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arXiv:0907.2531v1 [qfin.GN] 15 Jul 2009
A quantum statistical approach to simplified stock
markets
F. Bagarello
Dipartimento di Metodi e Modelli Matematici, Facolt` a di Ingegneria,
Universit` a di Palermo, I  90128 Palermo, Italy
Email: bagarell@unipa.it
home page: www.unipa.it\˜bagarell
Abstract
We use standard perturbation techniques originally formulated in quantum (statis
tical) mechanics in the analysis of a toy model of a stock market which is given in
terms of bosonic operators. In particular we discuss the probability of transition
from a given value of the portfolio of a certain trader to a different one. This com
putation can also be carried out using some kind of Feynman graphs adapted to the
present context.
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I Introduction and motivations
In some recent papers, [1, 2, 3], we have discussed why and how a quantum mechanical
framework, and in particular operator algebras and the number representation, can be
used in the analysis of some simplified models of stock markets. These models are just
prototypes of real stock markets because, among the other simplifications, we are not
considering financial derivatives. For this reason our interest looks different from that
widely discussed in [4], even if the general settings appear to be very close (and very close
also to the framework used in [5]). The main reason for using operator algebras in the
analysis of these simplified closed stock markets comes from the following considerations:
in the closed market we have in mind the total amount of cash stays constant. Also, the
total number of shares does not change with time. Moreover, when a trader τ interacts
with a second trader σ, they change money and shares in a discrete fashion: for instance,
τ increments his number of shares of 1 unit while his cash decrements of a certain number
of monetary units (which is the minimum amount of cash existing in the market: 1 cent
of dollar, for example), which is exactly the price of the share. Of course, for the trader
σ the situation is just reversed. So we have at least two quantities, the cash and the
number of shares, which change discontinuously as multiples of two fixed quantities. In
[1, 2, 3] we also have two other quantities defining our simplified market: the price of the
share (in the cited papers the traders can exchange just a single kind of shares!) and the
market supply, i.e. the overall tendency of the market to sell a share. It is clear that also
the price of the share must change discontinuously, and that’s why we have assumed that
also the market supply is labeled by a discrete quantity.
Operator algebras and quantum statistical mechanics provide a very natural settings
for discussing such a system. Indeed they produce a natural way for: (a) describing
quantities which change with discrete steps; (b) obtaining the differential equations for
the relevant variables of the system under consideration, the socalled observables of the
system; (c) finding conserved quantities; (d) compute transition probabilities.
For these reasons we have suggested in [1, 2, 3] an operatorvalued scheme for the de
scription of such a simplified market. Let us see why, neglecting here all the many math
ematical complications arising mainly from the fact that our operators are unbounded,
and limiting our introduction to few important facts in quantum mechanics and second
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quantization which will be used in the next sections. More details can be found, for
instance, in [6, 7] and [8], as well as in [1, 2, 3].
Let H be an Hilbert space and B(H) the set of all the bounded operators on H.
Let S be our (closed) physical system and A the set of all the operators, which may be
unbounded, useful for a complete description of S, which includes the observables of S.
The description of the time evolution of S is driven by a selfadjoint operator H = H†
which is called the hamiltonian of S and which in standard quantum mechanics represents
the energy of S. In the Heisenberg picture the time evolution of an observable X ∈ A is
given by
X(t) = eiHtXe−iHt
(1.1)
or, equivalently, by the solution of the differential equation
dX(t)
dt
= ieiHt[H,X]e−iHt= i[H,X(t)], (1.2)
where [A,B] := AB − BA is the commutator between A and B. The time evolution
defined in this way is usually a one parameter group of automorphisms of A. The wave
function Ψ of S is constant in time.
In the Scr¨ odinger picture the situation is just reversed: an observable X ∈ A does not
evolve in time (but if it has some explicit dependence on t) while the wave function Ψ of
S satisfies the Scr¨ odinger equation i∂Ψ(t)
not depend on time, Ψ(t) = e−iHtΨ.
In our paper a special role is played by the so called canonical commutation relations
(CCR): we say that a set of operators {al, a†
following hold:
[al,a†
[al,an] = [a†
∂t
= H Ψ(t), whose formal solution is, if H does
l,l = 1,2,...,L} satisfy the CCR if the
n] = δln1 1,
l,a†
n] = 0(1.3)
for all l,n = 1,2,...,L. Here 1 1 is the identity operator of B(H). These operators, which
are widely analyzed in any textbook in quantum mechanics, see [6] for instance, are those
which are used to describe L different modes of bosons. From these operators we can
construct ˆ nl= a†
number operator for the lth mode, whileˆ N is the number operator of S.
The Hilbert space of our system is constructed as follows: we introduce the vacuum
of the theory, that is a vector ϕ0which is annihilated by all the operators al: alϕ0= 0
lalandˆ N =?L
l=1ˆ nlwhich are both selfadjoint. In particular ˆ nlis the
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for all l = 1,2,...,L. Then we act on ϕ0with the operators a†
land their powers:
ϕn1,n2,...,nL:=
1
√n1!n2!...nL!(a†
1)n1(a†
2)n2···(a†
L)nLϕ0,(1.4)
nl= 0,1,2,... for all l. These vectors form an orthonormal set and are eigenstates of
both ˆ nlandˆ N: ˆ nlϕn1,n2,...,nL= nlϕn1,n2,...,nLandˆ Nϕn1,n2,...,nL= Nϕn1,n2,...,nL, where N =
?L
interpretation is given: if the L different modes of bosons of S are described by the vector
ϕn1,n2,...,nL, this implies that n1bosons are in the first mode, n2in the second mode, and
so on. The operator ˆ nlacts on ϕn1,n2,...,nLand returns nl, which is exactly the number of
bosons in the lth mode. The operatorˆ N counts the total number of bosons. Moreover,
the operator aldestroys a boson in the lth mode, while a†
mode. This is why aland a†
The Hilbert space H is obtained by taking the closure of the linear span of all the
vectors in (1.4).
l=1nl. Moreover using the CCR we deduce that ˆ nl(alϕn1,n2,...,nL) = (nl−1)(alϕn1,n2,...,nL)
and ˆ nl
?
a†
lϕn1,n2,...,nL
?
= (nl+ 1)(a†
lϕn1,n2,...,nL), for all l. For these reasons the following
lcreates a boson in the same
lare usually called the annihilation and the creation operators.
An operator Z ∈ A is a constant of motion if it commutes with H. Indeed in this case
equation (1.2) implies that˙Z(t) = 0, so that Z(t) = Z for all t.
The vector ϕn1,n2,...,nLin (1.4) defines a vector (or number) state over the algebra A
as
ωn1,n2,...,nL(X) =< ϕn1,n2,...,nL,Xϕn1,n2,...,nL>,(1.5)
where <, > is the scalar product in H. As we have discussed in [1, 2], these states may
be used to project from quantum to classical dynamics and to fix the initial conditions of
the market.
The paper is organized as follows:
In Section II we introduce a new model, slightly different from the one proposed in
[2], and we deduce some of its features and the related equations of motion, working in
the Heisenberg picture. One of the main improvements with respect to [2] is that several
kind of shares (and not just one!) will be considered here.
In Section III we adopt a different point of view, using the Scr¨ odinger picture to deduce
the transition probability from a given initial situation to a final state, corresponding to
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two different values of the portfolios of the various traders. Since the reader might not be
familiar with the tools adopted, we will be rather explicit in the derivation of our results.
Section IV is devoted to the conclusions and to our plans for the future.
II The model and first considerations
Let us consider N different traders τ1, τ2, ..., τN, exchanging L different kind of shares σ1,
σ2, ..., σL. Each trader has a starting amount of cash, which is used during the trading
procedure: the cash of the trader who sells a share increases while the cash of the trader
who buys that share consequently decreases. The absolute value of these variations is
the price of the share at the time in which the transaction takes place. Following our
previous results we start introducing a set of bosonic operators which are listed, together
with their economical meaning, in the following table. We adopt here latin indexes to
label the traders and greek indexes for the shares: j = 1,2,...,N and α = 1,2,...,L.
the operator and..
aj,α
a†
j,α
ˆ nj,α= a†
...its economical meaning
annihilates a share σαin the portfolio of τj
creates a share σαin the portfolio of τj
counts the number of share σαin the portfolio of τj
j,αaj,α
cj
c†
annihilates a monetary unit in the portfolio of τj
creates a monetary unit in the portfolio of τj
counts the number of monetary units in the portfolio of τj
j
ˆkj= c†
jcj
pα
p†
lowers the price of the share σαof one unit of cash
increases the price of the share σαof one unit of cash
gives the value of the share σα
α
ˆPα= p†
αpα
Table 1.– List of operators and of their economical meaning.
These operators are bosonic in the sense that they satisfy the following commutation
rules
[cj,c†
[pα,p†
k] = 1 1δj,k,
β] = 1 1δα,β
[aj,α,a†
k,β] = 1 1δj,kδα,β, (2.1)
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while all the other commutators are zero.
As discussed in the Introduction, in [2, 3] we have also introduced another set of
operators related to the market supply which was used to deduce the dynamics of the
price of the single kind of share considered there. However, the mechanism proposed in
those papers, thought being reasonable, is too naive and gives no insight on the nature
of the market itself. For this reason in [3, 9] we have also considered a different point
of view, leaving open the problem of finding the dynamics of the price and focusing the
attention on the time evolution of the portfolio of a fixed trader. This is the same point of
view which we briefly consider in this section, while we will comment on other possibilities
in the rest of the paper. More precisely, as we have discussed in the Introduction, the
dynamical behavior of our market is driven by a certain hamiltonianˆH. We assume that
ˆH can be written asˆH = H + Hprices, where
H = H0+ λHI, with
H0=?
HI=?
j,αωj,αˆ nj,α+?
i,j
jωjˆkj
i,j,αp(α)
?
a†
i,αaj,αc
ˆPα
ic†
j
ˆPα+ h.c.
?
.
(2.2)
Here h.c. stands for hermitian conjugate, c
and p(α)
values depending on the possibility of τito interact with τjand exchanging a share σα:
for instance p(1)
this does not exclude that, for instance, they could exchange a share σ2, so that p(2)
With this in mind it is natural to put p(α)
Going back to (2.2), we observe that H0is nothing but the standard free hamiltonian
which is used for manybody systems like the ones we are considering here (where the
bodies are nothing but the traders, and the cash). More interesting is the meaning of the
interaction hamiltonian HI. To understand HIwe consider its action on a vector like
ˆPα
i
and c†
j
ˆPαare defined as in [2], and ωj,α, ωj
i,jare positive real numbers. In particular these last coefficients assume different
2,5= 0 if there is no way for τ2and τ5to exchange a share σ1. Notice that
2,5?= 0.
i,i= 0 and p(α)
i,j= p(α)
j,i.
ϕ{nj,α};{kj};{Pα}:=a†
1,1
n1,1···a†
?n11!···nN,L!k1!···kL!P1!···PL!
N,L
nN,Lc†
1
k1···c†
N
kNp†
1
P1···p†
L
PL
ϕ0, (2.3)
where ϕ0is the vacuum of all the annihilation operators involved here, see Section I.
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Because of the CCR we deduce that the action of a single contribution of HI, a†
on ϕ{nj,α};{kj};{Pα}is proportional to another vector ϕ{n′
quantum numbers. In particular nj,α, ni,α, kjand kiare replaced respectively by nj,α−1,
ni,α+ 1, kj+ Pαand ki− Pα(if this is larger or equal than zero, otherwise the vector is
annihilated). This means that τjis selling a share σαto τiand earning money from this
operation. For this reason it is convenient to introduce the following selling and buying
operators:
xj,α:= aj,αc†
j
i,αaj,αc
ˆPα
ic†
j
ˆPα,
j,α};{k′
j};{P′
α}with just 4 different
ˆPα,x†
j,α:= a†
j,αcj
ˆPα
(2.4)
With these definitions and using the properties of the coefficients p(α)
as
i,jwe can rewrite HI
HI= 2
?
i,j,α
p(α)
i,jx†
i,αxj,α⇒ H =
?
j,α
ωj,αˆ nj,α+
?
j
ωjˆkj+ 2λ
?
i,j,α
p(α)
i,jx†
i,αxj,α
(2.5)
In [9] we have discussed the role of Hpricewhich should be used to deduce the time evolution
of the operatorsˆPα, α = 1,2,...,L, and which will not be fixed in these notes. This will
be justified below, after getting the differential equations of motion for our system. Again
in [9] we have also shown thatˆH corresponds to a closed market where the money and
the total number of shares of each type are conserved. Indeed, callingˆ Nα:=?N
ˆK are integrals of motion, as expected. Of course, something different may happen in
realistic markets. For instance, a given company could decide to issue more stocks in the
market or to split existing stocks. The relatedˆ Nα, sayˆ Nα0, is no longer a constant of
motion. HenceˆH should be modified in order to have [ˆH,ˆNα0] ?= 0. For that it is enough
to add inˆH some source or sink contribution, trying to preserve the selfadjointness ofˆH.
However, this is not always the more natural choice. For instance, if we want to include
in the model interactions with the environment (economical, political, social inputs), it
could be more convenient to use nonhermitean operators, like the generators appearing
in the analysis of quantum dynamical semigroups used to describe open systems, [10].
This aspect will not be considered here.
l=1ˆ nl,α
andˆK :=?N
l=1ˆklwe have seen that, for all α, [ˆH,ˆ Nα] = [ˆH,ˆK] = 0. Henceˆ Nαand
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Let us now define the portfolio operator of the trader τlas
ˆΠl(t) =
L
?
α=1
ˆPα(t) ˆ nl,α(t) +ˆkl(t).(2.6)
This is a natural definition, since it is just the sum of the cash and of the total value of
the shares that τlpossesses at time t. Once again, we stress that in our simplified model
there is no room for the financial derivatives.
Using (1.2) and the commutation rules assumed so far we derive the following system
of equations:
dˆ nl,α(t)
dt
dˆkl(t)
dt
dxl,α(t)
dt
= 2iλ?N
= −2iλ?N
+2iλ?N
j=1p(α)
i,j
?
x†
j,α(t)xl,α(t) − x†
αp(α)
l,α(t)xj,α(t)
?
,
j=1
?L
?L
i,jˆPα(t)
?
x†
j,α(t)xl,α(t) − x†
l,α(t)xj,α(t)
?
,
= ixl,α(t)(ωlˆPα(t) − ωl,α)+
j=1
β=1p(β)
l,j[xl,β(t)†(t),xl,α(t)]xj,β(t),
(2.7)
which, together with their adjoints, produce a closed system of differential equation.
Notice that these equations imply that?L
dependent operatorsˆPα(t) with L classical fields Pα(t) which are deduced by empirical
data. This is the reason why we were not interested in fixing HpricesinˆH:ˆPα(t) is replaced
by Pα(t) which are nolonger internal degrees of freedom of the model but, rather than
this, simple external classical fields. Neglecting all the details, which can be found in [9],
we observe that the first non trivial contribution in λ is
α=1ˆPα(t)d
dtˆ nl,α(t) +
d
dtˆkl(t) = 0. This system
is now replaced by a semiclassical approximation which is obtained replacing the time
nl,α(t) := ω{nj,α};{kj};{Pα}(ˆ nl,α(t)) =
= nl,α− 8λ2?N
= kl+ 8λ2?N
where we have defined:
j=1
?
p(α)
l,j
?2
?
˜ Mj,l;αℜ(Θ(2)
j,l;α(t)) =: nl,α+ δnl,α(t),
kl(t) := ω{nj,α};{kj};{Pα}(ˆkl(t)) =
j=1
?L
α=1
p(α)
l,j
?2
˜ Mj,l;αℜ(Θ(3)
j,l;α(t)) =: kl+ δkl(t),
(2.8)
?
˜ Mj,l;α= Mj,l;α− Ml,j;α,
Mj,l;α= nj,αnl,α
(kj+Pα)!
kj!
(kl+Pα)!
kl!
− nj,α(1 + nl,α)(kj+Pα)!
kj!
kl!
(kl−Pα)!,
(2.9)
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and
Θ(3)
Θ(2)
Θ(1)
Θ(0)
j,l;α(t) =?t
j,l;α(t) :=?t
0Pα(t′)Θ(1)
0Θ(1)
0e−iΘ(0)
j,l;α(t′)e−iΘ(0)
j,l;α(t′)e−iΘ(0)
j,l;α(t′)dt′,
j,l;α(t′)dt′,
j,l;α(t) :=?t
j,l;α(t) := (ωj− ωl)?t
j,l;α(t′)dt′,
0Pα(t′)dt′− (ωj,α− ωl,α)t
(2.10)
The time dependence of the portfolio can now be written as
Πl(t) := ω{nj,α};{kj};{Pα}(ˆΠl(t)) = Πl(0) + δΠl(t), (2.11)
with
δΠl(t) =
L
?
α=1
nl,α(Pα(t) − Pα(0)) +
L
?
α=1
Pα(t)δnl,α(t) + δkl(t). (2.12)
It should be emphasized that, even under the approximations we are considering here, we
still find?L
which produces the semiclassical version of (2.7) anymore, and therefore they are not
constant in time. This imposes some strict constraint on the validity of our expansion,
as we have discussed already in [3] and suggests the different approach to the problem
which we will discuss in the next section. Again we refer to [9] for more comments on
these results.
α=1Pα(t) ˙ nl,α(t) +˙kl(t) = 0. However, we are loosing the conservation laws
we have discussed before: bothˆ NlandˆK do not commute with the effective hamiltonian
IIIA time dependent point of view
In this section we will consider a slightly different point of view. Our hamiltonian has no
Hpricecontribution at all, since the price operatorsˆPα, α = 1,...,L, are now replaced from
the very beginning by external classical fields Pα(t), whose time dependence describes, as
an input of the model, the variation of the prices of the shares. Incidentally, this implies
that possible fast changes of the prices are automatically included in the model through
the analytic expressions of the functions Pα(t). Hence the interaction hamiltonian HIin
(2.5) turns out to be a time dependent operator, HI(t). More in details, the hamiltonian
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H = H0+ λHI(t) of the model looks like the one in (2.5) but with the following time
dependent selling and buying operators:
xj,α(t) := aj,αc†
j
Pα(t),x†
j,α(t) := a†
j,αcjPα(t)
(3.1)
The L price functions Pα(t) will be taken piecewise constant, since the price of a share
changes discontinuously: it has a certain value before the transaction and (in general)
a different value after the transaction. This new value does not change until the next
transaction takes place. More in details, we introduce a time step h which we call the
time of transaction, and we divide the interval [0,t[ in subintervals of duration h: [0,t[=
[t0,t1[∪[t0,t1[∪[t1,t2[···[tM−1,tM[, where t0= 0, t1= h, ..., tM−1= (M − 1)h = t − h,
tM= Mh = t. Hence h = t/M. As for the prices, for α = 1,...L we put
Pα(t) =
Pα,0,
Pα,1,
......,
Pα,M−1,
t ∈ [t0,t1[,
t ∈ [t1,t2[,
t ∈ [tM−1,tM[
(3.2)
An orthonormal basis in the Hilbert space of the model H is now the set of vectors defined
as
ϕ{nj,α};{kj}:=a†
?n11!···nN,L!k1!···kL!
where ϕ0is the vacuum of all the annihilation operators involved here. They differ from
the ones in (2.3) since the price operators disappear, of course. To simplify the notation
we introduce a set F = {{nj,α};{kj}} so that the vectors of the basis will be simply
written as ϕF.
The main problem we want to discuss here is the following: suppose that at t = 0 the
market is described by a vector ϕF0. This means that, since F0= {{no
the trader τ1has no
the trader τ2has no
on. We want to compute the probability that at time t the market has moved to the
configuration Ff= {{nf
of σ1, nf
1,1
n1,1···a†
N,L
nN,Lc†
1
k1···c†
N
kN
ϕ0,(3.3)
j,α},{ko
j}}, at t = 0
11shares of σ1, no
21shares of σ1, no
12shares of σ2, ..., and ko
22shares of σ2, ..., and ko
1units of cash. Analogously,
2units of cash. And so
j,α},{kf
j}}. This means that, for example, τ1has now nf
12shares of σ2, ..., and kf
11shares
1units of cash.
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Similar problems are very well known in ordinary quantum mechanics: we need to
compute a probability transition from the original state ϕF0to a final state ϕFf, and
therefore we will use here the standard timedependent perturbation scheme for which
we refer to [11]. The main difference with respect to what we have done in the previous
section is the use of the Schr¨ odinger rather than the Heisenberg picture. Hence the market
is described by a timedependent wave function Ψ(t) which, for t = 0, reduces to ϕF0:
Ψ(0) = ϕF0. The transition probability we are looking for is
PF0→Ff(t) :=??< ϕFf,Ψ(t) >??2
(3.4)
The computation of PF0→Ff(t) is a standard exercise, [11]. In order to make the paper
accessible also to those people who are not familiar with quantum mechanics, we give
here the main steps of its derivation.
Since the set of the vectors ϕF is an orthonormal basis in H the wave function Ψ(t)
can be written as
Ψ(t) =
?
where EFis the eigenvalue of H0defined as
F
cF(t)e−iEFtϕF,(3.5)
H0ϕF= EFϕF,⇒EF=
?
j,α
ωj,αnj,α+
?
j
ωjkj. (3.6)
This is a consequence of the fact that ϕFin (3.3) is an eigenstate of H0in (2.2). Using the
quantum mechanical terminology, we sometimes call EF the free energy of ϕF. Putting
(3.5) in (3.4), and recalling that < ϕF,ϕG>= δF,G, we have
PF0→Ff(t) :=??cFf(t)??2
(3.7)
The answer to our original question is therefore given if we are able to compute cFf(t) in
(3.5). Due to the analytic form of our hamiltonian, this cannot be done exactly. However,
several possible perturbation schemes exist in the literature. We will adopt here a simple
perturbation expansion in the interaction parameter λ appearing in the hamiltonian (2.5).
In other words, we look for the coefficients in (3.5) having the form
cF(t) = c(0)
F(t) + λc(1)
F(t) + λ2c(2)
F(t) + ···(3.8)
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Each c(j)
that Ψ(t) satisfies the Schr¨ odinger equation i∂Ψ(t)
equation and using the orthonormality of the vectors ϕF’s, we find that
F(t) satisfies a differential equation which can be deduced as follows: first we recall
∂t
= H(t)Ψ(t). Replacing (3.5) in this
˙ cF′(t) = −iλ
?
F
cF(t)ei(EF′−EF)t< ϕF′,HI(t)ϕF> (3.9)
Replacing now (3.8) in (3.9) we find the following infinite set of differential equations,
which we can solve, in principle, up to the desired order in λ:
˙ c(0)
˙ c(1)
˙ c(2)
.........,
F′(t) = 0,
F′(t) = −i?
Fc(0)
Fc(1)
F(t)ei(EF′−EF)t< ϕF′,HI(t)ϕF>,
F(t)ei(EF′−EF)t< ϕF′,HI(t)ϕF>,
F′(t) = −i?
(3.10)
The first equation, together with the assumed initial condition, gives c(0)
δF′,F0. When we replace this solution in the differential equation for c(1)
again that Ψ(0) = ϕF0,
F′(t) = c(0)
F′(t) we get, recalling
F′(0) =
c(1)
F′(t) = −i
?t
0
ei(EF′−EF0)t1< ϕF′,HI(t1)ϕF0> dt1
(3.11)
Using this in (3.10) we further get
c(2)
F′(t) = (−i)2?
where we have introduced the shorthand notation
F
?t
0
??t2
0
ei(EF−EF0)t1hF,F0(t1)dt1
?
ei(EF′−EF)t2hF′,F(t2)dt2, (3.12)
hF,G(t) :=< ϕF,HI(t)ϕG> (3.13)
III.1 First order corrections
We continue our analysis computing PF0→Ff(t) in (3.7) up to the first order corrections
in λ and assuming that Ffis different from F0. Hence we have
????
12
PF0→Ff(t) =
???c(1)
Ff(t)
???
2
= λ2
?t
0
ei(EFf−EF0)t1hFf,F0(t1)dt1
????
2
(3.14)
Page 13
Using (3.2) and introducing δE = EFf− EF0, after some algebra we get
PF0→Ff(t) = λ2
?δE h/2
δE/2
?2?????
M−1
?
k=0
hFf,F0(tk)eitkδE
?????
2
(3.15)
The computation of the matrix elements hFf,F0(tk) is easily performed. Indeed, because
of some standard properties of the bosonic operators, we find that
a†
i,αaj,αcPα,k
i
c†
jPα,kϕF0= Γ(k)
i,j;αϕF(i,j,α)
0,k
where
Γ(k)
i,j;α:=
?
(ko
j+ Pα,k)!
ko
j!
ko
i!
(ko
i− Pα,k)!no
j,α(1 + no
i,α) (3.16)
and F(i,j,α)
no
assuming that ko
have not enough money to buy a share σα!
We find that
0,k
differs from F0only for the following replacements: no
j→ ko
i≥ Pα,k, for all i, k and α. This is because otherwise the trader τiwould
j,α→ no
j,α− 1, no
i,α→
i,α+1, ko
j+Pα,k, ko
i→ ko
i−Pα,k. Notice that in our computations we are implicitly
hFf,F0(tk) = 2
?
i,j,α
p(α)
i,jΓ(k)
i,j;α< ϕFf,ϕF(i,j,α)
0,k
> (3.17)
Of course, due to the orthogonality of the vectors ϕF’s, the scalar product < ϕFf,ϕF(i,j,α)
is different from zero (and equal to one) if and only if nf
kf
coincide.
For concreteness sake we now consider two simple situations: in the first example
below we just assume that the prices of the various shares do not change with t. In the
second example we consider the case in which only few changes occur, and we take M = 3.
0,k
i,α+ 1,
>
j,α= no
j,α− 1, nf
i,α= no
i= ko
i− Pα,k and kf
j= ko
j+ Pα,k, and all the other new and old quantum numbers
Example 1: constant prices
Let us assume that, for all k and for all α, Pα,k = Pα(tk) = Pα. This means that
Γ(k)
0,k
and the related vectors ϕF(i,j,α)
0,k
independent of k. After few computation we get
i,j;α, F(i,j,α)
do not depend on k. Hence hFf,F0(tk) is also
PF0→Ff(t) = λ2
?sin(δEt/2)
δE/2
?2??hFf,F0(0)??2
(3.18)
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Page 14
to which corresponds a transition probability per unit of time
pF0→Ff= lim
t,∞
1
tPF0→Ff(t) = 2πλ2δ(EFf− EF0)??hFf,F0(0)??2, (3.19)
which shows that, in this limit, a transition between two states is possible only if the two
states have the same free energy. The presence of hFf,F0(0) in the final result shows, using
our previous remark, that at the order we are considering here a transition is possible only
if ϕF0does not differ from ϕFffor more than one share in two of the nj,α’s and for more
than Pαin two of the kj’s. All the other transitions are forbidden.
Example 2: few changes in the price
Let us now fix M = 3. Formula (3.15) can be rewritten as
PF0→Ff(t) = 4λ2
?sin(δEh/2)
δE/2
?2

?
i,j,α
p(α)
i,j(Γ(0)
i,j;α< ϕFf,ϕF(i,j,α)
0,0
> +
+Γ(1)
i,j;α< ϕFf,ϕF(i,j,α)
0,1
> eihδE+ Γ(2)
i,j;α< ϕFf,ϕF(i,j,α)
0,2
> e2ihδE)2
(3.20)
The meaning of this formula is not very different from the one discussed in the previous
example: if we restrict to this order of approximation, the only possibilities for a transition
F0→ Ff to occur are those already discussed in Example 1 above. We will see that, in
order to get something different, we need to go to higher orders in λ. In other words,
even if the prices depend on time, not new relevant features appear in the transition
probabilities.
As for the validity of the approximation, let us consider the easiest situation: we have
constant prices (Example 1) and, moreover, in the summation in (3.17) only one contri
bution survives, the one with i0,j0and α0. Then we have hFf,F0(t0) = 2p(α0)
ϕFf,ϕF(i0,j0,α0)
0
PF0→Ff(t) exceeds one, it is necessary to have small λ, small p(α)
possible). However, due to the analytic expression for Γi0,j0;α0, see (3.16), we must pay at
tention to the values of the nj,α’s and of the kj’s, since, if they are large, the approximation
may likely break down very soon in t.
i0,j0Γi0,j0;α0<
>. Because of (3.18), and since our approximation becomes meaningless if
i,jand large δE (if this is
Let us now finally see what can be said about the portfolio of the trader τl. Since
we know the initial state of the system, then we know the value of its portfolio at time
14
Page 15
zero: extending our original definition in (2.6) we haveˆΠl(0) =?L
the portfolios of all the traders, clearly! Formula (3.15) gives the transition probability
from ϕF0to ϕFf. This probability is just a single contribution in the computation of the
transition probability from a givenˆΠl(0) to a certainˆΠl(t), since the same value of the
portfolio can be recovered at time t for very many different states ϕFf: all the sets G with
the same nf
all these sets, we just have to sum up over all these different contributions:
α=1Pα(0) ˆ nl,α(0)+ˆkl(0).
As a matter of fact, the knowledge of ϕF0implies that we know the time zero value of
l,αand kf
lgive rise to the same portfolio for τl. Hence, if we call˜F the set of
PˆΠo
l→ˆΠf
l(t) =
?
G∈˜ F
PF0→G(t)
III.2Second order corrections
We start considering the easiest situation, i.e. the case of a time independent perturbation
HI: the prices are constant in time. Hence the integrals in formula (3.12) can be easily
computed and the result is the following:
c(2)
Ff(t) =
?
F
hFf,F(0)hF,F0(0)EF,F0,Ff(t), (3.21)
where
EF,F0,Ff(t) =
1
EF− EF0
?
ei(EFf−EF0)t− 1
EFf− EF0
−ei(EFf−EF)t− 1
EFf− EF
?
Recalling definition (3.13), we rewrite equation (3.21) as c(2)
ϕF,HIϕF0> EF,F0,Ff(t) which explicitly shows that up to this order in our perturbation
expansion transitions between states which differ, e.g., for 2 shares are allowed: it is
enough that some intermediate state ϕF differs for (plus) one share from ϕF0and for
(minus) one share from ϕFf.
If the Pα(t)’s depend on time the situation is a bit more complicated but not very
different. Going back to Example 2 above, and considering then a simple (but not trivial)
situation in which the prices of the shares really change, we can perform the computation
and we find
c(2)
{hFf,F(t0)hF,F0(t0)J0(F,F0,Ff;t1)+
Ff(t) =?
F< ϕFf,HIϕF><
Ff(t) = (−i)2?
F
15
Page 16
+hFf,F(t1)(hF,F0(t0)I0(F,F0;t1)I1(Ff,F;t2) + hF,F0(t1)J1(F,F0,Ff;t2))+
+hFf,F(t2)[hF,F0(t0)I0(F,F0;t1) + hF,F0(t1)I1(F,F0;t2))I2(Ff,F;t3)+
+ hF,F0(t2)J2(F,F0,Ff;t3)]},
where we have introduced the functions
(3.22)
Ij(F,G;t) :=
?t
tj
ei(EF−EG)t′dt′=
1
i(EF− EG)
?ei(EF−EG)t− ei(EF−EG)tj?
and
Jj(F,G,L;t) :=
?t
tj
Ij(F,G;t′)ei(EL−EF)t′dt′.
Needless to say, this last integral could be explicitly computed but we will not show here
the explicit result, since it will not be used.
The same comments as above about the possibility of having a non zero transition
probability can be repeated also for equation (3.22): it is enough that the timedepending
perturbation connect ϕF0to ϕFfvia some intermediate state ϕF in a single time sub
interval in order to permit a transition. If this never happens in [0,t], then the transition
probability is zero. We can see the problem from a different point of view: if some
transition takes place in the interval [0,t[, there must be another state, ϕF′
from ϕFf, such that the transition probability PF0→F′
f, different
f(t) is non zero.
III.3Feynman graphs
Following [11] we now try to connect the analytic expression of a given approximation
of cFf(t) with some kind of Feynman graph in such a way that the higher orders could
be easily written considering a certain set of rules which we will obviously call Feynman
rules.
The starting point is given by the expressions (3.11) and (3.12) for c(1)
which is convenient to rewrite in the following form:
Ff(t) and c(2)
Ff(t),
c(1)
Ff(t) = −i
?t
0
eiEFft1< ϕFf,HI(t1)ϕF0> e−iEF0t1dt1
(3.23)
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Page 17
and
c(2)
Ff(t) = (−i)2?
F
?t
0
dt2
?t2
0
dt1eiEFft2< ϕFf,HI(t2)ϕF> e−iEFt2×
× eiEFt1< ϕF,HI(t1)ϕF0> e−iEF0t1
Ff(t) is given in the figure below: at t = t0 the state
of the system is ϕF0, which evolves freely (and therefore e−iEF0t1ϕF0appears) until the
interaction occurs, at t = t1. After the interaction the system is moved to the state
ϕFf, which evolves freely (and therefore e−iEFft1ϕFfappears, and the different sign in
(3.23) is due to the antilinearity of the scalar product in the first variable.). The free
evolutions are the upward arrows, while the interaction between the initial and the final
states, < ϕFf,HI(t1)ϕF0>, is described by an horizontal wavy line. Obviously, since the
interaction may occur at any time between 0 and t, we have to integrate on all these
possible t1’s and multiply the result for −i.
6
t
(3.24)
A graphical way to describe c(1)
t0
t1
A
A
A
A
A
A K A
A
?
? ?
?
?
?
?
?
?
ϕF0
ϕFf
< ϕFf,HI(t1)ϕF0>
Figure 1: graphical expression for c(1)
Ff(t)
In a similar way we can construct the Feynman graph for c(2)
example c(2)
occur, the first at t = t1and the second at t = t2:
Ff(t), c(3)
Ff(t) and so on. For
Ff(t) can be deduced by a graph like the one in Figure 2, where two interactions
17
Page 18
6
t0
t1
t2
t
A
A
A
A
A
A K A
A
?
? ?
?
?
A
A K
A
A
A
A
ϕF0
ϕFf
ϕF
< ϕF,HI(t1)ϕF0>
< ϕFf,HI(t2)ϕF>
Figure 2: graphical expression for c(2)
Ff(t)
Because of the double interaction we have to integrate twice the result, since t1∈ (0,t2)
and t2∈ (0,t). For the same reason we have to sum over all the possible intermediate
states, ϕF. The free time evolution for the various free fields also appear, as well as a
(−i)2. Following these same rules we could also give at least a formal expression for the
other coefficients, as c(3)
for instance, a double sum on the intermediate states, allowing in this way a transition
from a state with, say, no
integral and a factor (−i)3.
Ff(t), c(4)
Ff(t) and so on: the third order correction c(3)
Ff(t) contains,
i,αshares to a state with nf
i,α= no
i,α+ 3 shares, a triple time
IV Conclusions
We have shown how quantum statistical dynamics can be adopted to construct and an
alyze simplified models of closed stock markets, where no derivatives are considered. In
particular we have shown that both the Schr¨ odinger and the Heisenberg pictures can be
successfully used in the perturbative analysis of the time evolution of the market: how
ever, the approximations considered in the Heisenberg picture are not completely under
control because of the many assumptions adopted as it happens using the Schr¨ odinger
wave function of the market and looking for transition probabilities.
We have also shown that the Feynman graphs technique can be adopted for the pertur
18
Page 19
bative analysis of our market, and some simple rules to write down the integral analytic
expression for the transition probabilities have been deduced.
Of course a more detailed analysis of the model should be performed, in particular
looking for those adjustments which can make more realistic the market we have de
scribed so far. These should include for instance source and sink effects to mimikate non
conservation of the number of shares, short terms exchanges, financial derivatives and so
on. Incidentally, we believe that the hamiltonian H in (2.5) and the related differential
equations in (2.7), could be interesting by themselves, hopefully in the description of some
(realistic?) manybody system. We hope to consider this aspect in the next future.
Acknowledgements
This work has been financially supported in part by M.U.R.S.T., within the project
Problemi Matematici Non Lineari di Propagazione e Stabilit` a nei Modelli del Continuo,
coordinated by Prof. T. Ruggeri.
References
[1] F. Bagarello, An operatorial approach to stock markets, J. Phys. A, 39, 68236840
(2006)
[2] F. Bagarello, Stock Markets and Quantum Dynamics: A Second Quantized Descrip
tion, Physica A, 386, 283302 (2007)
[3] F. Bagarello, Simplified Stock markets and their quantumlike dynamics, Rep. on
Math. Phys., in press
[4] B.E . Baaquie, Quantum Finance, Cambridge University Press (2004)
[5] M. Schaden, A Quantum Approach to Stock Price Fluctuations, Physica A 316, 511,
(2002)
[6] E. Merzbacher, Quantum Mechanics, Wiley, New York (1970),
19
Page 20
[7] M. Reed and B. Simon, Methods of Modern Mathematical Physics, I, Academic Press,
New York (1980)
[8] O. Bratteli and D. W. Robinson, Operator Algebras and Quantum Statistical Me
chanics, I, Springer Verlag, New York, (1979)
[9] F. Bagarello Heisenberg picture in the description of simplified stock markets, Bollet
tino del Dipartimento di Metodi e Modelli Matematici, in press
[10] R. Alicki and M. Fannes, Quantum Dynamical Systems, Oxford University Press,
Oxford, (2001)
[11] A. Messiah, Quantum mechanics, vol. 2, North Holland Publishing Company, Ams
terdam, (1962)
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