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Unformatted text preview: arXiv:cond-mat/0008103 7 Aug 2000 Power-laws in economy and ﬁnance: some ideas
† Science & Finance, 109-111 rue Victor Hugo, 92532 Levallois cedex, FRANCE;
Service de Physique de l’Etat Condens´, Centre d’´tudes de Saclay,
Orme des Merisiers, 91191 Gif-sur-Yvette cedex, FRANCE August 7, 2000
We discuss several models in order to shed light on the origin of powerlaw distributions and power-law correlations in ﬁnancial time series. From
an empirical point of view, the exponents describing the tails of the price
increments distribution and the decay of the volatility correlations are
rather robust and suggest universality. However, many of the models that
appear naturally (for example, to account for the distribution of wealth)
contain some multiplicative noise, which generically leads to non universal exponents. Recent progress in the empirical study of the volatility
suggests that the volatility results from some sort of multiplicative cascade. A convincing ‘microscopic’ (i.e. trader based) model that explains
this observation is however not yet available. It would be particularly important to understand the relevance of the pseudo-geometric progression
of natural human time scales on the long range nature of the volatility
correlations. 1 Introduction Physicists are often fascinated by power-laws. The reason for this is that complex, collective phenomenon do give rise to power-laws which are universal, that
is to a large degree independent of the microscopic details of the phenomenon.
These power-laws emerge from collective action and transcends individual speciﬁcities. As such, they are unforgeable signatures of a collective mechanism. Examples in the physics literature are numerous. A well known example is that of
phase transitions, where a system evolves from a disordered state to an ordered
state: many observables behave as universal power laws in the vicinity of the
transition point . This is related to an important property of power-laws, 1 namely scale invariance:1 the characteristic length scale of a physical system at
its critical point is inﬁnite, leading to self-similar, scale-free ﬂuctuations. Another example is ﬂuid turbulence, where the statistics of the velocity ﬁeld has
scale invariant properties, to a large extent independent of the nature of the
ﬂuid, of the power injected, etc. .
Power-laws are also often observed in economic and ﬁnancial data [3, 4, 5,
6, 7]. Compared to physics, however, much less eﬀort has been devoted to understand these power-laws in terms of ‘microscopic’ (i.e. agent based) models
and to relate the value of the exponents to generic mechanisms. The aim of
this contribution is to give a short review of diverse power-laws observed in
economy/ﬁnance, and to discuss several simple models (most of them inspired
from physics) which naturally lead to power-laws and could serve as a starting
point for further developments. It should be stressed that none of the models
presented here are intended to be fully realistic and complete, but are of pedagogical interest: they nicely illustrate how and when power-laws can arise. This
paper is furthermore written in the spirit of a conference proceedings, and many
rather speculative ideas are put forward to fuel further discussions. 2
2.1 Empirical power-laws: a short review
Distributional power-laws The oldest and most famous power-law in economy is the Pareto distribution of
wealth . The distribution of individual wealths P (W ) is often described, in
its asymptotic tail, by a power law:
P (W ) µ
W 1+µ W W0 , (1) where µ characterize the decay of the distribution for large W ’s: the smaller the
value of µ, the slower the decay, and the larger the contrast between the richest
and the poorest. More precisely, in a Pareto population of size N , the ratio of
the largest wealth to the typical (e.g. median) wealth grows as N 1/µ . In the
case µ < 1, the average wealth diverges: this corresponds to an economy where
a ﬁnite fraction of the total wealth is in the hands of a few individuals, even
when N → ∞. On the contrary, when µ > 1, the richest individual only holds
a zero fraction of the total wealth (again in the limit N → ∞). Empirically, the
exponent µ is in the range 1 – 2. This Pareto tail also describes the distribution
of income, the size of companies, of pension funds, etc. [8, 9, 10].
In ﬁnancial markets, the distribution of the price increments δxt = x(t ) −
x(t) over diﬀerent time scales t − t is important both for risk control purposes
1 Power-law distributions are scale invariant in the sense that the relative probability to
observe an event of a given size and an event ten times larger is independent of the reference
scale. 2 and for derivative pricing models . The availability of long series of high
frequency data has motivated many empirical studies in the past few years. By
pooling together the statistics of a thousand U.S. stocks, it is possible to study
quite accurately the far tail of the distribution of intra-day price increments δx,
which can be ﬁtted as a power law :
|δx|1+µ P (δx) (2) with µ in found to be close to 3. Similar values have also been reported for
Japanese stocks , German stocks , currencies [14, 15], bond markets ,
and perhaps also the distribution of the (time dependent) daily volatility σ,
deﬁned as an average over high frequency returns  (although other works
report a log-normal distribution [22, 25]).2 This suggests that the value of µ
could be universal. Note however that the value of µ depends somewhat on the
stock and on the period of time studied. For example, the value of µ for the
S&P500 during the years 1991-1995 is found to be close to 5. Furthermore, as the
time lag used for the deﬁnition of δx increases, the eﬀective exponent describing
the tail of the distribution increases as the distribution becomes more and more
Gaussian like [11, 12]. 2.2 Temporal power-laws Actually, the price increment at time t can usefully be thought of as the product
of a sign part t and an amplitude part (or volatility) σt :
δxt = t × σt . (3) The random variable t has short ranged temporal correlations, extending over
a few minutes or so on liquid markets [11, 6]. The volatility, on the contrary,
has very long ranged correlations, which can be ﬁtted as a power-law with a
small exponent ν [18, 19, 14, 20, 21, 22]:
C1 (τ ) = σt σt+τ − σt 2 c1
τν (4) with ν approximately equal to 0.1. This behaviour is, again, seen on many
diﬀerent types of markets, and quantiﬁes the intermittent activity of these markets: volatility tends to cluster in bursts which persist over very diﬀerent time
scales, from minutes to months. A similar power-law behaviour of the temporal
correlation of the volume of transactions (number of trades) is also observed
[23, 17]. This is not surprising, since volatility and volume are strongly correlated. From a empirical point of view, the intermittent nature of the activity in
ﬁnancial markets is similar to the energy dissipation in a turbulent ﬂuid [2, 24].
2 Note that the tails of a log-normal distribution can be ﬁtted (over a restricted interval)
by a power-law. Therefore it is not always easy to distinguish between true power-laws and
eﬀective power-laws. 3 In fact, the distribution of log σ is not far from being Gaussian [22, 25].
Therefore, it is natural to study the temporal correlation of log σ [26, 27]:
C0 (τ ) = log σt log σt+τ − log σt 2 . (5) This correlation is also found to decay very slowly with τ . This decay can be
ﬁtted by a logarithm [26, 27]: C0 (τ ) = λ2 log(T /τ ), with a rather small value of
λ2 0.05. This, together with the assumption that log σ is exactly Gaussian,
leads to a strict multifractal model for the price changes , in the sense that
diﬀerent moments of the price increments scale with diﬀerent powers of time
[24, 28, 29]. Within this model, one ﬁnds :
|x(t + τ ) − x(t)|q ∝ τ ζq ζq = q
1 − λ2 (q − 2) ,
2 (6) for τ
T and qλ2 < 1 (for higher values of q , the corresponding moment is
divergent). It is easy to show that in this model, the exponent ν deﬁned by Eq.
(4) is equal to λ2 . On the other hand, one can also ﬁt C0 (τ ) by a power-law
with a small exponent, in which case the model would only be approximately
multifractal, in the sense that the quantity |x(t + τ ) − x(t)|q is the sum of
diﬀerent powers of τ , which can also be ﬁtted by an eﬀective, q -dependent
exponent ζq < q/2 . 3
3.1 Simple models of wealth distribution
A model with trading and speculative investment As a simple dynamical model of economy, one can consider a stochastic equation
for the wealth Wi (t) of the ith agent at time t, that takes into account the
exchange of wealth between individuals through trading, and is consistent with
the basic symmetry of the problem under a change of monetary units. Since
the unit of money is arbitrary, one indeed expects that the equation governing
the evolution of wealth should be invariant when all Wi ’s are multiplied by a
common (arbitrary) factor. A rather general class of equation which has this
property is the following [31, 32]:
= ηi(t)Wi +
dt Jij Wj −
j (=i) JjiWi , (7) j (=i) where ηi (t) is a Gaussian random variable of mean m and variance 2σ 2 , which
describes the spontaneous growth or decrease of wealth due to investment in
stock markets, housing, etc. The term involving the (asymmetric) matrix Jij
describes the amount of money that agent j spends buying the production of
agent i. We assume that this production is consumable, and therefore must not 4 be counted as part of the wealth of i once it is bought. The equation (7) is
obviously invariant under the scale transformation Wi → λWi .
The simplest trading network one can think of is when all agents exchange
with all others at the same rate, i.e Jij ≡ J/N for all i = j . Here, N is the total
number of agents, and the scaling J/N is needed to make the limit N → ∞ well
deﬁned. In this case, the equation for Wi (t) becomes:
= ηi(t)Wi + J (W − Wi ),
dt (8) where W = N −1 i Wi is the average overall wealth. This is called, in the
physics language, a ‘mean-ﬁeld’ model since all agents feel the very same inﬂuence of their environment. It is useful to rewrite eq. (8) in terms of the
normalised wealths wi ≡ Wi /W . This leads to:
= (ηi (t) − m − σ2 )wi + J (1 − wi ),
dt (9) to which one can associate the following Fokker-Planck equation for the evolution of the density of wealth P (w, t):
∂ [J (w − 1) + σ2 w]P
∂w (10) The equilibrium, long time solution of this equation is easily shown to be:
µ−1 (µ − 1)µ e− w
Peq (w) =
Γ[µ] w1+µ µ≡1+ J
σ2 (11) Therefore, one ﬁnds in this model that the distribution of wealth indeed
exhibits a Pareto power-law tail for large w’s. Interestingly, however, the value
of the exponent µ is not universal, and depends on the parameter of the model
(J and σ2 , but not on the average growth rate m). In agreement with intuition,
the exponent µ grows (corresponding to a narrower distribution), when exchange
between agents is more active (i.e. when J increases), and also when the success
in individual investment strategies is more narrowly distributed (i.e. when σ 2
decreases). In this model, the exponent µ is always found to be larger than
one. This means that the wealth is not too unevenly distributed within the
Let us now describe more realistic trading network, where the number of
economic neighbours to a given individual is ﬁnite. We will assume that the
matrix Jij is still symmetrical, and is either equal to J (if i and j trade), or equal
to 0. A reasonable assumption is that the graph describing the connectivity of
the population is completely random, i.e. that two points are neighbours with
probability c/N and disconnected with probability 1 − c/N . In such a graph, the
average number of neighbours is equal to c. We have performed some numerical
5 simulations of Eq. (7) for c = 4 and have found that the wealth distribution
still has a power-law tail, with an exponent µ which only depends on the ratio
J/σ2 . However, one ﬁnds that the exponent µ can now be smaller than one
for suﬃciently small values of J/σ2 . In this model, one therefore expects
‘wealth condensation’ when the exchange rate is too small, in the sense that a
ﬁnite fraction of the total wealth is held by only a few individuals.
Although not very realistic, one could also think that the individuals are
located on the nodes of a d-dimensional hyper-cubic lattice, trading with their
neighbours up to a ﬁnite distance. In this case, one knows that for d > 2 there
exists again a phase transition between a ‘social’ economy where µ > 1 and
a rich dominated phase µ < 1. On the other hand, for d ≤ 2, and for large
populations, one is always in the extreme case where µ → 0 at large times. In
the case d = 1, i.e. operators organized along a chain-like structure (as a simple
model of intermediaries), one can actually compute exactly the distribution of
wealth [33, 34]. One ﬁnds for example that the ratio of the maximum wealth to
the typical (e.g. median) wealth behaves as exp N , where N is the size of the
population, instead of N 1/µ in the case of a Pareto distribution with µ > 0.
The conclusion of the above results is that the distribution of wealth tends
to be very broadly distributed when exchanges are limited, either in amplitude
(i.e. J too small compared to σ2 ) or topologically (a s in the above chain
structure). Favoring exchanges (in particular with distant neighbours) seems to
be an eﬃcient way to reduce inequalities. 3.2 Two related models Let us now interpret Wi as the size of a company. The growth of this company
can take place either from internal growth, depending on its success or failure.
This leads to a term ηi(t)Wi much as above. Another possibility is merging with
another company. If the merging process between two companies is completely
random and takes place at a rate Γ per unit time, then the model is exactly
the same as the one considered by Derrida and Spohn  in the context of
‘directed polymers in random media’, and bears strong similarities with the
model discussed in the previous section. One again ﬁnds that the distribution
of W ’s is a power-law with a non universal exponent, which depends on the
values of Γ and σ2 , and can be smaller than one.3
One can also consider a model where companies grow at a random rate η ,
but may also suddenly die at a rate Γ per unit time, and be replaced by a new
(small) company. There again, one ﬁnds a stationary Pareto distribution, with
a non universal exponent which depends continuously on m, σ and Γ .
3 In the language of disordered systems, this corresponds to the ‘glassy’ phase of the directed
polymer, where the partition function is dominated by a few paths only. 6 4 Simple models for herding and copy-cats We now turn to simple models for thick tails in the distribution of price increments in ﬁnancial markets. An intuitive explanation is herding: if a large
number of agents acting on markets coordinate their action, the price change
is likely to be huge due to a large unbalance between buy and sell orders .
However, this coordination can result from two rather diﬀerent mechanisms.
• One is the feedback of past price changes onto themselves, which we will
discuss in the following section. Since all agents are inﬂuenced by the very
same price changes, this can induce non trivial collective behaviour: for
example, an accidental price drop can trigger large sell orders, which lead
to further downward moves.
• The second is direct inﬂuence between the traders, through exchange of
information that leads to ‘clusters’ of agents sharing the same decision to
buy, sell, or be inactive at any given instant of time. 4.1 Herding and percolation A simple model of how herding aﬀects the price ﬂuctuations was proposed in .
It assumes that the price increment δx depends linearly on the instantaneous
oﬀset between supply and demand [37, 38]. More precisely, if each operator in
the market i wants to buy or sell a certain ﬁxed quantity of the ﬁnancial asset,
one has :4
i where ϕi can take the values − 1, 0 or + 1, depending on whether the operator
i is selling, inactive, or buying, and λ is a measure of the market depth. Note
that the linearity of this relation, even for small arguments, has been questioned
by Zhang . Suppose now that the operators interact among themselves in
an heterogeneous manner: with a small probability c/N (where N is the total
number of operators on the market), two operators i and j are ‘connected’, and
with probability 1 − c/N , they ignore each other. The factor 1/N means that
on average, the number of operator connected to any particular one is equal
to c (the resulting graph is precisely the same as the random trading graph of
Section 3.1). Suppose ﬁnally that if two operators are connected, they come to
agree on the strategy they should follow, i.e. ϕi = ϕj .
It is easy to understand that the population of operators clusters into groups
sharing the same opinion. These clusters are deﬁned such that there exists a
connection between any two operators belonging to this cluster, although the
4 This can alternatively be written for the relative price increment δx/x, which is more
adapted to describe long time scales. On short time scales, however, an additive model is
preferable. See the discussion in . 7 connection can be indirect and follow a certain ‘path’ between operators. These
clusters do not have all the same size, i.e. do not contain the same number of
operators. If the size of cluster C is called S (C ), one can write:
δx = 1
λ S (C )ϕ(C ), (13) C where ϕ(C ) is the common opinion of all operators belonging to C . The statistics of the price increments δx therefore reduces to the statistics of the size of
clusters, a classical problem in percolation theory . One ﬁnds that as long
as c < 1 (less than one ‘neighbour’ on average with whom one can exchange
information), then all S (C )’s are small compared to the total number of traders
N . More precisely, the distribution of cluster sizes takes the following form in
the limit where 1 − c =
P (S ) ∝S 1 1
exp − 2 S
S 5/2 S N. (14) When c = 1 (percolation threshold), the distribution becomes a pure power-law
with an exponent µ = 3/2, and the Central Limit Theorem tells us that in this
case, the distribution of the price increments δx is precisely a pure symmetrical
L´vy distribution of index µ = 3/2  (assuming that ϕ = ± 1 play identical
roles, that is if there is no global bias pushing the price up or down). If c < 1, on
the other hand, one ﬁnds that the L´vy distribution is truncated exponentially,
leading to a larger eﬀective tail exponent µ . If c > 1, a ﬁnite fraction of the
N traders have the same opinion: this leads to a crash. 4.2 Avalanches of opinion changes The above model is interesting but has one major drawback: one has to assume
that the parameter c is smaller than one, but relatively close to one such that
Eq. (14) is valid, and non trivial statistics follows. One should thus explain
why the value of c spontaneously stabilises in the neighbourhood of the critical
value c = 1. Furthermore, this model is purely static, and does not specify how
the above clusters evolve with time. As such, it cannot address the problem of
volatility clustering. Several extensions of this simple model have been proposed
[41, 42], in particular to increase the value of µ from µ = 3/2 to µ ∼ 3 and to
account for volatility clustering.
One particularly interesting model is inspired by the recent work of Dahmen
and Sethna , that describes the behaviour of random magnets in a time dependent magnetic ﬁeld. Transposed to the present problem (as ﬁrst suggested
in ) , this model describes the collective behaviour of a set of traders exchanging information, but having all diﬀerent a priori opinions. One trader can
however change his mind and take the opinion of his neighbours if the coupling
is strong, or if the strength of his a priori opinion is weak. More precisely, the
8 opinion ϕi (t) of agent i at time t is determined as: ϕi(t) = sign hi (t) + N Jij ϕj (t) , (15) j =1 where Jij is a connectivity matrix describing how strongly agent j aﬀects agent
i, and hi (t) describes the a priori opinion of agent i: hi > 0 means a propensity
to buy, hi < 0 a propensity to sell. We assume that hi is a random variable
with a time dependent mean h(t) and root mean square ∆. The quantity h(t)
represents for example conﬁdence in long term economy growth (h(t) > 0), or
fear of recession (h(t) < 0, leading to a non zero average pessimism or optimism.
∆, J , such
Suppose that one starts at t = 0 from a ‘euphoric’ state, where h
that ϕi = 1 for all i’s.5 Now, conﬁdence is decreased progressively. How will
sell orders appear ?
Interestingly, one ﬁnds that for small enough inﬂuence (or strong heterogeneities of agents’ anticipations), i.e. for J
∆, the average opinion O(t) =
i ϕi (t)/N evolves continuously from O (t = 0)) = +1 to O (t → ∞) = −1.
The situation changes when imitation is stronger since a discontinuity then appears in O(t) around a ‘crash’ time tc , when a ﬁnite fraction of the population
simultaneously change opinion. The gap O(t− ) − O(t+ ) opens continuously as
J crosses a critical value Jc (∆) . For J close to Jc , one ﬁnds that the sell
orders again organise as avalanches of various sizes, distributed as a power-law
with an exponential cut-oﬀ. In the ‘mean-ﬁeld’ case where Jij ≡ J/N for all
ij , one ﬁnds µ = 5/4. Note that in this case, the value of the exponent µ is
universal, and does not depend, for example, on the shape of the distribution
of the hi ’s, but only on some global properties of the connectivity matrix Jij .
A slowly oscillating h(t) can therefore lead to a succession of bull and bear
markets, with a strongly non Gaussian, intermittent behaviour, since most of
the activity is concentrated around the crash times tc . Some modiﬁcations of
this model are however needed to account for the empirical value µ ∼ 3 observed
on the distribution of price increments (see the discussion in ).
Note that the same model can be applied to other interesting situations,
for example to describe how applause end in a concert hall  (here, ϕ = ±1
describes, respectively, a clapping and a quiet person, and O(t) is the total
clapping activity). Clapping can end abruptly (as often observed, at least by
the present author) when imitation is strong, or smoothly when many fans are
present in the audience. A static version of the same model has been proposed
to describe rational group decision making .
5 Here J denotes the order of magnitude of j 9 Jij 5 Models of feedback eﬀects on price ﬂuctuations 5.1 Risk-aversion induced crashes The above average ‘stimulus’ h(t) may also strongly depend on the past behaviour of the price itself. For example, past positive trends are, for many
investors, incentives to buy, and vice-versa. Actually, for a given trend amplitude, price drops tend to feedback more strongly on investors’ behaviour than
price rises. Risk-aversion creates an asymmetry between positive and negative
price changes . This is reﬂected by option markets, and pushes the price of
out-of-the-money puts up.
Similarly, past periods of high volatility increases the risk of investing in
stocks, and usual portfolio theories then suggest that sell orders should follow.
A simple mathematical transcription of these eﬀects is to write Eq. (12) in a
linearized, continuous time form:6
≡ u = h(t),
λ (16) and write a dynamical equation for h(t) which encodes the above feedback eﬀects
= au − bu2 − ch + η (t),
where a describes trends following eﬀects, b risk aversion eﬀects (breaking the
u → −u symmetry), c is a mean reverting term which arises from market clearing
mechanisms, and η is a noise term representing random external news .
Eliminating h from the above equations leads to:
= −γu − βu2 + η (t) ≡ −
+ η (t)
λ (18) where γ = c − a/λ and β = b/λ. Equation (18) represents the evolution of the
position u of a viscous ﬁctitious particle in a ‘potential’ V (u) = γu2 /2 + βu3 /3.
If trend following eﬀects are not too strong, γ is positive and V (u) has a local
minimum for u = 0, and a local maximum for u∗ = −γ/β , beyond which the
potential plummets to −∞.7 A ‘potential barrier’ V ∗ = γu∗2 /6 separates the
(meta-)stable region around u = 0 from the unstable region. The nature of
the motion of u in such a potential is the following: starting at u = 0, the
particle has a random harmonic-like motion in the vicinity of u = 0 until an
‘activated’ event (i.e. driven by the noise term) brings the particle near u∗ .
6 In the following, the herding eﬀects described by Jij are neglected, or more precisely, only
their average eﬀect encoded by h is taken into account.
7 If γ is negative, the minimum appears for a positive value of the return u∗ . This corresponds to a speculative bubble. See . 10 Once this barrier is crossed, the ﬁctitious particle reaches −∞ in ﬁnite time. In
ﬁnancial terms, the regime where u oscillates around u = 0 and where β can be
neglected, is the ‘normal’ ﬂuctuation regime. This normal regime can however
be interrupted by ‘crashes’, where the time derivative of the price becomes very
large and negative, due to the risk aversion term b which destabilizes the price
by amplifying the sell orders. The interesting point is that these two regimes
can be clearly separated since the average time t∗ needed for such crashes to
occur is exponentially large in V ∗ , and can thus appear only very rarely. A
very long time scale is therefore naturally generated in this model.
Note that in this line of thought, a crash occurs because of an improbable
succession of unfavorable events, and not due to a single large event in particular.
Furthermore, there are no ‘precursors’ in this model: before u has reached u∗, it
is impossible to decide whether it will do so or whether it will quietly come back
in the ‘normal’ region u 0. Solving the Fokker-Planck equation associated to
the Langevin equation (18) leads to a stationary state with a power law tail for
the distribution of u (i.e. of the instantaneous price increment) decaying as u−2
for u → −∞. More generally, if the risk aversion term took the form −bu1+µ ,
the negative tail would decay as u−1−µ. 5.2 Dynamical volatility models The simplest model that describes volatility feedback eﬀects is to write an arch
like equation , which relates today’s activity to a measure of yesterday’s
activity, for example:
σk = σk −1 + K (σ0 − σk −1) + g|δxk −1 |, (19) where σ0 is an average volatility level, K a mean-reverting term, and g describes
how much yesterday’s observed price change aﬀects the behaviour of traders
today. Since |δxk −1| is a noisy version of σk −1 , the above equation is, in the
continuous time limit, a Langevin equation with multiplicative noise:
= K (σ0 − σ) + gση (t),
dt (20) which is, up to notation changes, exactly the same equation as (9) above. Therefore, the stationary distribution of the volatility in this model is again given by
Eq. (11), with the tail exponent now given by µ − 1 ∝ K/g2 : over-reactions to
past informations (i.e. large values of g) decreases the tail exponent µ. Therefore, one again ﬁnds a non universal exponent in this model, which is bequeathed
to the distribution of price increments if one assumes that the ‘sign’ contribution
to δxk (see Eq. (3)) has thin tails.
Note that the temporal correlations of the volatility σ can be exactly calculated within this model , and is found to be exponentially decaying, at
variance with the slow power-law (or logarithmic) decay observed empirically.
11 Furthermore, the distribution (11) does not concur with the nearly log-normal
distribution of the volatility that seems to hold empirically [22, 25].
At this point, the slow decay of the volatility can be ascribed to two rather
diﬀerent mechanisms. One is the existence of traders with many diﬀerent time
horizons, as suggested in [50, 21]. If traders are aﬀected not only by yesterday’s price change amplitude |δxk −1|, but also by price changes on coarser time
scales |xk − xk −p|, then the correlation function is expected to be a sum of exponentials with decay rates given by p−1 . Interestingly, if the diﬀerent p’s are
uniformly distributed on a log scale, the resulting sum of exponentials is to a
good approximation decaying as a logarithm. More precisely:8
C (τ ) = 1
log(pmax /pmin ) pmax d(log p) exp(−τ /p) pmin log(pmax /τ )
log(pmax /pmin ) (21) whenever pmin
pmax . Now, the human time scales are indeed in a natural
pseudo-geometric progression: hour, day, week, month, trimester, year .
Yet, some recent numerical simulations of traders allowed to switch between diﬀerent strategies (active/inactive, or chartist/fundamentalist) suggest
strongly intermittent behaviour [51, 42, 52], and a slow decay of the volatility
correlation function without the explicit existence of logarithmically distributed
time scales. An intuitive, semi-analytical explanation of this numerical ﬁnding
is however still lacking. Note that from a purely phenomenological point of view,
one can deﬁne a model that assumes that the volatility today is the retarded
result of past inﬂuences:
∞ σk = M (p)ηk −p, (22) p=0 where the η ’s are uncorrelated shocks and M (p) a memory kernel describing
how much the past is remembered. If one chooses a power-law decay for M (p)
with an exponent α, then the decay of the correlation function of σ is also
a power-law with an exponent ν (deﬁned in Eq. (4)) given by ν = 2α − 1.
The value ν = 0 corresponds to a memory kernel decaying as 1/ p, i.e., as the
probability of not returning to the origin for a random walk of length p. Whether
this is a mere coincidence is left for future investigations. Note however that a
decomposition such as (22) naturally leads to a normal distribution for σ, very
diﬀerent from the empirical log-normal behaviour of the volatility. A consistent
market model leading simultaneously to a nearly log-normal distribution and
a nearly logarithmic (in time) decay of the volatility correlation remains to be
built. In this respect, the cascade construction of Mandelbrot et al.  does
indeed have these two properties exactly, but is non-causal (the volatility today
depends on future events) and lacks an intuitive interpretation.
8 This mechanism is well known in the physics of slow, glassy systems, where the relaxation
times p are the exponential of some local activation energy E . At low temperatures, any small
dispersion of E will generate a 1/p distribution for p over a wide time interval, and eventually
to a logarithmic relaxation. 12 6 Concluding remarks Many ideas have been presented in this rather hairy paper. Most of them will
perhaps turn out to be wrong, but will hopefully motivate some further work to
understand the origin of power-law distributions and power-law correlations in
ﬁnancial time series. From an empirical point of view, the exponents describing the tails of the price increments distribution and the decay of the volatility
correlations are rather robust and suggest some kind of universality, probably
related to the fact that all speculative markets obey common rules where simple
human psychology (greed and fear) coupled to basic mechanisms of price formation ultimately lead to the emergence of scaling and power-laws. Still, many
points remain obscure: we have seen above that models that appear naturally
in the context of economy and ﬁnance contain multiplicative noise, which is a
simple mechanism for power-law distributions (as emphasized in, e.g. [32, 41]).
However, these models generically lead to non universal exponents (as discussed
above in the context of the Pareto tails) that depends continuously on the value
of the parameters. Recent progress in the empirical study of the volatility (using, e.g., wavelets [26, 27]) suggests that the volatility results from some sort
of multiplicative cascade, as postulated in . A convincing ‘microscopic’ (i.e.
trader based) model that explains this observation would at this stage be very
valuable, and would shed light on the possible relevance of the pseudo geometric
human time scales on the decay of the volatility correlations. Acknowledgments
I want to warmly thank all my collaborators for sharing with me their knowledge:
P. Cizeau, R. Cont, I. Giardina, L. Laloux, A. Matacz, M. M´zard, M. Meyer
and M. Potters. Several discussions with E. Bacry, R. da Silvera, M. Dacorogna,
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