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arXiv:0907.2531v1 [q-fin.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

E-mail: 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 so-called observables of the

system; (c) finding conserved quantities; (d) compute transition probabilities.

For these reasons we have suggested in [1, 2, 3] an operator-valued 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 self-adjoint 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 l-th 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 self-adjoint. 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 l-th mode. The operatorˆ N counts the total number of bosons. Moreover,

the operator aldestroys a boson in the l-th 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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