets, form a vector space over the reals. A portfolio is defined as an equivalence class of baskets containing assets in the same proportions. Therefore portfolios form a projective space. Cross ratios, being invariants of projective maps, form key structures in the proposed model. Quotation with respect to an asset X (i.e. in units of X) are given by linear maps. Among various types of metrics that have financial interpretation, the min-max metrics on the space of quotations can be introduced. This metrics has an interesting interpretation in terms of rates of return. It can be generalized so that to incorporate a new numerical parameter (called temperature) that describes agent's lack of knowledge about the state of the market. In a dual way, a metrics on the space of market quotation is defined. In addition, one can define an interesting metric structure on the space of portfolios/quotation that is invariant with respect to hyperbolic (Lorentz) symmetries of the space of portfolios. The introduced formalism opens new interesting and possibly fruitful fields of research.

Edward W. Piotrowski

Jan Sladkowski

English

Research Papers in Economics

Geometry of Financial Markets – Towards Information Theory Model of Markets
(RePEc:sla:eakjkl:26 26-VII-2006)
Edward W. Piotrowski∗
Institute of Mathematics, University of Bia  lystok
Lipowa 41, Pl 15424 Bia  lystok, Poland
Jan S  ladkowski†
Institute of Physics, University of Silesia
Uniwersytecka 4, Pl 40007 Katowice, Poland
(Dated: September 9, 2006)
Most of parameters used to describe states and dynamics of ﬁnancial market depend on propor 
tions of the appropriate variables rather than on their actual values. Therefore, projective geometry
seems to be the correct language to describe the theater of ﬁnancial activities. We suppose that the
object of interest of agents, called here baskets, form a vector space over the reals. A portfolio is
deﬁned as an equivalence class of baskets containing assets in the same proportions. Therefore port 
folios form a projective space. Cross ratios, being invariants of projective maps, form key structures
in the proposed model. Quotation with respect to an asset Ξ (i.e. in units of Ξ) are given by linear
maps. Among various types of metrics that have ﬁnancial interpretation, the min max metrics on
the space of quotations can be introduced. This metrics has an interesting interpretation in terms
of rates of return. It can be generalized so that to incorporate a new numerical parameter (called
temperature) that describes agent’s lack of knowledge about the state of the market. In a dual way,
a metrics on the space of market quotation is deﬁned. In addition, one can deﬁne an interesting
metric structure on the space of portfolios/quotation that is invariant with respect to hyperbolic
(Lorentz) symmetries of the space of portfolios. The introduced formalism opens new interesting
and possibly fruitful ﬁelds of research.
PACS numbers: 89.65.Gh, 89.70.+c, 02.40.Dr
Keywords: ﬁnance, projective geometry, portfolio theory, econophysics
I. INTRODUCTION
In majority of the models considered in economics one cannot ask questions about symmetries of the considered
phenomena, especially if one put the stress on group theoretical aspects. The reason is that one can hardly speak
about invariance (covariance) of terms used in analysis or numerical values returned by most of models [1]. We
would like to argue that projective geometry, equipped with an appropriate metric structure and some measure of
investors performance, might form a precise formalism that allows us to carry out objective (quantitative) analysis of
investment processes and symmetries of their market context. We describe a simple geometrical model of a ﬁnancial
market – we call it Information Theory Model of Markets (ITMM) – that explores ideas of projective geometry. Our
model presents in some sense a picture of ﬁnancial markets dual to that assumed in the most popular ones, Capital
Asset Pricing Model and Arbitrage Pricing Models [2]. Investors, due to their lack of knowledge, wrong prognosis for
the future or simple fear, behave in an unpredictable, chaotic way. Prices are determined by their decisions – in the
same way as the gas pressure is determined by (chaotic) particles dynamics. A non random pricing of capital assets
follows from investors knowledge and possible random factors cancel themselves due to variety of strategies adopted
by investors if the market is liquid enough. The formalism of projective geometry allows us to carry out analysis of
invariant and covariant quantities. A detailed axiomatic formulation of the model will be given elsewhere [3], here
we would like to present only some basic features. The paper is organized as follows. In the next section we give
some basic deﬁnitions and describe mathematical tools we are going to use. Then we show the importance of metric
structures and give two exemplary metrics. It follows that some important analogies with physical theories can be
expected. Finally, we discuss a possible connection between investors performance and knowledge about markets
measured by information theory means.
∗Electronic address: ep@alpha.uwb.edu.pl; URL: http://alpha.uwb.edu.pl/ep/sj/index.shtml
†Electronic address: sladk@us.edu.pl2
II. PROJECTIVE GEOMETRY AS A FORMALISM DESCRIBING INVESTMENTS
The market determines what goods are made and what products are bought and sold. We assume that objects of
investors interest span a (N+1) dimensional vector space G over the reals. Elements of this vector space are called
baskets. Let us ﬁx some basis {g0,g1,...,gN} in G. gµ∈ G, the µ th element of the basis, is called the µ th asset
(market good). Assets, although selected in an arbitrary way, are distinguished because they are used for eﬀective
bookkeeping, accounting, market analysis and so on. For any basket p∈G we have a unique representation
p =
N X
µ=0
pµgµ.
The coeﬃcient pµ∈ R is called the µ th coordinate of the basket. A portfolio is deﬁned as an equivalence class of
non empty baskets (that is in G \ {0}) [4]. Two baskets p′ and p′′ are equivalent if and only if there exists λ ∈ R,
such that
N X
µ=0
p′
µgµ =
N X
µ=0
λ p′′
µgµ.
Equivalently,
(p′
0,...,p′
N) = (λp′′
0,...,λp′′
N).
If for a given portfolio we have pµ = 0, then there exists such a basket representing this portfolio that it contains
exactly a unit of asset gµ. Coordinates of this basket, p = (p0,...,pµ−1,1,pµ+1,...,pN), are called inhomogeneous
coordinates of the portfolio p with respect to µ th asset. If pµ= 0, p = (p0,...,pµ−1,0,pµ+1,...,pN), then we say
that that the portfolio p is improper for the µ th asset. Market quotation U in units of ν th asset is a linear map
U(gν,   ): G → R. The map U associates with a given portfolio p its current value in units of gν:
(Up)ν = U(gν,p) =
N X
µ=0
U(gν,gµ)pµ, (1)
where U(gν,gµ) is the price of a unit of µ th asset given in units of ν th asset.
A. Basic deﬁnition and ideas
We require that
U(gµ,U(gν,p)gν)gµ = U(gµ,p)gµ
for p and gµ and gν being exchangeable assets (that is U(gµ,gν)  = 0 and U(gµ,gν)  = ±∞, so inserting p = gρ we get
U(gµ,gν)U(gν,gρ) = U(gµ,gρ) (2)
for any µ, ν, ρ. Therefore quotations are transitive [5]. If we set µ=ν =ρ in Eq.2 then we see that there are two
possibilities U(gµ,gµ)=1 or U(gµ,gµ)=0. The case U(gµ,gµ)=1 implies projectivity of U: U((Up)µgµ)µ = (Up)µ.
The case U(gµ,gµ) = 0 means that the µ th asset is not subjected to quotation in the market (one can only, for
example, present somebody with such an asset). For µ=ρ we get U(gµ,gµ)=1 and therefore
U(gµ,gν) =
￿
U(gν,gµ)
￿−1
.
In general, the quotation map can be represented by a (N+1)×(N+1) matrix with (µ,ν) th entry given by Uµν :=
U(gµ,gν). The simplest way of determining this matrix consist in selecting some asset that is called the currency.
Suppose that the asset g0 is selected as the currency. The matrix Uµν is deﬁned uniquely by N values uk := U(g0,gk)
for k=1,...,N. (Note that U00=1). If u0 := 1, due to the transitivity (Eq.2) all entries of (Uµν) are determined by
the formula:
Uµν = u−1
µ uν. (3)3
After this Eq.1 simpliﬁes to:
(Up)ν =
N X
µ=0
uµpµu−1
ν .
For uk=0 Eq. 3 remains valid if we set u
−1
k := 0. Sometimes we have to rescale the prices uk in units proportional
to g0, ( e.g. if g0 represents shares, after split, after currency denomination and so on). Therefore it is convenient to
identify quotations U = (λ,λu1,...,λuN) for all λ∈R\{0}, that is introduce homogeneous coordinates. We say that
the portfolio p is balanced for the quotation U if there is such an asset gµ, so that the value of p in units of gµ is 0,
that is
(Up)µ =
N X
ν=0
U(gµ,gν)pν =
N X
ν=0
uνpνu−1
µ = 0.
For quotation denominated in currency this formula simpliﬁes to
PN
µ=0 uµpµ = 0. The linearity of these equations
allows for simple interpretations: portfolio p is balanced if the corresponding point belongs to the hyperplane repre 
senting quotation U.
MARKET PROJECTIVE GEOMETRY
p portfolio point
U quotation hyperplane
Up = 0 portfolio is balanced for U point p lies in quotation hyperplane
TABLE I: Projective geometry dictionary
               
r p r p
′
U                  
````````````
r p
′′
U
′ U
′′
FIG. 1: Two diﬀerent portfolios p, p
′ balanced for the same quotation U and a portfolio p
′′ balanced for two diﬀerent quotations
U
′, U
′′ (DUALITY!).
An important invariant can be deﬁned in projective geometry – a cross ratio of four points [6]. For exchange ratios
it describes the relative change of quotation (cf Fig.2):
{$,Q,Q′,e} :=
c′
$→e
c$→e
=
q′
$ qe
q′
e q$
=
|Q′e||Q$|
|Q′$||Qe|
=
P(△Q′eO)P(△Q$O)
P(△Q′$O)P(△QeO)
,
where c$→e :=
q$
qe is the exchange ratio $ → e (one obtains for q$ dollar qe euro) etc and P(△abc) denotes the area
of the triangle with vertices a,b, and c. In Fig.2 lengths of the segments Q$ and Qe are proportional to q$ and qe,
respectively. The invariance cross ratios of is crucial to our model.
B. Example: trading in a single asset
Let us consider the cross ratio [G,U→G,UG→,$] for U→G := (v,v ep→G,...) and UG→ := (w,wepG→,...) and the
points G and $ given by crossing of the prime line U→GUG→ and one asset portfolios: G i $ corresponding to assets G
and $. p→G and pG→ are the logarithmic quotations for buying and selling, respectively and the dots (...) represent
quotations for the remaining assets and need not be the same for both quotations. The logarithm of the cross ratio
[G,U→G,UG→,$] on the straight line U→GUG→ is equal to:
ln[G,U→G,UG→,$] = ln[
wepG→
wepG→ − v ep→G ,1,0,
w
w − v
] = ln
v wepG→
v ep→G w
= pG→ − p→G.4
Z
Z
Z
Z
Z
Z
Z
Z
Z
Z
Z
Z
Z
Z
Z
Z
Z
Z
Z
Z
Z Z
￿
￿
￿
￿
￿
￿
￿
￿
￿
￿
￿
￿ ￿ h h h h h h h h h h h h h h h h h h h h h h
hhhhh h
B
B
B
B
B
B
B
B
B
B
B
B
B B
￿
￿
￿
￿
￿
￿
￿
￿
￿
Z
Z
Z Z
e
$
Q
Q$
O
Qe
Q
′
FIG. 2: Exchange ratios.
III. METRIC STRUCTURES
It is a common lore that price movements are best described by diﬀusion processes. Diﬀusion equations of various
types involve Laplace operator and therefore metric structure. Metric structures are to some extent independent of the
conﬁguration (phase) space structure. One of our aims is to ﬁnd a suitable metrics on the projective space. Various
premises rooted in ﬁnance theory can be used to select a metric structure on the space of portfolios. For example,
often we would like to know which market movements are equivalent to portfolio modiﬁcations. Below we describe
two classes of metrics that we were able to construct in an explicit way. Both have interesting physical connotations.
There probably is quite a lot of other interesting metrics yet to be found.
A. Exemplary metric structure
Let us try to deﬁne a metrics on the space of quotations. Any two diﬀerent quotations U′ and U′′ determine a
projective prime line. To deﬁne a cross ratio we need two additional points lying in that line. It seems natural to
select them, let us consider two hyperplanes of improper quotations for two basic assets. These hyperplanes cut the
projective space RP
N into 2N N dimensional simplexes. Suppose that the quotations belong to the same simplex –
only then the distance would be ﬁnite. Each hyperplane of improper quotation for an asset gµ is cut by the prime
line. In this way we select N +1 points but only two of them, say Pb and Pc, lie in the vicinity of U′ and U′′ –
and only these two points belong to the boundary of the simplex that contains U′ and U′′, cf Fig.3. The cross ratio
[Pb,U′,U′′,Pc] can be used to deﬁne the distance (metrics):
d(U′,U′′) = ln([Pb,U′,U′′,Pc]) = ln
|U′Pc||U′′Pb|
|U′Pb||U′′Pc|
, (4)
where |P1P2| denotes euclidean distance of points P1 and P2. After some tedious but elementary calculations the
metrics can be given in a more transparent form:
d(U′,U′′) = ln([Pb,U′,U′′,Pc]) = ln
|U′Pc||U′′Pb|
|U′Pb||U′′Pc|
= ln
￿
max
µ
￿u′′
µ
u′
µ
￿￿
− ln
￿
min
µ
￿u′′
µ
u′
µ
￿￿
= max
µ
￿
rµ(U′,U′′)
￿
− min
µ
￿
rµ(U′,U′′)
￿
= max
µ
￿
rµ(U′,U′′)
￿
+ max
µ
￿
rµ(U′′,U′)
￿
.
(5)
The function rµ(U′,U′′) is known in ﬁnance as the interval interest rate. We have already proposed a method that
allows us to measure quantitatively investors qualiﬁcations [7]. Inspired by previous results and statistical physics,
we can introduce a temperature like parameter in the metrics given by Eq.5. Such a generalized metrics take the5
￿
￿
￿
￿
￿
￿
￿
￿
￿
￿
￿
￿
￿
￿
￿
￿
b
b a
T
T
T
T
T
T
T
T
T
T
T
T
T
b
c
b c
b b b b
￿
￿
￿
￿
￿
￿
￿
￿
￿
￿
￿
￿
￿ ￿
b
m
b m
r U2 r U0 I
II
r U1
III
IV
r
Pb r
Pa
r
Pc
r U
′
r U
′′
FIG. 3: Quotation in a three assets market (RP
2).
following form:
d(U′,U′′,T) :=
P
µ
rµ(U′,U′′)e
rµ(U′,U′′)
T
P
µ
e
rµ(U′,U′′)
T
+
P
µ
rµ(U′′,U′)e
rµ(U′′,U′)
T
P
µ
e
rµ(U′′,U′)
T
.
It should be possible to deﬁne canonical ensembles of portfolios, the temperature (entropy) of portfolios and, possibly,
various thermodynamics like potentials in a way analogous to that of Ref. [7].
B. Hyperbolic (Lorentz) geometry
We were able to identify another interesting metrics. Consider quotations at two diﬀerent times t′ and t′′ in a
simpliﬁed, two assets market. Let the homogeneous coordinates are ˆ p∗′ = (gˆ p′
0,g ˆ p′
1) and ˆ p∗′′ = (gˆ p′′
0,g ˆ p′′
1), respectively.
Suppose the quotations are not equal, ˆ p∗′  = ˆ p∗′′. The linear transformation:
ˆ S = ˆ S(ˆ p∗′, ˆ p∗′′) :=
1
gˆ p′
0 gˆ p′′
1 − gˆ p′′
0 gˆ p′
1
￿
− gˆ p′
1 + gˆ p′′
1 gˆ p′
0 − gˆ p′′
0
− gˆ p′
1 − gˆ p′′
1 gˆ p′
0 + gˆ p′′
0
￿
changes the basis in such a way that the quotations ˆ p∗′ and ˆ p∗′′ have coordinates fˆ p′ := (1,−1) and fˆ p′′ := (1,1).
From the physicist point of view, the directions (1,−1) and (1,1) deﬁne the propagation of light in a two dimensional
spacetime. We can accept this directions us absolute directions (light cone). The underlying metric structure can also
be found. In the dual representation, that is in the space of portfolios, two portfolios balanced on quotations ˆ p∗′ and
ˆ p∗′′ are inﬁnitely separated. Explicit form of the metrics on the space of portfolios is as follows:
d(p∗′,p∗′′) = |arctan(v′) − arctan(v′′)|,
where
v(p∗) = v(p∗, ˆ p∗′, ˆ p∗′′) =
gp0(gˆ p′′
0 − gˆ p′
0) + gp1(gˆ p′′
1 − gˆ p′
1)
gp0(gˆ p′′
0 + gˆ p′
0) + gp1(gˆ p′′
1 + gˆ p′
1)
.
Note that if we neglect details of the economic processes that make capital then one can change the content of a
portfolio only if one ”travels with speed of light” in the market.
C. Information theory context
The projective geometry structure of clear cut market model with a metrics that respects symmetries of the modelled
processes should yet be completed by discussion/construction of algorithms that governs the supply and demand as 
pects of agents behaviour. These algorithms should be optimal from the metrical structure point of view and, of course,6
respect speciﬁc regulations laid down by authorities. For example, in the simple Merchandising Mathematician Model
[8] and Kelly optimal bets [9] the optimal market strategies have direct connections with the Boltzmann/Shannon
entropy. These examples suggest that there might be a uniﬁed description of market phenomena that involves tools
from geometry, statistical physics and information theory. And the key ingredients would probably follow from the
underlying metric structure.
IV. CONCLUSIONS: TOWARDS INFORMATION THEORETICAL DESCRIPTION OF MARKETS
We have attempted at formulation of kind of Market Symmetry Principle: Conclusions drawn from a logically
complete market model are invariant with respect to projective symmetry transformations. We anticipate that metric
structures might play a key role that would pave the way for information theoretical description of market phenomena.
This point of view is supported by the explicit examples given in the paper. The presented projective geometry
formalism although simpliﬁed, is, to the best of our knowledge, the only one that attempts to introduce metric structure
to ﬁnance theory models that respect observed market processes symmetries, eg preselected absolute directions. This
would allow for analysis of hyperplanes of equilibrium temperature, entropy, various thermodynamical potentials,
Legendre transforms and, possibly identiﬁcation of conservation laws with tools borrowed from information theory
and (quantum) game theory [10].
[1] An invariant of a process (phenomenon, transformation etc.) is a numerical parameter whose value remains constant during
that process. Analogously, a covariant is a parameter whose numerical value has only a (relative) sense in a preselected
coordinate system but changes in a speciﬁed way if the coordinate systems is changed (e.g. given by some symmetry
transformation).
[2] D. G. Luendbeger, Investment Science, Oxford University Press, New York 1998.
[3] Most of the axioms takes the same form as in standard models because they simple deﬁne market organization.
[4] We could also take the empty basket into consideration but this would spoil the projective space interpretation.
[5] Note that this means that we treat all taxes, brokerages etc. as liabilities and therefore as separate assets. Models that
have scale eﬀects (projective symmetry is broken) should have a dual description in terms of nontransitive quotations.
[6] H. Busemann, P. J. Kelly. Projective Geometry and Projective Metrics, Academic Press, New York, 1953.
[7] E. W. Piotrowski, J. S  ladkowski, Acta Phys. Pol. B32 597 (2001).
[8] E. W. Piotrowski, J. S  ladkowski, Physica A 318 496 (2003).
[9] E. W. Piotrowski, M. Schroeder, Kelly Criterion revisited: optimal bets, talk given at the APFA5 Conference, Torino,
2006; physics/0607166.
[10] E. W. Piotrowski, J. S  ladkowski, Int. J. Theor. Phys. 42 1101 (2003).

Geometry of Financial Markets - Towards Information Theory Model of Markets

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Geometry of Financial Markets - Towards Information Theory Model of Markets

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