Derivation of one-dimensional hydrodynamic

model for stock price evolution

C. Vamo¸s

N. Suciu

W. Blaj

”T. Popoviciu” Institute of Numerical A nalysis, Romanian Academy,

P.O.Box 68, 3400 Cluj-Napoca 1, Romania

Globinvest SA Investment Company, Eroilor 2, 3400 Cluj-Napoca, Romania

Abstract

It was proved that balance equations for systems with corpuscular structure can

be derived if a kinematic description by piece-wise analytic functions is available

[4]. This article presents a rigorous derivation of an one-dimensional hydrodynamic

mode l f or the stock price evolution. The kinematic desc ription is given by a set of

time functions describing the evolution of the stock price.

PACS : 05.90.+m, 47.90.+a.

Key words:

Econophysics; Statistical mechanics; Hydrodynamics

1Introduction

The movement of the molecules of a ﬂuid as an ensemble is described b y

macroscopic continuous ﬁelds (e.g. the density, velocity or temperature of a

ﬂuid) with a deterministic evolution given by hydrodynamic equations. Thus,

even if, on a microscopic scale, the motion of each molecule is very complex,

at a macroscopic scale the molecules ensemble has a deterministic common

motion. In comparison, a ﬁnancial market contains a set of assets with a

v e ry complex price evolution and one can assume that a hydrodynamic-type

descriptionisalsopossibleinthiscase.Suchanapproachcanbefoundin[1].

The derivation of a hydrodynamic description for a ﬁnancial market cannot

be obtained by means of the usual methods of the nonequilibrium statisti-

cal physics. The derivation of the hydrodynamic equations has, as a starting

point, the Hamiltonian of all the molecules [2,3]. From the Hamilton equa-

tion through the BBGKY chain one derives the Boltzmann e quation, and the

hydrodynamic equations are obtained by averaging with respect to velocity.

Preprint submitted to Elsevier Preprin t 29 June 2000

This approach cannot be applied for a ﬁnancial market since the evolution of

the asset price is not described by a Hamiltonian. That is, for the evolution

of the asset price we have no dynamical law, the only available information

being a kinematic one by means of time functions.

However, as shown in [4,5], the hydrodynamic t ype description can be obtained

for any corpuscular system with a kinematic description. The continuous ﬁelds

are obtained as space-time coarse-grained averages, and they satisfy the hy-

drodynamic equations. This method can be used for the molecules of a ﬂuid,

as well as for the asset price. If the number of the particles is very large and

they form a thermodynamic system satisfying local equilibrium, then the usual

hydrodynamic description is obtained. But the coarse-grained averages can be

calculated for an arbitrary number of particles (even for a single p article) and

they give a hydrodynamic description even for mechanical systems having no

statistical properties characteristic to a thermodynamic system [6].

In the following we derive the one-dimensional h ydrodynamic model of a ﬁ-

nancial market. Then we exemplify this method for the price of a single stock

so that the signiﬁcance of the various continuous ﬁelds could be better under-

stood. At the end, we discuss the possible applications of this new method in

ﬁnance.

2 Hydrodynamic model for ﬁnancial markets

Consider a ﬁnancial market formed by N assets. We study the ev olution of

this system during the temporal interval I =[0,T]. The position of an asset

is given by a quantity directly related to its price (for example, the price

itself, the logarithm of the price or the normalized value of the price) and it is

denoted by the coordinate x. We assume that the system kinematics is known,

i.e. the position of each asset i ≤ N is a given function of time x

: I → R ,

and its temporal change (“velocity”) is ξ

= dx

/dt : I → R.

Usually the price of an asset is not available as an analytic function but as

values given at equal temporal intervals ∆t. Then the function x

(t)must

approximate the discrete sequence of the price values, for example b y spline

functions. In the following, we assume that the position x

(t) has a linear

variation bet ween the moments when the price has well deﬁned values. Its

derivative ξ

(t)hasaconstantvalueexceptforthemomentst = n ∆t when it

has discon tinuous variations.

Let ϕ

(t), t ∈ I, be the real function of time describing the variation of an

arbitrary quantity ϕ attributed to the i-th asset. In the following, ϕ

will

represent a constant unit function (1) and the price ch ange ( ξ

). Since the

only variations of the velocity ξ

are the jumps from one constant value to

another, the temporal derivative of ϕ

identically vanishes ˙ϕ

≡ 0almost

everywhere. Consider two real parameters τ and a with 0 < τ <T/2and

a>0, and deﬁne the function

hϕi(x, t)=

4 τ a

i=1

t+τ

t−τ

(x, t

) dt

, (1)

where

(x, t)=ϕ

(t) H(a − |x

(t) − x|) , (2)

and H is the left continuous Heaviside function. A nonvanishing contribution

to hϕi is only due to assets lying in the spatial in terval (x − a, x + a) during

the temporal interval (t−τ,t+τ ). Therefore, hϕi(x, t) characterizes the mean

distribution of ϕ about the point x and the time t . It is a coarse-grained

average over the price and time intervals deﬁned by a and τ , i.e. the density

of ϕ .Obviously, hϕi also depends on the parameters a and τ , but we do

not write explicitly this dependence. The average hϕi is non vanishing only if

the integral interval in Eq. (1) is contained in I , i.e. t ∈ (τ,T − τ ).

For a given x , the integrand (2) is a con tinuous function, except at a ﬁnite

number of poin ts where it has discontinuities of jump type. Hence G

is Rie-

mann integrable and the partial derivative with respect to t of hϕi is

∂

hϕi =

4 τ a

i=1

[ G

(x, t + τ ) − G

(x, t − τ )]. (3)

The function hϕi depends on x through the instants u when the i-th asset

enters or leaves the interval (x − a, x + a). These instants are given by the

zeros of the equations

(u) − x ± a =0,

and using the implicit function theorem we obtain du / dx =1/ ξ

(u). If

u ∈ (t − τ,t+ τ ), then u occurs as integration limit in (1) and the derivative

of hϕi with respect to x is

∂

hϕi =

4 τ a

i=1





u∈U

(u)

−

u∈U

(u)





, (4)

where U

) is the set containing the instants when the i-th asset leaves

(enters) the interval (x − a, x + a) during the interval (t − τ,t+ τ ). One can

prove that the partial derivatives (3) and (4) are almost eve rywhere continuous

[4].

Relation (3) show s that ∂

hϕi is related to the change of G

from t − τ to

t + τ.Since ˙ϕ

≡ 0and

H ≡ 0 almost everywhere, the changes of G

are

only jumps. When the i-th asset enters (leaves) the interval (x − a, x + a), the

change of G

is +ϕ

(u) (respectiv ely −ϕ

(u) ). Comparing with (4), the

corresponding part of ∂

hϕi is equal to −∂

hϕξi. The change of G

due to

occurs at t = n ∆t, when the assets price has a discontinuous variation.

The discontinuous part of ∂

hϕi is

ϕ =

4 τ a

i=1

T/∆t

n=0

H(τ − |n∆t − t|)[G

(x, n∆t − 0) − G

(x, n∆t +0)],

(5)

where G

(x, n∆t+0) is the limit from above, and G

(x, n∆t−0) from below.

We deduce that the relation

∂

hϕi + ∂

hϕξi = δ

ϕ (6)

is always true. In the following we show that this identity is the general form

of the balance equations for the corpuscular system considered.

First we apply the iden tity (6) to the number of assets, i.e. ϕ

≡ 1. The price-

time average (1) becomes the asset n umber density or the concen tration c =

h1i .Themeanvelocityﬁeld v is deﬁned by hξi = cv if c 6=0 andiszero

otherwise. The term δ

ϕ deﬁned by (5) vanishes, because the discontinuous

variations do not imply a variation of the number of assets. Then relation (6)

becomes

∂

c + ∂

(cv)=0, (7)

which is the continuity equation.

For velo city we choose ϕ

= ξ

and get hϕi = cv. The second term in

the left-hand side of (6) can be written as the sum of two terms, namely

h ξ

i = cv

+ h(ξ−v)

i.Theﬁrst term is due to the average motion of the assets

in price space and the second one can be interpreted as the ”microscopic” ﬂux

of velocity. In this case (6) represents the balance equation of velocity

∂

(cv)+∂

(cv

)+∂

(c θ)=δ

ξ. (8)

Here, we have introduced the kinetic temperature θ = h(ξ − v)

i /c (when

c 6= 0 ) which is the continuous ﬁeld related to the volatility. As it follows from

the deﬁnition (6), the term δ

ξ is the analogue of the exterior force being a

measure of the causes which ha ve induced the discontinuous variations of the

velocity ξ

3 Hydrodynamic description of a single stoc k price

To discuss the meaning of the continuous ﬁelds deﬁned in the previous section,

we presen t the hydrodynamic model related to a single asset. We use the daily

close price of the stocks issued by the Aluminium Compan y of America for

100 days beginning with the 27-th of December 1979 (Fig. 1). In order to

maketheresultmoreintuitiveweuseascoordinatex the price and not its

logarithm. The value of the coarse-grained average hϕi for an arbitrary time t

and position x is giv en by the price trajectory within the rectangle in Fig. 1.

Fig. 1. Stock price recording and a veraging domain.

To ﬁnd the meaning of the concentration c = h1i, ﬁrst w e discuss the general

case of N assets. If t ∈ (τ,T − τ ), then from Eq. (1) it follows

+∞

−∞

c(x, t) dx = N.

Hence c describes the distribution of the N assets on the coordinate axis. As a

veriﬁcation w e consider the particular case τ ¿ ∆t, a ¿ 1andτ <a.Thenfor

t = n ∆t, the concentration is nonvanishing only in the neighborhood of the

points x = x

(n ∆t) and it coincides with the discrete repartition of the assets

price at time n ∆t. When the values of the parameters τ and a increase, the

repartition of the assets price becomes more smeared and the concen tration c

approaches a continuous ﬁeld. The parameters a and τ characterize the price

time scale of the continuous ﬁelds. For a single stock, the concentration c(x, t)

is proportional to the time interval when the stock price is con tained within

the rectangle in Fig.1.

Figure 2 presents the support of the concentration for τ =5danda =0.5,

that is the points (t, x)wherec 6= 0. The stock mean price at time t is given

+∞

−∞

xc(x, t) dx and corresponds to the circles in Fig. 2. It is easy to verify

that when a ¿ 1, this mean price becomes the usual moving average on the

interval (t − τ,t+ τ ).

Fig. 2. The support of the concentration ﬁeld is a price-time band that follows the

trend of the mean (

◦

) and the recorded (

)prices.

The v elocity ﬁeld v = hξi/c is non vanishing only if c 6= 0 , thus it has the same

support as the concen tration ﬁeld.InFig.3wepresenttheregionswherethe

velocity ﬁeldhasthesamesign.Onecanobservethattherearesituationswhen

the velocity has the same sign for all the prices, whereas in other situations

diﬀerent price trends occur at diﬀerent values of the price.

Fig. 3. The support of the velocity ﬁeld and the areas with rising (grey) and lowering

(black) prices.

In Figs. 4 and 5 the four continuous ﬁelds in equations (7) and (8) are presented

for t =27andt = 30 (corresponding to sections ’a’ and ’b’ in Figs. 2 and 3).

Fig. 4. Hydrodynamic ﬁelds at t=27 da ys: concentration (¦), velocity (+), temper-

ature (4)andforce(×)

Fig. 5. Hydrodynamic ﬁelds at t=30 days.

One can notice how, under the inﬂuence of a nonuniform velocity, the concen-

tration develops a complex shape with two maxima. The temperature is not

constant having a rather complex variation. As we expect, the exterior force

is positive (negative) in the inferior (superior) part of the ”ﬂuid”, so that it

retains the stock price in a bounded domain.

4Conclusion

As recen tly sho wn [7,8], only a small part of the asset price correlation can be

attributed to the market evolution, the major part being noise. The method

presented in this article could render evident the market evolution in a similar

way with the hydrodynamic description of the macroscopic ﬂow of a ﬂuid

obtained from the disorderly molecular motion. Moreo ver, the coarse-grained

averages can be calculated from the a vailable information of the asset prices.

In this article we derived only the balance equations for concentration and

velocity. The same method can be used to obtain the equation for the kinetic

temperature.

Relations (6)-(8) are either identities or equations, according to the available

information on the microscopic structure. If the motion of eac h particle is ex-

plicitly known, then (6)-(8) are simple iden tities containing only known func-

tions. Otherwise, they become the balance equations for the coarse-grained

averages hϕi, which now are unknown functions. In general, the number of

continuous ﬁelds is greater than the number of balance equations and to obtain

a solvable problem, additional relations are needed (e.g. the expression of the

stress tensor for a speciﬁed material). In continuum mechanics, such relations

are referred to as ”constitutive relations” [9] and represen t the second part

of the hydrodynamic description. Thus,thehydrodynamicequationsalways

consist of balance equations and constitutive relations. The constitutive rela-

tions describe the macroscopic properties of the material and are related to

the speciﬁc microscopic structure of the corpuscular system. If from the study

of a ﬁnancial mark et one could form ulate such constitutiv e relations, then the

hydrodynamic equations might allow a forecast of the market evolution.

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