We continue the analysis of quantum-like description of market phenomena and economics. We show that it is possible to define a risk inclination operator acting in some Hilbert space that has a lot of common with quantum description of the harmonic oscillator. The approach has roots in the recently developed quantum game theory and quantum computing. A quantum anthropic principle is formulated. (final version published in Acta Phys.Pol.B,32(2001)3873-3879)

Edward W. Piotrowski

Jan Sladkowski

English

Research Papers in Economics

Quantum-like approach to ﬁnancial risk:
quantum anthropic principle
(RePEc:sla:eakjkl:8 5-X-2001)
Edward W. Piotrowski
Institute of Theoretical Physics, University of Białystok,
Lipowa 41, Pl 15424 Białystok, Poland
e-mail: ep@alpha.uwb.edu.pl
Jan Sładkowski
Institute of Physics, University of Silesia,
Uniwersytecka 4, Pl 40007 Katowice, Poland
e-mail: sladk@us.edu.pl
Abstract
We continue the analysis of quantum-like description of market
phenomena and economics. We show that it is possible to deﬁne a
risk inclination operator acting in some Hilbert space that has a lot
of common with quantum description of the harmonic oscillator. The
approach has roots in the recently developed quantum game theory
and quantum computing. A quantum anthropic principle is formulated.
1 Noncommutative quantum mechanics
The term noncommutative quantum mechanics is misleading as it sug-
gest there might exists something like commutative quantum mechanics. In
fact quantum theory has built in various types of noncommutativity since
its infancy. Nowadays the adjective noncommutative reﬂects the possibility
that the space variables
^
x
i might not commute. This sort of generalizations
1has it roots in M-theory (branes), deformation quantization, Weyl quanti-
zation and curiosity [1]. This is of course connected with the A. Connes
program [2, 3, 4]. Among various possible generalizations the most popular
one are
[
^
x
i
;
^
x
j
]
=
i
￿
i
j
;
￿
i
j
2
C
;
(
1
a
)
[
^
x
i
;
^
x
j
]
=
i
C
i
j
k
^
x
k
;
C
i
j
k
2
C
;
(
1
b
)
^
x
i
^
x
j
=
q
￿
1
^
R
i
j
k
l
^
x
k
^
x
l
;
^
R
i
j
k
l
2
C
:
(
1
c
)
In all these cases the index
i takes values from
1 to
N.W es h a l ls u p p o s e
that the appropriate algebra
A
x has a unit element [2, 3]. Eqs. (1b) and (1c)
describe the so called Lie algebraic and quantum group generalizations,
respectively. Here we would be interested in a generalization of the form
given by Eq. (1a) which may traced back to works of H. Weyl [5]a n dM o y a l
[6].
2 Quantum market games
It was a complete surprise to us to ﬁnd the University of Bologna preprint
entitled Quantum Mechanics and Mathematical Economics are Isomor-
phic and written by Lambertini [7]. We were actually working on the direct
application of recently developed quantum game theory [8, 9, 10] in eco-
nomics [11]-[ 16]. The Lambertini’s paper was a further incentive to work
harder. In the ”standard” quantum game theory one tries in some sense
to quantize an operational description of ”classical” versions of the game
being analyzed. It usually enlarges the set admissible strategies in a non-
trivial way. We follow a diﬀerent rout. The market players strategies are
described in terms state vectors
j
 
i belonging to some Hilbert space
H
[12, 13, 15]. The probability densities of revealing the players, say Alice and
Bob, intentions are described in terms of random variables
p and
q:
j
h
q
j
 
i
A
j
2
A
h
 
j
 
i
A
j
h
p
j
 
i
B
j
2
B
h
 
j
 
i
B
d
q
d
p
; (1)
where
h
q
j
 
i
A is the probability amplitude of oﬀering the price
q by Alice
who wants to buy and the demand component of her state is given by
j
 
i
A
2
H
A. Bob’s amplitude
h
p
j
 
i
B is interpreted in an analogous way
(opposite position). Of course, the ”intentions”
q and
p n o ta l w a y sr e s u l ti n
2the accomplishment of the transaction [13]. If one considers the following
facts[11, 17, 18]:
￿ error theory: second moments of a random variable describe errors
￿ M. Markowitz’s portfolio theory (Nobel Prize 1990)
￿ L. Bachelier’s theory of options: the random variable
q
2
+
p
2 measures
joint risk for a stock buying-selling transaction ( Merton & Scholes won
Nobel Prize in 1997)
then it seems reasonable to deﬁne the observable the risk inclination
operator:
H
(
P
k
;
Q
k
)
:
=
(
P
k
￿
p
k
0
)
2
2
m
+
m
!
2
(
Q
k
￿
q
k
0
)
2
2
;
(
2
)
where
p
k
0
:
=
k
h
 
j
P
k
j
 
i
k
k
h
 
j
 
i
k ,
q
k
0
:
=
k
h
 
j
Q
k
j
 
i
k
k
h
 
j
 
i
k ,
!
:
=
2
￿
￿ .
￿ denotes the character-
istic time of transaction [11] which is, roughly speaking, an average time
spread between two opposite moves of a player (e. g. buying and selling the
same asset). The parameter
m
>
0 measures the risk asymmetry between
buying and selling positions. Analogies with quantum harmonic oscillator
allow for the following characterization of quantum market games. The con-
stant
h
E describes the minimal inclination of the player to risk. It is equal
to the product of the lowest eigenvalue of
H
(
P
k
;
Q
k
) and
2
￿.
2
￿ is in fact
the minimal interval during which it makes sense to measure the proﬁt. Ex-
cept the ground state all the adiabatic strategies
H
(
P
k
;
Q
k
)
j
 
i
=
c
o
n
s
t
j
 
i
are giﬀens [11, 13] that is goods that do not obey the law of demand and
supply. It should be noted here that in a general case the operators
Q
k do
not commute because traders observe moves of other players and often act
accordingly. One big bid can inﬂuence the market at least in a limited time
spread. Therefore it is natural to apply the formalism of noncommutative
quantum mechanics where one considers
[
x
i
;
x
k
]
=
i
￿
i
k
:
=
i
￿
￿
i
k
:
(
3
)
The analysis of harmonic oscillator in more then one dimensions [18]i m p l y
that the parameter
￿ modiﬁes the constant
}
E
!
p
}
2
E
+
￿
2 and, accord-
ingly, the eigenvalues of
H
(
P
k
;
Q
k
). This has the natural interpretation that
moves performed by other players can diminish or increase one’s inclina-
tion to taking risk.
33 Market as a measuring apparatus
When a game allows a great number of players in it is useful to consider it
as a two-players game: the trader
j
 
i
k against the Rest of the World (RW).
The concrete algorithm
A may allow for an eﬀective strategy of RW (for
a suﬃciently large number of players the single player strategy should not
inﬂuence on the form of the RW strategy). If one considers the RW strategy
it make sense to declare its simultaneous demand and supply states because
for one player RW is a buyer and for another it is a seller. To describe
such situation it is convenient to use the Wigner formalism. The following
subsection describe shortly various aspects of quantum markets.
3.1 Quantum Zeno eﬀect
If the market continuously measures the same strategy of the player, say
the demand
h
q
j
 
i, and the process is repeated suﬃciently often for the
whole market, then the prices given by some algorithm do not result from
the supplying strategy
h
p
j
 
i of the player. The necessary condition for
determining the proﬁt of the game is the transition of the player to the state
h
p
j
 
i. If, simultaneously, many of the players changes their strategies then
the quotation process may collapse due to the lack of opposite moves. In
this way the quantum Zeno [20] eﬀects explains stock exchange crashes.
Another example of the quantum market Zeno eﬀect is the stabilization of
prices of an asset provided by a monopolist.
3.2 Eigenstates of
Q and
P
Let us suppose that the amplitudes for the strategies
h
q
j
 
i
k or
h
p
j
 
i
k that
have inﬁnite integrals of squares of their modules, (
h
q
j
 
i
k
6
2
L
2)h a v et h e
natural interpretation as the will of the
k-th player of buying (selling) of the
amount
d
k (
s
k) of the asset
G.S ot h es t r a t e g y
h
q
j
 
i
k
=
h
q
j
a
i
=
Æ
(
q
;
a
) means
that in the case of classifying the player to the set
f
k
d
g,r e f u s a lo fb u y i n g
cheaper than at
c
=
e
a and the will of buying at any price equal or above
e
a.
In the case of ”measurement” in the set
f
k
d
g the player declares the will of
selling at any price. The above interpretation is consistent with the Heisen-
berg uncertainty relation. The strategies
h
q
j
 
i
2
=
h
q
j
a
i (or
h
p
j
 
i
2
=
h
p
j
a
i)
do not correspond to the RW behaviour because the conditions
d
2
;
s
2
>
0,
if always fulﬁlled, allow for unlimited proﬁts (the readiness to buy or sell
G
4at any price). The demand and supply functions give probabilities of com-
ing oﬀ transactions in the game when the player use the strategy
h
p
j
c
o
n
s
t
i
or
h
q
j
c
o
n
s
t
i and RW, proposing the price, use the strategy
￿.
3.3 Correlated coherent strategies
We will deﬁne correlated coherent strategies as the eigenvectors of the
annihilation operator
C
k
C
k
(
r
;
￿
)
:
=
1
2
￿
￿
1
+
i
r
p
1
￿
r
2
￿
Q
k
+
i
￿
P
k
;
(
4
)
where
r is the correlation coeﬃcient
r
2
[
￿
1
;
1
],
￿
>
0. In these strategies
buying and selling transactions are correlated and the product of disper-
sions fulﬁlls the Heisenberg-like uncertainty relation
￿
p
￿
q
p
1
￿
r
2
￿
}
E
2
and is minimal. The annihilation operators
C
k and their eigenvectors may
be parameterized by
￿
p
=
}
E
2
￿ ,
￿
q
=
￿
p
1
￿
r
2 and
r.
3.4 Mixed states and thermal strategies
According to classics of game theory the biggest choice of strategies is
provided by the mixed states
￿
(
p
;
q
). Among them the most interesting are
the thermal ones. They are characterized by constant inclination to risk,
E
(
H
(
P
;
Q
)
)
=
c
o
n
s
t and maximal entropy. The Wigner measure for the
n-th exited state of harmonic oscillator have the form
W
n
(
p
;
q
)
d
p
d
q
=
(
￿
1
)
n
￿
}
E
e
￿
2
H
(
p
;
q
)
}
E
!
L
n
￿
4
H
(
p
;
q
)
}
E
!
￿
d
p
d
q
;
(
5
)
where
L
n is the
n-th Laguerre polynomial. The mixed state
￿
￿ deter-
mined by the Wigner measures
W
n
d
p
d
q weighted by the Gibbs distribution
w
n
(
￿
)
:
=
e
￿
￿
n
}
E
!
P
1
k
=
0
e
￿
￿
k
}
E
! have the form
￿
￿
(
p
;
q
)
d
p
d
q
:
=
1
X
n
=
0
w
n
(
￿
)
W
n
(
p
;
q
)
d
p
d
q
=
!
2
￿
x
e
￿
x
H
(
p
;
q
)
￿
￿
￿
x
=
2
}
E
!
t
a
n
h
(
￿
}
E
!
2
)
d
p
d
q
:
53.5 Market cleared by quantum computer
When the algorithm
A calculating in a separable Hilbert space
H
k does
not know the players strategies it must choose the basis in an arbitrary
way. This may result in arbitrary long representations of the amplitudes
of strategies. Therefore the algorithm
A should be looked for in the NP
(non-polynomial) class and quantum markets may be formed provided the
quatum computation technology is possible. Is the quantum arbitrage pos-
sible only if there is a unique correspondence between
} and
h
E?W a st h e
hypothetical evidence given by Robert Giﬀen in the British Parliament the
ﬁrst ever description of quantum phenomenon? The commonly accepted
universality of quantum theory should encourage physicist in looking for
traces quantum world in social phenomena.
4Co n c l u s i o n s
We think that the formalism of quantum theory may provide us with tools of
unexpected power that combined with methods of game theory may allow
for much deeper understanding of ﬁnancial phenomena than it is usually
expected. Let us quote the Editor’s Note to Comlexity Digest 2001.27(4)
(http://www.comdig.org) ”It might be that while observing the due ceremo-
nial of everyday market transaction we are in fact observing capital ﬂows
resulting from quantum games eluding classical description. If human deci-
sions can be traced to microscopic quantum events one would expect that
nature would have taken advantage of quantum computation in evolving
complex brains. In that sense one could indeed say that quantum comput-
ers are playing their market games according to quantum rules”. During
the past decade options gained a signiﬁcant position in capital turnover all
over the world. It may mean that simple methods of maximization of proﬁt
became unsatisfactory. At present minimization of ﬁnancial risk is playing
the key role. Lower risk means better prognosises for industrial and mar-
ket development and this results in higher proﬁts. Methods minimazing the
ﬁnancial risk will become a focus of investors attention. All this tempt us
into formulating the quantum anthropic principle of the following form.
At earlier civilization stages markets are governed by classical laws (as
classical logic prevailed in reasoning) but the incomparable eﬃcacy
of quantum algorithms in multiplying proﬁts 1 will result in quan-
6tum behaviour prevailing over the classical one. We envisage markets
cleared by quantum algorithms (computers) [11, 21].
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8

Quantum-Like Approach to Financial Risk: Quantum Anthropic Principle

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