# Financial Markets and the
Phase Transition between Water and Steam

Christof Schmidhuber
Zurich University of Applied Sciences
School of Engineering, Technikumstrasse 9
CH-8401 Winterthur, Switzerland
<EMAIL_ADDRESS>

###### Abstract

A lattice gas model of financial markets is presented that has the potential
to explain previous empirical observations of the interplay of trends and
reversion in the markets, including a Langevin equation for the time evolution
of trends. In this model, the shares of an asset correspond to gas molecules
that are distributed across a hidden social network of investors. Neighbors in
the network tend to align their positions due to herding behavior,
corresponding to an attractive force of the gas molecules.

This model is equivalent to the Ising model on this network, with the
magnetization in the role of the deviation of the market price of an asset
from its long-term value. Moreover, in efficient markets the system should
naturally drive itself to its critical temperature, where it undergoes a
second-order phase transition. There, it is characterized by long-range
correlations and universal critical exponents, in analogy with the phase
transition between water and steam.

Applying scalar field theory and the renormalization group, we show that these
critical exponents imply predictions for the auto-correlations of financial
market returns and for Hurst exponents. For a network topology of $R^{D}$,
consistency with observation implies a fractal dimension of the network of
$D\approx 3$, and a correlation time of at least the length of the economic
cycle. However, while this simplest model agrees well with market data on very
long and on short time scales, it does not explain the observed market trends
on intermediate time horizons from one month to one year.

In a next step, the approach should therefore be extended to other models of
critical dynamics, to general network topologies, and to the vicinity of the
critical point. It allows us to indirectly measure universal properties of the
hidden social network of investors from the empirically observable interplay
of trends and reversion.

Keywords: Financial Markets, Trends, Renormalization Group, Critical Dynamics,
Fractal Dimension, Hurst exponents

Highlights

  * •

The shares of financial assets are modeled by the molecules of a lattice gas

  * •

The lattice represents the invisible underlying social network of investors

  * •

In efficient markets, arbitrageurs drive the system to its critical
temperature

  * •

Quantum field theory is used to derive scaling laws in analogy with phase
transitions

  * •

Consistency with empirical observations implies a fractal network dimension
near 3

##  1 Introduction and Summary

Trends in financial markets have long been exploited by the tactical trading
industry [1], and they have also been well-studied in the literature (see,
e.g., [2, 3, 4, 5, 6, 7, 8, 9, 10]). Traditional trend-followers are “long” a
given market when its current trend is positive, and “short” when the trend is
negative. Thus, they suffer draw-downs when trends revert.

In [11], we have carefully measured the interplay of trends and reversion in
financial markets with high precision by aggregating in new ways across 30
years of daily futures returns for equity indices, interest rates, currencies
and commodities, and across trends over time horizons $T$ ranging from 2 days
to 4 years. The key results are shown in fig. 1. A brief summary of how they
were derived is given in the appendix.

We have found that trends tend to revert before they become statistically
strongly significant. In other words, by the time a trend has become so
obvious that everybody can see it in a price chart, it is already over. This
is consistent with the hypothesis that any obvious market inefficiencies are
quickly eliminated by investors. More precisely, we have defined the strength
$\phi_{T}$ of a trend (36) measured over a time horizon $T$ as its statistical
significance. We could accurately model tomorrow’s return $R(t+1)$ in a market
(normalized to have variance 1) by a quartic potential $V_{T}(\phi_{T})$ of
today’s trend strength $\phi_{T}(t)$ in that market:

$R(t+1)\ =\ a\
-{\partial\over\partial\phi_{T}}V_{T}\big{(}\phi_{T}(t)\big{)}+\epsilon(t)\ \
\ \text{with}\ \ \ V_{T}(\phi_{T})=-{b_{T}\over
2}\cdot\phi_{T}^{2}+{g_{T}\over 24}\cdot\phi_{T}^{4},\vspace{5pt}$ (1)

where $a$ is a risk premium, $\epsilon$ represents random noise, and $b_{T}$
and $g_{T}$ are kinetic coefficients ($g_{T}$ is related to $c_{T}$ of [11] by
$g_{T}=-6c_{T}$). From this, we have derived a Langevin equation for the time
evolution of the trend strength, which describes a damped motion in the
potential $V_{T}(\phi)$ (see fig. 2). We have interpreted $b_{T}$ as the
persistence of trends, and $g_{T}$ as the strength of trend reversion. Within
the limits of statistical significance, we have found the following (I thank
J.-P. Bouchaud for pointing out that related observations were made in [12]):

  * •

The parameters $b_{T},g_{T}$ are universal in the sense that they are the same
for all assets

  * •

While $g_{T}$ has been fairly stable over time, $b_{T}$ appears to have
vanished over the decades

  * •

As a function of the time scale $T$, $g_{T}$ is approximately constant

  * •

$b_{T}$ depends on the time scale: it peaks at $T\sim$ 3-12 months and decays
for longer or shorter horizons. It appears to become negative for $T<1$ week
or $T>$ several years.

Figure 1:  Upper left (a): The expectation value $E(r)$ of the next day’s
return of a futures market can be modeled by a cubic polynomial of the trend
strength $\phi$, whose linear term $b\phi$ represents trend-persistence, and
whose cubic term $-(g/6)\ \phi^{3}$ represents trend-reversion. Upper right
(b): As confirmed by bootstrapping, the regression coefficients $b$ and $g$
corresponding to the linear and cubic terms are statistically highly
significant. Lower left (c): A heat map shows how the expectation value of
tomorrow’s return depends both on today’s trend strength $\phi$ and its time
horizon $T$. (a) can be thought of as a cross-section of (c) along the dashed
line. Lower right (d): The coefficients $b$ and $g$ as a function of the time
horizon $T$, corresponding to the dashed line in (c) as it moves from 2 days
to 4 years.

These results are reminiscent of physical systems at critical points.
Stochastic processes with a quartic potential such as in (1) are well-known to
also describe the dynamics of the order parameter of many statistical
mechanical systems near second-order phase transitions [13]; see [14] for a
review. In this analogy, $V(\phi)$ is the Landau potential as a function of
the order parameter. E.g., for a magnet, the order parameter is the overall
magnetization.

Figure 2:  Landau potential of the Ising model as a function of the
magnetization at high temperature (left), low temperature (center), and the
critical temperature (right).

The classical such system is the Ising model [15], which consists of magnetic
spins on a square lattice. Spins can point either up or down, and neighboring
spins tend to align (fig. 3, right). The effective potential $V(\phi)$
describes the energy of the system as a function of the magnetization $\phi$,
i.e., of the average value of the spins (fig. 2). There is a low-temperature
magnetized phase, where the potential has two minima corresponding to spins
pointing either up or down, and a high-temperature disordered phase, where the
average magnetization at the minimum of the potential is zero. The two phases
are seperated by a second-order phase transition at some critical temperature.
The time evolution of the magnetization follows a similar Langevin equation as
we have derived for the trend strength $\phi$.

Of course, this could be a coincidence. On the other hand, analogies between
financial markets and critical phenomena, such as scaling relations, have long
been observed [16, 17, 18]. Much of the literature has focused on the scaling
of volatilities rather than autocorrelations; see [19] for a summary of
empirical results and, e.g., [20] for a general review. Such analogies are
plausible, if financial markets are regarded as social networks, whose nodes
correspond to idividual investors (fig. 3), who interact with each other in
analogy with spins on a lattice, thereby creating the macroscopic phenomena of
trends (herding behavior) and reversion (contrarian behavior). To replicate
these phenomena, various spin- and agent models have been proposed (see, e.g.,
[21, 22, 23]).

In fact, the idea of using the Ising model and its cousins to model social
networks and financial markets has a long history (see [24] for an extensive
review). What is new about the results of [11] is that they make these long-
standing ideas very specific, in the sense that we can now directly observe
such a quartic Landau-like potential in financial markets, and we can
empirically measure its coefficients and their scaling behavior. This allows
us to make this conjectured relation between financial markets and phase
transitions on a social network precise, and to test quantitatively, whether
it agrees with financial market data.

Figure 3: Left: the social network of investors. Center: investors that are
over/under-weight in a given asset relatively to the CAPM allocation are
replaced by spins pointing up/down. Right: excerpt of an Ising model with 5
spins pointing up and energy $E=-4/2=-2$.

The first question is what exactly the microscopic degrees of freedom
represent, which sit on the vertices of the network, corresponding to
individual investors. If we would identify up/down spins with buy/sell orders,
as in the review [24], this would imply that the role of the order parameter
(the magnetization) is played by the net order size, and not by the trend
strength, as in our empirical observations.

We therefore take a different route. We identify the microscopic degrees of
freedom with the relative portfolio weights, by which the investor sitting on
a given node over- or underweighs a given asset. By “relative”, we mean
relative to the market capitalization of the asset, as in the CAPM model. If
we represent these relative weights by spins that can point either up (over-
weight) or down (under-weight), and if we also assume that neigboring spins
tend to align due to herding behavior, then we have all the ingredients that
make a phase transition plausible, in analogy with the magnet. Moreover, we
will argue that, in efficient markets [25, 26, 27], this system is driven to
the critical temperature, at which the phase transition becomes second-order.
It thus represents a form of self-organized criticality [28].

We will show that this implies that the order parameter $\pi(t)$, integrated
over the whole lattice (i.e., the overall magnetization), corresponds to the
deviation of the asset price from its long-term value. This deviation thus
follows a Langevin equation with potential $V(\pi)$:

$\dot{\pi}(t)\ =\ a\ -{\partial\over\partial\pi}V(\pi)+\epsilon(t)\ \ \
\text{with}\ \ \ V(\pi)=-{b\over 2}\cdot\pi^{2}+{g\over 24}\cdot\pi^{4}.$ (2)

We will further show how this can indeed be translated into an equation of the
form (1) for the trend strength $\phi_{T}$, with coefficients $b_{T},g_{T}$
that depend on the time scale.

Other models, in which the deviation of the asset price from its long-term
mean moves in a polynomial potential have previously been considered [29, 30,
31, 32]. There, various potentials were found for shorter time windows, whose
coefficients depend on the asset and the time window. By contrast, our quartic
potential is measured over the whole 30-year time window, and is universal
across assets. Indeed, if we assume that (2) arises from the dimensional
reduction of a higher-dimensional system near its critical point, this implies
strong constraints on stochastic price processes such as (2) and their scaling
behavior: it allows only processes that derive from renormalizable higher-
dimensional field theories.

Thus, the network leaves its imprint on the asset price dynamics even after it
is integrated over. This allows us to infer the properties of the presumed
unobservable social network that makes up financial markets from the
observable interplay of trends and reversion. In the case where the network
topology is assumed to be that of $R^{D}$, we find that the correlation time
in financial markets must be at least of the same order of magnitude as the
length of the economic cycle. At shorter time horizons, there must be a
“scaling regime” with a power-law fall-off of autocorrelations. Consistency
with empirical observations, such as Hurst exponents, then implies a (fractal)
network dimension of approximately $D\approx 3$.

However, upon closer look at the auto-correlations of market returns, we also
find that they cannot be fully explained by a simple Ising model on a network
with the topology of $R^{D}$. In a next step, this approach should therefore
be extended to other models of critical dynamics, as well as to more general
network topologies. Candidates for the latter include small-world networks
[33], scale-free networks [34], or the Feynman diagrams of large-N field
theory [35]. For a recent review, see [36]. To our knowledge, no convincing
specific model has emerged as a consensus so far. However, our results provide
an empirical basis for accepting or rejecting such candidates: any
statistical-mechanical model of financial markets, if accurate, must replicate
the interplay of trends and reversion observed in [11].

We hope that this will lead us to a specific and precise statistical model of
the presumed social network of investors, thereby lead to a deeper
understanding of financial markets, and - along the way - make the powerful
tools of field theory accessible to finance.

##  2 Lattice Gas Model of Financial Markets

We begin with a rudimentary lattice gas model of financial markets. It is
based on the following highly simplifying assumptions, to be refined later:

  * •

The social network of investors is modeled by a hypercubic lattice of given
dimension $D$. Each node corresponds to an investor.

  * •

There are only two assets in the world: (i) a bond, which can also be thought
of as cash, and (ii) a single stock, whose shares represent the molecules of
the lattice gas

  * •

Each investor can hold either one bond, or one share of the stock. The fact
that an investor can hold at most one share represents a “hard core” of the
gas molecules, or in other words, two molecules cannot sit at the same lattice
site.

  * •

The long-term equity allocation of the average investor is 50%, i.e., if there
are $N$ lattice sites (investors), the average number of shares is
$n_{0}=N/2$.

  * •

Neighbors tend to align their positions: if your “friend” ownes the stock, you
tend to also be interested in it. This represents an attractive force between
gas modelules.

A sample configuration of how shares may be distributed is shown in fig. 3
(right). Needless to say, the real network of investors is not a hypercubic
lattice, and more generally, one may say that this is at best a rough cartoon
of any real physical or financial system. However, it is well-known that this
primitive lattice gas model can precisely describe real gases, such as water
and steam or carbon dioxide, near their second-order phase transitions. There,
all the microscopic details of these systems become irrelevant, resulting in a
universal macroscopic behavior. Let us therefore assume that this phenomenon
of universality also occurs on social networks. Later, we will allow for the
possibility that these universality classes must be refined for nontrivial
network topologies. For now, let us quantify the above model.

For a given configuration, we define the occupation numbers
$n_{i}\in\\{0,1\\}$ as the number of shares the investor $i$ holds. The total
number of shares is $n=\sum_{i}n_{i}$. E.g., in fig. 3 (right), $n=5$. We also
define the energy $E$ of a configuration $\vec{n}$ of occupation numbers as

$E(n)=-{1\over D}\sum_{\langle ij\rangle}n_{i}n_{j},$ (3)

where $\langle ij\rangle$ denote all pairs of neigbors, i.e., all links on the
lattice. The normalization factor $1/D$ accounts for the fact that each node
appears in $2D$ links. E.g., in fig. 3 (right), $E=-4/2=-2$, as there are four
links with one share on both ends. This term represents the attractive force,
as shares tend to lie next to each other to minimize the energy.

We now imagine that the occupation numbers fluctuate while the total number of
shares is kept fixed. In other words, shares can change owner (i.e., their
location on the lattice). In addition, we assume that new shares can be issued
or existing shares can be redeemed at a given cost $\mu$, which we set equal
to 1. So we add a second term to the energy (3):

$E(n)=-{1\over D}\sum_{\langle ij\rangle}n_{i}n_{j}+\mu\sum_{i}n_{i}\ \ \
\text{with}\ \mu=1.$ (4)

This term allows us to also let the total number of shares fluctuate, as in a
system that is in equilibrium with a larger system. It is a counterweight to
the attractive force (if $\mu$ was zero, the network would fill up with shares
to minimize the energy). $\mu$ (or $-\mu$, depending on the convention) is
called the “chemical potential” in statistical mechanics.

Where in this model is the time-dependent share price $P(t)$ that corresponds
to real market prices? Since we have implicitly assumed that new shares are
issued or redeemed at price 1 to satisfy any increase or decrease of demand,
the share price is not dynamical. However, there is a dynamical market
capitalization, namely the total number of shares $n(t)$, which is equivalent
to a dynamical share price.

To make this equivalence precise, suppose that the firm did not issue or
repurchase shares. Thus, $n=n_{0}$ would be fixed, rather than fluctuate with
the demand. Instead, there would be a share price $P(t)$ that dynamically
adjusts to satisfy the demand. Then the market capitalization would be
$n_{0}\cdot P(t)$. Putting both equations for the market capitalization
together, we identify

$P(t)={n(t)/n_{0}}$ (5)

as the dynamical share price, where $n_{0}=N/2$ is the long-term value of the
firm.

As is well-known, the lattice gas model is equivalent to the Ising model.
Replacing the occupation numbers $n_{i}$ by spins $s_{i}$, the energy (4)
simply becomes

$E(n)=-{1\over D}\sum_{\langle ij\rangle}s_{i}s_{j}\ \ \ \text{with}\
s_{i}=n_{i}-1/2\in\\{+1/2,-1/2\\},$ (6)

up to an irrelevant constant that depends only on the total number of lattice
sites. The spins represent the over- or under-weight of a given investor in
the stock (setting $\mu\neq 1$ in (4) would correspond to introducing an
external magnetic field in the Ising model; here, we focus on the case
$\mu=1$). From (5,6), the total magnetization

$M(t)=\sum_{i}s_{i}=n(t)-{N\over 2}={N\over 2}\big{(}P(t)-1\big{)}$ (7)

of the Ising model is proportional to the deviation of the share price from
its long-term value.

To summarize, we have re-interpreted the well-known Ising model in terms of
financial markets. The order parameter (the overall magnetization) plays the
role of the deviation of the share price from its long-term equilibrium value
as defined, e.g., in the CAPM model. In the above, we have assumed a long-term
market weight of 50% for the asset; this can be generalized to an arbitrary
market weight by assigning different probabilities $p$ and $(1-p)$ to the
occupation numbers 0 and 1.

For $k$ assets, the spins have $k$ components. E.g., for $k=3$, they can point
in three different directions, corresponding to relative weights of a
portfolio of three assets. It will be interesting to empirically measure
whether there is an $O(k)$ symmetry for uncorrelated assets, in which case the
Ising model would be replaced by a $O(k)$ vector model, such as a Heisenberg
model for $k=3$.

As usual in statistical mechanics, we now consider the ensemble of all
possible spin configuration $\vec{s}$, weighted with relative probability
$G(\vec{s})/Z$, where

$G(\vec{s})=\exp\\{-{1\over kT}E(\vec{s})\\}\ \ \ \text{and}\ \ \
Z=\sum_{\vec{s}}G(\vec{s})$ (8)

where $G$ is the “Gibbs factor”, and the “partition function” $Z$ acts as a
normalization factor. The parameter $T$ (which is unrelated to the time
horizon) controls how important the statistical fluctuations are. In physics,
$T$ is interpreted as the temperature. $T$ is multiplied by Boltzmann’s
constant $k$ to give it the dimension of an energy. Below, we will relate $T$
to the persistence of trends $b$. Given (8), the expectation value of a spin
$s_{i}$ is

$\langle s_{i}\rangle={1\over Z}\sum_{\vec{s}}s_{i}\exp\\{-{1\over
kT}E(\vec{s})\\}.$

Figure 4: Computer simulation of the lattice gas / Ising model. The upper
graphs show the Landau potential. The lower graphs show typical spin
configurations. If interpreted as assets in the network of investors, black
dots represent shares, and white dots represent cash.

Fig. 4 shows images from a computer simulation of this model. Black dots
represent shares, white dots represent cash. There is a high-temperature
phase, corresponding to a strongly positive quadratic potential term, where
all spin configurations are equally likely according to (8), and the spins are
thus randomnly distributed. On the other hand, at low temperature, the energy
(6) is much more important than the statistical fluctuations in (8). So the
lowest energy configurations dominate: all spins point either up or down,
corresponding to the two minima of the potential with a strongly negative
quadratic term.

As the temperature rises, statistical fluctuations become more important in
(8) and bubbles - so-called Peierls droplets - start to form. Likewise, if the
temperature is reduced starting from the high-temperature phase, spins start
to feel the aligning force (6) and clusters start to form. Both the clusters
and the bubbles have a typical size, the correlation length $\xi$. As we
approach the critical point, where the quadratic term in the potential
vanishes and the second-order phase transition occurs, the correlation length
diverges: structures of all sizes can be seen, and the system is scale
invariant. At and above the critical temperature, the distinction between the
two ground states disappears: one cannot distinguish whether one observes
clusters of shares in a sea of cash, or clusters of cash in a sea of shares.

Let us now argue that financial markets are driven towards this critical
point. First, suppose that the persistence of trends $b$, which corresponds to
the quadratic term of the Landau potential, was significantly larger than
zero, as in the low-temperature phase. Then trend-following would be so
profitable that large amounts of capital would flow into this strategy. This
would drive $b$ back to zero. On the other hand, if $b$ was significantly
smaller than zero, markets would revert strongly und reliably. As a result,
mean reversion strategies (such as statistical arbitrage) would be so
successful that a large amount of capital would flow into them. This would
again drive $b$ back to zero.

The important conclusion is that $b$ should be close to its critical value
zero in efficient markets. In fact, a key empirical observation of [11] was
that $b$, which was already small (of the order of a few percent) several
decades ago, has further decreased since then. It now appears to be so close
to zero that even a diversified portfolio of pure trends can no longer be
expected to yield significant excess return. This can indeed be explained by
the fact that hundreds of billions, if not trillions of dollars, have flown
into trend-following, thereby eliminating much of this market inefficiency. It
gives insight into how markets dynamically become efficient over time. The
precise dynamical feedback mechanism that drives $b$ to 0 will be studied in
seperate work.

Returning to the lattice gas, as we approach the critical temperature $T_{c}$
from below, the expectation value $\langle M(T)\rangle$ of the magnetization
(7) goes to zero. If one measures how exactly it goes to zero, one finds a
power law. Likewise, the correlation length $\xi$ diverges as a power:

$M(T)\sim|\hat{t}|^{\beta}\ ,\ \ \ \xi(T)\sim|\hat{t}|^{-\nu}\ \ \
\text{with}\ \ \hat{t}\equiv(T-T_{c})/T$ (9)

with so-called “critical coefficients” $\beta,\nu$ that can be measured or
computed. E.g., in $D=3$ dimensions one finds $\beta=0.33$ and $\nu=0.63$. The
crucial property of $\beta,\nu$ is that they are “universal”. E.g., our
primitive lattice gas model in $D=3$ dimensions yields exactly the same values
$\beta,\nu$ as are observed for the second-order phase transitions of water
and steam, of $CO_{2}$, and of all other systems in the same universality
class. In fact, $\beta$ and $\nu$ depend only on 2 parameters: on the
dimension $D$ of the lattice (but not on whether it is a square, triangular,
or similar lattice), and on the symmetries. Here, the symmetry is $Z_{2}$,
corresponding to flipping all spins.

A potential caveat is that this only applies to networks with the trivial
topology of $D-$dimensional Euclidean space. For more general network
topologies, it is conceivable that the universality classes are characterized
by additional parameters, such as the exponent $\gamma$ of the so-called
“degree distribution” $p(k)\sim k^{-\gamma}$ of nodes with $k$ neighbors in
the case of scale-free networks [34]. It will be important to investigate this
in the future.

In either case, even though our lattice gas model of financial markets may
seem very far from reality on a microscopic scale, the phenomenon of
universality may allow it to precisely model the macroscopic behavior of real
markets in the vicinity of the phase transition. In particular, the universal
coefficients $\beta$ and $\nu$ in (9) also determine how the system behaves
under a rescaling in spare and time. E.g., at the critical point, the
covariance of spins at different points $\vec{x},\vec{y}$ scales with their
distance like

$\langle s(\vec{x})s(\vec{y})\rangle\sim|\vec{x}-\vec{y}|^{2-D-\eta}\ \ \
\text{with}\ \ \eta=2{\beta\over\nu}+2-D.$ (10)

In the next section, we will see how $\eta$ can also be used to predict the
auto-correlations of financial market returns at different points in time.
Those can be compared with auto-correlations in real markets. This establishes
a direct link between observable properties of asset prices, and the critical
exponents of the presumed social network that makes up financial markets but
cannot be observed directly.

##  3 Field Theory of Financial Markets

In the limit of infinitely many lattice sites with lattice spacing
$a\rightarrow 0$, the lattice becomes a continuous manifold of dimension $D$.
Since networks generally have fractal dimensions, we formally allow $D$ to be
any positive real number. The index $i$ becomes a $D$-dimensional vector
$\vec{x}$, and the spins $s_{i}(t)$ become a field $\phi(\vec{x},t)$ that
measures the local magnetization, corresponding to the excess demand for an
asset in a given neighborhood of the social network of investors. Let us first
consider the system at a fixed point $t=0$ in time. One replaces the partition
function $Z$ in (8) near the critical temperature $T\sim T_{c}$ by a path
integral

$Z=\int D\phi(\vec{x})\exp\\{-S[\phi(\vec{x})]\\}$

with action $S[\phi]$ and Landau potential $V(\phi)$:

$\displaystyle S[\phi(\vec{x})]$ $\displaystyle=$ $\displaystyle\int
d^{D}\vec{x}\ \Big{\\{}{1\over 2}\ (\vec{\nabla}\phi)^{2}+V(\phi)\Big{\\}}\ \
\ \text{with}\ \ V(\phi)={1\over 2}\ r\phi^{2}+{1\over 24}\ g\phi^{4}.$ (11)

Here, the factor $1/kT$ in (8) has been absorbed such that $\phi$ has a
standard kinetic term. The temperature then re-appears in the potential: $r$
is proportional to the deviation $(T-T_{c})$ from the critical temperature.
For a general introduction to field theory and critical phenomena, see, e.g.,
[37, 38]. Here, we merely list some results that we need in the following:

  * •

The parameters $r(\lambda)$ and $g(\lambda)$ in (11) “flow” under so-called
renormalization group transformations [39], in which all distances are
rescaled by a factor $\lambda$, and all objects with a scaling dimension are
rescaled accordingly (see fig. 5).

  * •

In particular, for dimensions $D\geq 4$, $g(\lambda)$ flows towards zero at
large scales, so the quartic term of the potential becomes weaker and weaker.
For $D>4$, the macroscopic scaling behavior of the system is the same as for a
free theory (i.e., $g=0$).

  * •

For dimensions $D<4$, $g(\lambda)$ flows to a nontrivial fixed point value
$g^{*}$ at large scales (fig. 5). The coefficients of all other possible terms
in the action, such as $\phi^{6},(\vec{\nabla}\phi)^{2}$, …, flow to zero.
This explains why systems that are very different on a microscopic scale have
the same macroscopic behavior with the same critical coefficients.

  * •

At this fixed point, the field $\phi$ scales as
$\phi\rightarrow\lambda^{\eta/2}\phi$ with the so-called “anomalous dimension”
$\eta=2(\beta/\nu)+2-D$. $\eta$ can be computed perturbatively in $g^{*}$ via
Feynman diagrams [37, 38]. Table 1 lists estimated values of $\eta$ for
various dimensions $D$ [37, 40].

Figure 5: Left: Renormalization Group Flow in $\phi^{4}$ theory. At large
scales, the coefficient of the quartic term flows to a fixed point value
$g^{*}$. The coefficient of the quadratic term must normally be fine-tuned to
zero to reach the critical point. In our model, it is attracted to zero in
efficient markets. Right: time evolution of Peierls droplets of positive
magnetization.

Thus, the mere fact that we empirically observe a quartic potential in (1) may
already be a first indication that the fractal dimension of the social network
of investors is less than 4.

Table 1 | Critical coefficients vs. network dimension $D$
---|---
Dimension |  4 |  3.5 |  3 |  2.5 |  2 |  1.5
$\eta$ |  0.00 |  0.002 |  0.036 |  0.106 |  0.250 |  0.523
$z$ |  2.000 |  2.001 |  2.024 |  2.071 |  2.167 |  2.352
$\kappa\equiv(2-\eta)/z$ |  1.000 |  0.998 |  0.970 |  0.915 |  0.808 |  0.628

The above points refer to static critical phenomena. Let us also discuss the
time evolution of the system near the critical point, when it is slightly out
of equilibrium. For a general introduction to this field of Critical Dynamics,
see [14]. In this paper, we focus on the simplest case - the so-called purely
dissipative “Model A” [13].

By emerging and decaying, Peierls droplets of positive or negative
magnetization create trends of the overall magnetization. In our model, where
the magnetization represents the deviation of an asset price from its value,
this is the origin of the price trends in financial markets. The droplets
represent groups of investors that collectively over- or underweigh an asset.
The diffusion of the field $\phi(\vec{x},t)$ in time is described by a
Langevin equation [37, 14]:

${\partial\over\partial t}\phi(\vec{x},t)={\Omega\over
2}\Big{(}\nabla^{2}\phi(\vec{x},t)-r\phi(\vec{x},t)-{1\over
6}g\phi(\vec{x},t)^{3}\Big{)}+\text{random noise},$ (12)

where $\Omega$ is a diffusion coefficient and the variance of the noise is
$\Omega$. By a rescaling of time, let us set $\Omega=1$. Fig. 5 (right) shows
the diffusion of a droplet of size $L$ in the case of one dimension,
corresponding to a local magnetization with Gaussian distribution. For $g=0$,
it decays over a time span $\Delta t\sim L^{2}$. Correspondingly, the spatial
correlation length $\xi$ and the correlation time $\tau$ are related by
$\tau\sim\xi^{2}$. For nonzero $g$, there is a correction [13]:

$\Delta t\sim L^{z}\ \ ,\ \ \tau\sim\xi^{z}\ ,$

where $z$ is a new critical exponent that depends on $D$ and the symmetry
group $O(N)$. Its estimated value for various dimensions and $N=1$ (the Ising
model) is shown in table 1. For a recent calulation, see [41]. For the range
of fractal dimensions $D$ that will turn out to be relevant for us, $2\leq
D\leq 3.5$, one may approximate $z=2+c\cdot\eta$ [13] with $c\approx 2/3$.

In the following, we focus on the spatially constant mode of the field $\phi$,
i.e., its integral over the whole network, corresponding to the overall
magnetization:

$\pi(t)=\int d^{D}x\ \phi(x,t).$

In the previous section, $\pi(t)$ has been identified with the deviation of
the share price from its long-term value. Since this “value” is not
observable, we cannot compare $\pi(t)$ directly with experiment. Fortunately,
the unknown “value” drops out of the market returns $R(t)$, which are
proportional to the price change $\dot{\pi}(t)$. In the classical action (11),
$\phi$ is normalized to have standard kinetic term, so $\dot{\pi}$ has
variance 1. On the other hand, the daily returns in our empirical analysis
were also normalized to have variance 1. Thus, we directly identify

$R(t)\equiv\dot{\pi}(t)={d\over dt}\pi(t).$

This immediately yields a first important prediction of $\phi^{4}$ theory for
the auto-correlation of market returns that can be compared with observation:

$\langle\dot{\pi}(t)\dot{\pi}(t^{\prime})\rangle\sim\
-\ddot{\Delta}(t-t^{\prime})\ \ \text{with}\ \
\Delta(t)\equiv\langle\pi(0)\pi(t)\rangle,$ (13)

where $\Delta(t)$ is the so-called two-point function or “propagator” of the
theory. Due to the huge amount of noise in financial markets, this
autocorrelation of daily returns is very difficult to measure empirically. For
this reason, in our empirical observations [11] we have aggregated the recent
returns of a given market over a time horizon $T=2/\omega$ in the form of the
strength $\phi_{\omega}$ (36) of the recent trend in that market. Its
continuum definition in terms of $\dot{\pi}$ is:

$\displaystyle\phi_{\omega}(t)$ $\displaystyle=$ $\displaystyle
2\omega^{3/2}\cdot\int_{-\infty}^{t-}d\zeta\ (t-\zeta)\ e^{-\omega(t-\zeta)}\
\dot{\pi}(\zeta)$ (14)

Here, $t-$ indicates that the point $t$ is excluded from the integration,
i.e., the return at time $t$ is predicted only from returns before time $t$.
As explained in the appendix, the weight function $ne^{-\omega n}$, by which
the return from $n\sim(t-\zeta)$ days ago is weighted in (14), peaks at
$n=1/\omega=T/2$, and the average lookback period is $\langle n\rangle=T$. The
pre-factor $2\omega^{3/2}$ ensures that the variance of the trend strength is
approximately one, so $\phi_{\omega}$ measures the statistical significance
(t-statistics) of the trend. The aggregate auto-correlation can then be
written as a (variant of the) Laplace transform of the propagator:

$\displaystyle\langle\phi_{\omega}(t),\dot{\pi}(t)\rangle=\langle\phi_{\omega}(0),\dot{\pi}(0)\rangle$
$\displaystyle=$ $\displaystyle-2\omega^{3/2}\int_{0+}^{\infty}d\zeta\
\zeta\cdot e^{-\omega\zeta}\ \ddot{\Delta}(\zeta).$ (15)

Fig. 6 (left) shows the empirically observed two-point function
$\langle\phi_{\omega},\dot{\pi}\rangle$ on a logarithmic scale, with time
horizon $T=2/\omega=2^{k}$ days. It shows a curious behavior: at intra-month
scales, the (very small) autocorrelation is negative. It turns positive on
scales from one month to about 1 year - the time horizons, at which trend-
following has been most profitable. At time scales of several years, it turns
negative again. The correct field theory of financial markets must reproduce
this propagator.

Figure 6:  Left: autocorrelations of market returns (white dots), aggregated
by the trend strength $\phi_{\omega}=\phi_{2/T}$. For intermediate time
horizons, they turn positive, corresponding to the regime where trend-
following works best. Right: the variance of the trend strength as a function
of the time horizon falls off at short scales, rises at intermediate scales,
and seems to fall off again at long scales. As a cross-check, similar analyses
were performed with the simpler version of the trend strength $\tilde{\phi}$
(black dots) for non-overlapping time windows.

As an additional cross check, we have repeated the analysis with a simpler
version $\tilde{\phi}_{T}$ of the trend strength (14), proportional to the
equally weighted average of the daily log returns in a time window of length
$T$ (this corresponds to the step function in fig. 9 of the appendix):

$\displaystyle\tilde{\phi}_{T}(t)$ $\displaystyle=$ $\displaystyle
T^{-1/2}\int_{-\infty}^{t-}d\zeta\ \dot{\pi}(\zeta)\ =\
T^{-1/2}\big{[}\pi(t)-\pi(t-T)\big{]}.$ (16)

To measure its auto-correlations empirically, we have slightly extended our
history of futures returns from 30 years to $2^{13}$ trading days $\approx
31.5$ years, and we have constructed $2^{13-k}$ non-overlapping time windows
of length $T=2^{k}$ for each $k$. Fig. 6 (left) shows the correlation of this
new trend strength (black dots) between neighboring time windows. Within
estimation errors, these empirical results are in line with the empirical
autocorrelations (15). The theoretical prediction for this correlation depends
on the propagator:

$\langle\tilde{\phi}_{T}(t)\tilde{\phi}_{T}(t-T)\rangle\ =\ -{1\over
T}\big{[}\Delta(0)-2\Delta(T)+\Delta(2T)\big{)}].$

Much of the literature on scaling in financial markets is concerned with the
scaling of the variance, or more generally, of the $q$th moments of returns
over the time period $T$ [42, 43]:

$K_{q}(T)=\langle|\pi(T)-\pi(0)|^{q}\rangle/\langle|\pi(0)|^{q}\rangle\ \
\Rightarrow\ \ K_{2}(T)=2-2\Delta(T)/\Delta(0).$ (17)

With a limited amount of data, these moments are easier to measure empirically
than the auto-correlations of returns. Here we focus on $q=2$. From (13, 14,
16), after some algebra we obtain formulas for the variance of our trend
strengths:

$\displaystyle\langle\phi_{\omega}(0)\phi_{\omega}(0)\rangle$ $\displaystyle=$
$\displaystyle-2\omega^{3}\int_{0+}^{\infty}du\ e^{-\omega u}\int_{0}^{u}dv\
v\dot{\Delta}(v).$ (18)
$\displaystyle\langle\tilde{\phi}_{T}(0)\tilde{\phi}_{T}(0)\rangle$
$\displaystyle=$ $\displaystyle{2\over T}\big{[}\Delta(0)-\Delta(T)\big{]}\ \
\ \text{with}\ \ T=2/\omega$ (19)

Note that these trend strengths were normalized to have variance 1, if the
small auto-correlations of financial market returns are neglected. (18, 19)
thus describes the precise effect of the auto-correlations on this
normalization, as predicted by field theory.

Fig. 6 (right) shows the empirical results. They mirror the curious behavior
found for the propagator: the variance of the trend strength as a function of
the time scale decreases at short scales, increases at intermediate scales,
then may decrease again. The correct field theory description of financial
markets must reproduce this empirically observed variance.

A further important test comes from the Langevin equation (12). For the
spatially constant mode $\pi(t)$, which we have identified with the asset
price minus its value, it implies:

$\dot{\pi}(t)=-{r\over 2}\cdot\pi(t)-{g\over 12}\cdot\pi(t)^{3}+\eta(t),$ (20)

where $\eta$ represents Gaussian noise. Again, $\pi$ is not directly
observable because of the unknown “value” of an asset. However, this “value”
cancels out of the trend strength. To compare with observation, we must
translate (20) into an equation involving only the trend strength. Generally,
the coefficients will then depend on the time horizon $T=2/\omega$:

$\displaystyle\dot{\pi}(t)$ $\displaystyle=$
$\displaystyle\beta_{\omega}\cdot\phi_{\omega}(t)+\gamma_{\omega}\cdot\phi_{\omega}(t)^{3}+\epsilon_{\omega}(t)$
(21)

The multiple regression coefficients $\beta_{\omega},\gamma_{\omega}$ can be
found by inverting the covariance matrix:

$\displaystyle\begin{pmatrix}\beta\\\ \gamma\end{pmatrix}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}\langle\phi^{2}\rangle&\langle\phi^{4}\rangle\\\
\langle\phi^{4}\rangle&\langle\phi^{6}\rangle\end{pmatrix}^{-1}\begin{pmatrix}\langle\phi,\dot{\pi}\rangle\\\
\langle\phi^{3}\dot{\pi}\rangle\end{pmatrix}\ =\
\begin{pmatrix}\langle\phi^{2}\rangle&\langle\phi^{4}\rangle\\\
\langle\phi^{4}\rangle&\langle\phi^{6}\rangle\end{pmatrix}^{-1}\begin{pmatrix}-{r\over
2}\langle\phi,\pi\rangle-{g\over 12}\langle\phi,\pi^{3}\rangle\\\ -{r\over
2}\langle\phi^{3}\pi\rangle-{g\over
12}\langle\phi^{3}\pi^{3}\rangle\end{pmatrix},$

where all fields sit at $t=0$ (we have suppressed the index $\omega$). This
reduces the problem to the (tedious but straightforward) computation of
various correlation functions in $\phi^{4}$ theory, which is left for future
work.

The correct field theory must replicate the empirically observed coefficients
$b_{T},g_{T}$ of (1), as shown in fig. 1d. These results can be regarded as a
refinement of the results for the propagator plotted in fig. 6 (left):
inserting (20) into the expectation value
$\langle\dot{\pi}(0)\dot{\pi}(t)\rangle$ decomposes the propagator into a
component that comes from the persistence of trends $r$, and another component
that comes from the strength of reversion $g$. In particular, this should
allow us to directly measure the fixed point value $g^{*}$, once we have found
the right model. As $g^{*}$ depends on the (possibly fractal) dimension $D$ of
the presumed underlying social network of investors, this will amount to a
direct empirical estimate of $D$.

##  4 Illustration: $D$-dimensional Lattice

In the previous section, we have derived three key requirements - yielding the
correct propagator, the correct variance of the trend strength, and the
correct Langevin equation - that a field theory of financial markets must
satisfy in order to agree with observation.

In this section, we illustrate how to apply these criteria in practise at the
simplest example, to be generalized in future work: the purely dissipative
model A on a hypercubic lattice as discussed in section 3. We do not
necessarily expect this model to closely match observation. One of the reasons
is that the social network of investors is surely much more connected than a
hypercubic lattice, where the shortest path between two sites is on average
very large. However, this example demonstrates how to connect the rather
abstract concepts of quantum field theory with the practical observation of
financial markets in order to derive two key properties of the presumed
underlying social network of investors: its fractal dimension $D$, and its
correlation length $\xi$. To this end, we discuss two distinct regimes:

  * •

the “scaling regime” at small time scales $T$ (i.e.,
$\omega=2/T\rightarrow\infty$). Here, the renormalization group dictates a
power-law fall-off of all correlation functions. Matching with observation
should yield an estimate of the anomalous dimension $\eta$, and thereby of
$D$.

  * •

the “exponential regime” at large time scales (i.e., $\omega\rightarrow 0$).
Here, all correlations fall off exponentially. Matching with observation
should yield an estimate of the correlation time $\tau$ of market prices, and
thereby of the correlation length $\xi\sim\tau^{-z}$ in the network.

These two regimes also appear in more general models of critical dynamics,
albeit with different scaling exponents. The precise interpolation between the
two regimes at intermediate time scales should hold the key to identifying the
correct model, beyond “model A” on a hypercubic lattice. This is left for
future work. In this paper, we only use the fact that the scaling regime must
approach the exponential regime, as the dimension approaches the critical
dimension $D=4$.

Sub-section 4.1 discusses the propagator of the model. Sub-section 4.2 turns
to the scaling of variances and compares the theoretical predictions with the
observed variances and Hurst exponents. Sub-section 4.3 discusses the results.

###  4.1 The Propagator

We consider “model A” on a Euclidean manifold of dimension $D$ (representing
the network of investors) and set the variance $\Omega$ of the random noise to
1. For the free theory ($g=0$, corresponding to $D>4$), the propagator of the
spatially constant mode $\pi(t)$ is [14]:

$G_{cl}(p,t)={1\over p^{2}+r}\ \exp\\{-{p^{2}+r\over 2}|t|\\}\ \ \Rightarrow\
\ \Delta(t)={1\over 2}\cdot{\xi^{2}}\exp\\{-|t|/\xi^{2}\\},$

where $\xi$ is the correlation length, which is equal to $\sqrt{2/r}$ for the
free theory with $g=0$. For $g\neq 0$ and $D<4$, the renormalization group
provides the following scaling relation for the exact two-point function in
the critical domain, slightly above the critical temperature (where $\xi$ in
(9) is finite) [14]:

$\Delta(t)=\langle\pi(0)\pi(t)\rangle=\xi^{2-\eta}\cdot g(t/\xi^{z}),$

with some function $g(x)$ that falls off exponentially at large $x$. Let us
first consider the scaling regime. Consistency with the static limit
$t\rightarrow 0$ (where $\Delta(0)\sim\xi^{2-\eta}$ from the spatial integral
over $\vec{x}$), and with the limit $D\rightarrow 4$ (where $\kappa\rightarrow
1$ as in table 1) implies:

$\Delta(t)={1\over 2}\cdot\tau^{\kappa}-{1\over 2}\cdot|t|^{\kappa}\ \ \
\text{for }t\ll\tau\ \ \ \text{with}\ \ \tau\equiv\xi^{z},\ \
\kappa\equiv{2-\eta\over z}<1$ (22)

On the other hand, in the exponential regime, consistency with the limit
$D\rightarrow 4$ implies

$\Delta(t)={1\over 2}\cdot\tau^{\kappa}\exp\\{-|t|/\tau\\}\ \ \ \text{for
}t\gg\tau$ (23)

Figure 7: Left: the propagator of the spatially constant mode falls off like a
power at small time scales, and exponentially at time scales larger than the
correlation time. The unknown behavior of the correct model in the
intermediate regime is indicated by a question mark.

In the low-temperature phase, these relations hold for the connected two-point
function $\langle\pi(0)\pi(t)\rangle-\langle\pi(0)\rangle^{2}$ instead. In
both cases, we have for $t>0$:

$\displaystyle\dot{\Delta}(t)$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}-{1\over
2}\tau^{\kappa-1}\cdot\exp\\{-t/\tau\\}\\\ -{1\over 2}\kappa\cdot
t^{\kappa-1}\\\ \end{array}\right.,\
\ddot{\Delta}(t)=\left\\{\begin{array}[]{ll}{1\over
2}\tau^{\kappa-2}\cdot\exp\\{-t/\tau\\}&\text{for }t\gg\tau\\\ {1\over
2}\kappa(1-\kappa)\cdot t^{\kappa-2}&\text{for }t\ll\tau\\\
\end{array}\right.$ (28)

Note that this holds both above and below the critical temperature: the
propagator in these cases differs only by the expectation value of $\pi$,
which is a constant that drops out of $\ddot{\Delta}$.

Fig. 7 schematically shows this propagator. The singularity of the orange line
is the imprint that the invisible higher-dimensional social network of
investors leaves on the stochastic process $\pi(t)$, even after it is
integrated out. As $D\rightarrow 4$ and $\kappa\rightarrow 1$, the propagator
in the scaling regime approaches that of the exponential regime, and the
singularity disappears.

The propagator (22, 23) predicts that auto-correlations $-\ddot{\Delta}$ (13)
of market returns are purely negative. Thus, it does not explain the positive
autocorrelations that we observe in fig. 6 (left) on intermediate scales from
one month to one year. These empirical results suggest that the true
propagator $\Delta$ has negative curvature in part of the intermediate regime,
which creates the phenomenon of market trends. Again, this should be a key
clue for finding the true model in the future.

###  4.2 Scaling of the Variance

For the time being, though, let us nevertheless proceed and discuss the
scaling regime and the exponential regime, where our model is in line with
observations. To this end, it is more convenient to consider the scaling of
the variances (18, 19) of the trend strengths instead of the propagator. Fig.
8 (left) extends the empirically estimated variance of fig. 6 to 16-year
windows. From (18, 19, 28), we obtain in the scaling regime
$T=2^{k}={2/\omega}\ll\tau$ for $\kappa\approx 1$:

$\displaystyle{\langle\phi_{\omega}^{2}\rangle}$ $\displaystyle\sim$
$\displaystyle\omega^{1-\kappa}=e^{(1-\kappa)\cdot\ln\omega}\ \approx\
1-(1-\kappa)\ln 2\cdot k$ (29)
$\displaystyle\langle\tilde{\phi}_{T}^{2}\rangle$ $\displaystyle=$
$\displaystyle T^{\kappa-1}=e^{(\kappa-1)\cdot\ln T}\approx 1-(1-\kappa)\ln
2\cdot k$ (30)

Thus, $\kappa$ can be read off from the slope $(1-\kappa)\ln 2$ of the
variance as a function of $k$ (orange line in fig. 7). We measure by
regression:

$\kappa=0.96\pm 0.01\ \ \ \Rightarrow\ \ \ D=2.9\pm 0.1.$ (31)

Here, we have interpolated the formally non-integer dimension $D$ from table 1
based on the approximation $z=2+\eta(D)\cdot 2/3$. We propose to identify $D$
with the fractal dimension of the network; although there are different
definitions of this fractal dimension, they coincide for homogenous networks
such as ours, where all nodes have the same number of links (for other network
topologies, this is generally not the case, though [45]).

Figure 8: Left: Fit of the scaling behavior of the variance in the scaling
regime (dashed orange line) by the field theory prediction with critical
coefficient $\kappa\approx 0.96$, corresponding to a fractal dimension of the
presumed underlying network of $D\approx 3$ . The correlation time, where the
variance falls off rapidly, must be at least as long as the economic cycle.
Right: Scaling of the $q$-th moment $M_{q}$ for $q=\\{1,2,3,4\\}$ as a
function of the time scale $T$ in our dataset of liquid daily futures returns.
The slopes of the lines are the Hurst coefficients $H_{q}$.

On the other hand, in the exponential regime $T={2/\omega}\gg\tau$ we get from
(18, 19, 28):

$\displaystyle{\langle\phi_{\omega}^{2}\rangle}\ =\
{\omega^{2}\over(\omega+1/\tau)^{2}}\cdot\tau^{\kappa-1}\ \ \ ,\ \ \
{\langle\tilde{\phi}_{T}^{2}\rangle}\ =\ {\tau\over
T}(1-e^{-T/\tau})\cdot\tau^{\kappa-1}$ (32)

In both cases, field theory predicts the variance of trends to fall off
rapidly as $T$ becomes larger than the correlation time. There is some
indication from fig. 8 (left) that this happens at $T\sim 8$ years, but we do
not have enough data to confirm this with significance, and we cannot even
rule out that the correlation time is infinite. All we can say is that the
correlation time must be larger than $4-8$ years. Interestingly, this lower
bound is of the same order of magnitude as the length of the economic cycle. A
more precise statement will require an analysis of many decades, if not
centuries of historical data.

Let us also comment on Hurst exponents. We use the definition of the $q$-th
generalized Hurst exponent $H_{q}$ in terms of the scaling of the $q$-th
moment of $\pi$ (see 17):

$M_{q}(T)\equiv\langle|\pi(T)-\pi(0)|^{q}\rangle\sim\langle|\pi(0)|^{q}\rangle\cdot
T^{qH_{q}}\ \ \ \text{with}\ \ q\in R,$

The process $\pi$ is called “mono-scaling”, if all $H_{q}$ are identical, and
“multi-scaling”, if $H_{q}$ depends on $q$. There is ample evidence that
financial market returns are “multi-scaling”, not “mono-scaling” (see [19] and
references therein, and [20] for a general review of scaling properties,
including in financial markets).

Our data set also supports this. Fig. 8 (right) shows the empirical
measurement of $M_{q}(T)$ for $T$ ranging from 1 day to 4 years for our
30-year history of daily futures returns across asset classes. From the slopes
of the lines, we can estimate the generalized Hurst exponents. The results are
consistent with those for liquid markets in [19] (our dataset only covers
liquid futures). Our Hurst exponents are not quite the same for all horizons,
though. E.g., $H_{2}$ increases from 0.48 at short scales to 0.5 at scales of
several years.

What does field theory predict for the generalized Hurst exponents? We focus
on even, integer $q=2n$. In the scaling regime, we get from (16, 30):

$M_{2n}(T)\ =\ T^{n}\langle|\tilde{\phi}_{T}^{2}|^{n}\rangle\ \sim\
T^{n\kappa}\ \ \ \text{with}\ \ \ n\in\\{1,2,3,...\\}$

Thus, for infinite correlation time, field theory predicts mono-scaling with
Hurst exponent

$H={\kappa\over 2}\ \ \ \text{with}\ \ \ \kappa\ =\ {2-\eta\over z}\ $ (33)

independently of $n$. Thus, the Hurst exponent is directly related to the
anomalous scaling dimension $\eta$ of the deviation $\pi$ of the asset price
from its value, which in turn is related to the dimension $D$ of the
underlying network of investors. E.g., for $D=(2,3,4)$, field theory predicts
$H\approx(0.40,0.485,0.50)$. However, any finite correlation time will destroy
this mono-scaling. How precisely the $H_{2n}$ depend on $n$ in this case, and
whether this can explain the observed multi-scaling in financial markets, must
be addressed in future work.

###  4.3 Discussion

To summarize this section, we have tried out the simplest model of critical
dynamics (“model A”) on the simplest network topology (a $D$-dimensional
hypercubic lattice) at the critical point (where $g=g^{*}$ is at its fixed
point) in order to model financial markets as a social network of investors.
Although these assumptions were radical simplifications that seem unrealistic,
the predictions of this model are not all that far from the reality of
financial markets.

At small scales, it predicts a power-law fall-off of autocorrelations and a
scaling behavior of volatilities that is reminscent of the one that has been
observed in financial markets. In our approach, the origin of the associated
scaling laws and of Hurst exponents that differ from 1/2 is the same as in
second-order phase transitions: they reflect anomalous dimensions that are
well-known from quantum field theory. We have demonstrated how this allows us
to infer the (fractal) dimension of the presumed underlying social network of
investors. It is encouraging that the estimated fractal dimension $D\approx 3$
is in line with reported fractal dimensions of other social networks, the
majority of which lie between 2 and 4 [45].

Moreover, it is highly nontrivial that the small typical values of the
anomalous dimension $\eta$ have just the right magnitude to the explain the
observed small negative 1-day market auto-correlations of a few percent. We
take this as a sign that we are on the right track by modeling financial
markets in analogy with critical phenomena. At the same time, we have seen
that “model A” does not describe the intermediate regime at horizons from a
few weeks to a few years well. We may need a more complex model, or a more
complex network topology, or we may see non-universal properties of the
network (see the concluding remarks).

At large scales, our model predicts an exponential fall-off of all
correlations. This should be verified by studying historical market data on a
scale of decades and centuries. Of course, the question is then whether
markets in previous centuries can already have been efficient enough to be
driven to the critical point by the mechanism described in section 2, and
whether the structure and dimension of the network of investors has been
stable in time.

##  5 Conclusion and Outlook

Motivated by the previous empirical observations [11] on trends and reversion
in financial markets, in this paper we have proposed a statistical-mechanical
model of the markets that has the potential to explain these observations. The
key clue is that financial market returns can accurately be described by the
stochastic process (1) with a quartic potential. Such a quartic potential
typically occurs in certain physical models near second-order phase
transitions, such as the Ising model or the lattice gas model.

This has led us to propose a lattice gas model of financial markets. The
lattice represents the social network of investors, who correspond to its
vertices. The microscopic degrees of freedom sitting at these vertices
represent the investors’ asset allocations. They can either be modeled by gas
molecules that correspond to the shares of an asset, or - equivalently - by
spins that denote whether the investor is over- or underweight in the asset,
relatively to its CAPM weight. For $N$ different assets, the spins have $N$
components.

We have identified the order parameter, corresponding to the magnetization in
the spin model, as the deviation of the asset price from its long-term value.
If we assume that neighboring spins tend to align due to herding behavior, the
interaction of the spins naturally creates the macroscopic phenomena of trends
and reversion of the asset price.

In addition, we have argued that efficient markets drive themselves to the
critical point, at which a second-order phase transition occurs. This assumes
that investors quickly exploit and thereby eliminate any obvious market
inefficiencies either by momentum strategies, such as trend-following, or by
reversion strategies, such as statistical arbitrage. This argument is
supported by the observation that the persistence of trends, corresponding to
the quadratic term in the quartic potential, has gradually disappeared over
the decades [11].

There is a wide variety of stochastic processes such as (1) that have been
studied in finance [46]. Different assets are modeled by different processes,
which are often complex and require the calibration of many parameters to fit
the historical data. What is currently missing is a unified theory that
determines from first principles, which stochastic process with which
parameters should be used to model a given asset. Our approach suggests such a
principle: these processes must arise from the dimensional reduction of a
higher-dimensional system near its critical point, at which it is described by
a renormalizable (self-consistent) quantum field theory. This key criterium of
renormalizability leaves us with only a small set of universal field theories,
and therefore of allowed stochastic processes.

More precisely, at this critical point, the system is characterized by a
divergent correlation length and by critical coefficients, which are universal
in the sense that - in the case of a trivial network topology - they depend
only on two parameters: the fractal dimension $D$ of the network, and the
symmetry group $O(N)$. We have used the methodology of field theory and the
renormalization group to relate these critical exponents to observable
autocorrelations and Hurst exponents of financial market returns. In
particular, the second Hurst exponent follows from the anomalous scaling
dimension $\eta$ and the dynamic critical exponent $z$ (33):

$H_{2}={\kappa\over 2}\ \ \ \text{with}\ \ \ \kappa={2-\eta\over z}$

This allows us to measure $\kappa$, and from its value infer the fractal
dimension of the presumed underlying social network of investors.

As an illustrative example, we have applied this procedure to the simplest
case: “Model A” on a network with the topology of $D$-dimensional Euclidean
space. Consistency with observation then yields the following “predictions”
for the fractal dimension $D$ of the network and for the correlation time
$\tau$ in financial markets (31):

$D\approx 3\ \ ,\ \ \tau\geq\ \text{length of the economic cycle}.$

This can in principle explain the scaling observed in financial markets at
short and very long time scales, and the small magnitude of market auto-
correlations. Encouragingly, $D$ is within the range of fractal dimensions
typically found in social networks [45].

On the other hand, by comparing the two-point function of this model with the
observed auto-correlations of financial market returns at intermediate time
scales, we have also seen that the universal properties of model A on a
network with Euclidean topology do not explain the trends observed in real
markets over time horizons from 1 month to 1 year. This leaves us with the
task of refining the model, such that it can explain the observed interplay of
trends and reversion also on intermediate time scales. Some natural
refinements are:

  1. 1.

Beyond “Model A”: one may try to model the asset allocations of investors by
other degrees of freedom than spins, e.g., by a binary fluid in the case of
two assets, leading to different universality classes. More generally, the
models “A-J” of critical dynamics and their generalizations provide a rich
laboratory of systems [13, 14]. The universal properties of some of them
indeed include two-point functions that switch between positive and negative
auto-correlations, as observed in fig. 6 (left).

  2. 2.

Nontrivial network topologies: we do expect the social network of investors to
be much more connected than a hypercubic lattice, for which the average
distance between two points is very long. Candidates for social networks
include scale-free networks [34], which are characterized by a “degree
distribution” $p(k)\sim k^{-\gamma}$ of nodes with $k$ neighbors. It is
currently not clear, which of these networks can be part of renormalizable
field theories (except for the planar diagrams of large-N field theory [35],
on which the renormalization group flow has been studied [47]). An important
next step is thus to classify their universality classes. Apart from the
dimension $D$ and the symmetry group $O(N)$, they may also depend on
parameters such as $\gamma$.

  3. 3.

Approach to the critical point: it is also conceivable that the presumed
underlying social network of investors is not exactly at the fixed point
$g=g^{*}$. In this case, we may be able to understand the oscillations of the
two-point function in fig. 6 (left) as non-universal phenomena that arise
during the approach to the critical point.

In all three cases, it will be fascinating to search for the right candidate
in analogy with the example of section 4, by using the different models of
critical dynamics to predict the scaling behavior of market returns, and by
then ruling out those models that are inconsistent with empirical observation.
Further work in this direction is underway.

## Acknowledgements

I would like to thank Uwe Täuber for very helpful remarks and explanations on
dynamic scaling, and Wolfgang Breymann for encouragement, valuable
discussions, and for pointing out the relation to self-organized criticality.
I also thank Didier Sornette for an inspiring conversation and Michel
Dacorogna for comments on the manuscript. This research is supported by grant
no. CRSK-2 190659 from the Swiss National Science Foundation.

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##  6 Appendix: Brief Review of Empirical Results

In [11], we have empirically observed the interplay of trends and reversion in
financial markets. The analysis is based on historical daily log-returns of a
set of 24 futures contracts [44]. This set is diversified across four asset
classes (equity indices, interest rates, currencies, commodities), three
regions (Americas, Europe, Asia) and three commodity sectors (energy, metals,
agriculture). We used futures returns, instead of the underlying market
returns, because futures returns are guaranteed to be marked-to-market daily.
This is a brief suammary of the approach and results. For detailed
explanations and references, we refer to [11].

###  6.1 Trend Definition

For each of the 24 markets $i$, we consider 30 years of daily price data
$P_{i}(t)$ from 1990 to 2019. We define normalized daily log-returns
$R_{i}(t)$:

$R_{i}(t)={r_{i}(t)\over\sigma_{i}}\ ,\ \ \ r_{i}(t)=\ln{P_{i}(t)\over
P_{i}(t-1)}\ ,\ \ \ \sigma_{i}^{2}=\text{var}(r_{i})\ ,\ \ \
\mu_{i}=\text{mean}(r_{i})\ ,$ (34)

where the long-term daily risk premium $\mu_{i}$ and the long-term daily
standard deviation $\sigma_{i}$ of a market $i$ are measured over the whole
30-year period. For each market and on each day, we compute 10 different trend
strengths on 10 different time scales $T_{k}$ with

$T_{k}=2^{k}\ \text{business days with}\ k\in\\{1,2,3,...,10\\}$

This represents time horizons of approximately 2 days to 4 years. At a given
point in time, a market may well be in an up-trend on one scale, but on a
down-trend on another scale. For a given horizon $T$, we define the trend
strength $\phi_{i,T}(t)$ of a given market $i$ at the close of trading of day
$t\in Z$ as a weighted average of past daily log-returns of that market (i.e.,
on or before day $t$), in excess of the long-term risk premium in that market:

$\phi_{i,T}(t)=\sum_{n=0}^{\infty}w_{T}(n)\cdot\hat{R}_{i}(t-n)\ \ \
\text{with}\ \ \ \hat{R}_{i}(t-n)=R_{i}(t-n)-{\mu_{i}\over\sigma_{i}},$

where $w_{T}(n)$ is a weight function for the time scale $T$ (to avoid any
biases, in the out-of-sample cross-validation of our results, the risk premia
were estimated only from the training samples, excluding the validation
samples).

The weight function $w_{T}(n)$ is always normalized such that the trend
strength $\phi_{i,T}$ has standard deviation 1. Assuming that market returns
on different days are independent from each other (which is true to high
accuracy), this implies:

$\sum_{n=0}^{\infty}w_{T}^{2}(n)=1.$

With this normalization, $\phi_{i,T}$ can be regarded as the statistical
significance of the trend. E.g., $\phi_{i,T}=2$ represents a highly
significant up-trend, while $\phi_{i,T}=-0.5$ represents a weakly significant
down-trend. This normalization makes all trend strengths comparable with each
other, and thus allows us to aggregate across different markets and time
scales.

The simplest weight function would be a step function (fig. 1, dotted line).
In this case, the trend strength $\phi_{i,T}$ is just proportional to the
average log-return over the past $T$ days. A wedge-like weight function (fig.
1, solid line) would correspond to a moving log-price average crossover. The
advantages and disadvantages of different weight functions are disussed in
[11]. For our study, we used two such definitions of the trend strength
$\psi_{T}$ and $\phi_{T}$:

$\displaystyle\psi_{T}:$ $\displaystyle\tilde{w}_{T}(n)=M_{T}\ e^{-2n/T}\ \ \
\text{with normalization}\ \ M_{T}=\sqrt{1-e^{-4/T}},$ (35)
$\displaystyle\phi_{T}:$ $\displaystyle
w_{T}(n)=N_{T}\cdot(n+1)\cdot\exp(-{2n\over T})\ \ \ \text{with}\ \ \
N_{T}={(1-e^{-4/T})^{2}\over\sqrt{1-e^{-8/T}}}.$ (36)

The ”average lookback period” of these trend strengths, i.e., the expectation
value $E[n+1]$ of the number of days we look back (where ”today”, i.e. $n=0$,
counts as a 1-day lookback), is $T/2$ for $\psi$ and $T$ for $\phi$. Actually,
for a given horizon $T$, the precise definition of the trend strengths does
not matter much - they all yield quite similar results in our regression
analysis, as long as the weight function rises gradually, decays gradually,
and the average lookback period is the same. However, (35,36) were best-suited
for our study, because they

  * •

have only a single free parameter, the horizon $T$ (to avoid overfitting
historical data)

  * •

can be computed recursively (which yields a Langevin equation for continuous
time)

(36) also has the advantage of reducing trading costs. It was originally
introduced by the author in 2008 at Syndex Capital Management, and has been
used to replicate Managed Futures indices as part of a UCITS fund from
2010-2014.

Figure 9: Our trend strength is defined as a weighted sum of past log-returns.
The grey area shows the weight function used in this paper, compared with
three standard alternatives. All four weight functions shown here have the
same average lookback period.

###  6.2 Prediction of next-day Return

The results shown in fig. 1 are aggregated across 30 years of daily returns
for each market, across all 24 markets, and across different time scales. This
aggregation was necessary in order to achieve results that are statistically
highly significant. In particular, for shorter time windows, or without
aggregating across markets, the observed patterns cannot be detected reliably.
This may explain why they have not been reported already a decade ago.

Guided by the patterns observed in the graphs in fig. 1, we have performed a
nonlinear regression of the next-day normalized log-return $R_{i}(t+1)$ (34)
against a polynomial of the current trend strength $\phi_{i,k}(t)$ (36) across
all markets $i$ and time scales $T=2^{k}$:

$R_{i}(t+1)=a+b\cdot\phi_{i,k}(t)+c\cdot\phi_{i,k}^{3}(t)+\epsilon_{i}(t+1),$
(37)

where $\epsilon$ represents random noise, and $a$ measures the average
normalized risk premium $\mu_{i}/\sigma_{i}$ across all assets. These risk
premia have not been not the focus of [11]. Instead, we have concentrated on
determining the coefficients $b,c,$ which measure how the expected return of
an asset varies in time. We interpret $b$ as the persistence of trends, and
$c$ as the strength of trend reversion. $d$ and the higher-order terms were
not statistically significant. The results of the overall regression analysis
are shown in table 2.

Table 2 | Regression with linear and cubic terms
---|---
Coefficient |  Value |  Error |  t-statistics
$a$ |  $1.33$% |  $\pm 0.41$% |  $3.3$
$b$ |  $1.29$% |  $\pm 0.43$% |  $3.0$
$c$ |  $-0.62$% |  $\pm 0.23$% |  $2.7$
R-squared |  Single time scales | Aggregated across time scales
$R^{2}$ |  $1.31$ bp |  $4.91$ bp
$R^{2}_{adj}$ |  $1.03$ bp |  $3.98$ bp

The coefficients $a,b,c$ and the R-squared’s are very small, but they must be
so by nature: if the patterns were much stronger, traders would immediately
see and exploit them and achieve stellar Sharpe ratios. In fact, it is shown
in [11] that the estimated coefficients are just strong enough to explain the
Sharpe ratios of order 1 that the Managed Futures (“CTA”) industry has
historically achieved.

[11] also reports further analyses, in which $b$ and $c$ are allowed to have a
simple polynomial dependence on the log of the trend’s time horizon
$k\equiv\log_{2}T$:

$R_{i}(t+1)=\alpha_{i}+b(k)\cdot\phi_{i,k}(t)+c(k)\cdot\phi_{i,k}^{3}(t)+\epsilon_{i}(t+1).$
(38)

Within the limits of statistical significance, we find that $b$ and $c$ are
universal, i.e., the same for all assets. While the strength of reversion $c$
seems to be approximately constant, we find that the persistence of trends $b$
depends on the time horizon of the trend. Within the range of time scales
considered here, and averaged over the past 30 years, it can be approximated
by a parabolic function of the log of the time scale:

$c\sim-0.6\%\ \ \ ,\ \ \ b(k)\sim 2.0\%\cdot\Big{(}1-{(k-k_{0})^{2}\over\Delta
k^{2}}\Big{)}\ \ \ \text{with}\ \ \ k_{0}\sim 6\ ,\ \ \ \Delta k\sim 5.$ (39)

This implies that trends may only be stable if the log of the time horizon is
within the range $k_{0}\pm\Delta k$, corresponding to time scales from a few
days to several years. The parameters $k_{0}$ and $\Delta k$ are also
universal. By bootstrapping and cross-validation, we have found that all four
parameters in (39) are statistically highly significant out-of-sample.

###  6.3 Statistical analysis

Given the small values of the coefficients, and the huge amount of noise in
financial market data, we had to pay great attention to a careful analysis of
the statistical significance of the results. In particular, since market
returns cannot be assumed to be independent, identically distributed normal
variables, we cannot trust the usual estimates of the t-statistics, adjusted
R-squared’s, and other statistics. Instead, the test statistics shown in table
2 have been measured empirically as follows:

  * •

The standard errors of the coefficients and their t-statistics were computed
by bootstrapping, using 5000 bootstrapping samples. Regression on each new
sample of days yielded the distribution of the 5000 regression coefficients
$b$ and $c$ shown in fig. 1 (upper right), from which the estimation errors in
table 2 were read off. These actual errors are 3-4 times as large as the
errors that standard regression tools reported. Nevertheless, the regression
coefficients are still statistically highly significant.

  * •

The adjusted R-squared was computed by 15-fold cross validation. The resulting
out-of-sample R-squared is reported as $R^{2}_{adj}$ in table 2. The actual
out-of-sample correction $R^{2}-R^{2}_{adj}$ is about 3 times as big as what
standard regression tools reported. Nevertheless, the remaining $R^{2}_{adj}$
is still highly significant despite being very small, and in fact agrees with
the one that is achieved in real trading.

  * •

Table 2 also reports ${R}^{2}$ and ${R}^{2}_{adj}$ ”aggregated across time
scales”. Those are based on using the equally-weighted mean of the 10 trend
strengths on each day to predict the next-day return for each market. I.e., we
combine the 10 different trend factors into a single one, which has a higher
predictive power than each single factor by itself.

The regression results of table 2 and further regression analyses in [11]
confirm and quantify our conclusions from fig. 1. In particular, we see that
the values of $b$ and $c$ \- although very small - are statistically highly
significant, despite the fact that market returns are neither normally
distributed, nor independent, nor identically distributed.