Application of non-equilibrium statisticalmechanics to the analysis of problems in

financial markets and economy

Andrey Sokolov

Submitted in total fulfilment of the requirements

of the degree of Doctor of Philosophy

School of Physics

The University of Melbourne

September, 2014

Produced on archival quality paper

Abstract

This thesis contributes to a growing body of work in the emerging inter-

disciplinary field of econophysics, where tools and techniques of statistical

mechanics and other branches of physics are applied to problems in economics

and finance. The thesis examines the following four topics: 1) money flows in

the interbank networks, 2) wealth distributions in multi-agent exchange sys-

tems, 3) variability and dynamics of the foreign exchange markets via lattice

gauge theories, 4) memory loss and patterns of change in Abelian sandpiles

via hidden Markov models.

In economic and financial systems, interactions between the agents oc-

cur on a network, whose properties depend on the system in question. Such

networks are not static but rather dynamic in character, since the links com-

prising the network frequently depend on the actions of the agents as they

interact with one another. Moreover, the links of the network typically repre-

sent flows, e.g. money flows in financial networks, which further complicates

their study. These issues are addressed in the thesis in the case of the financial

networks of money flows between Australian banks.

In most advanced economies, including Australia, high-value transactions

in the banking system are settled in real time via the so called real-time gross

settlement systems, which are controlled by the central banks. Such systems

have been introduced in many countries in the last ten to twenty years in

order to diminish the liquidity risk in the banking system. As such systems

are computerized, the information pertaining to all transactions, including

their source, destination, and value, is recorded by the central banks and can

be used to investigate the properties and dynamics of the interbank flows.

The flows of payments in the interbank networks are not homogeneous but

possess an intricate structure that reflects the nature of various payments. In-

deed, some of the payments recorded by the central bank represent overnight

loans extended from one bank to another and the return payments made on

the following day. The work undertaken in the thesis examines the flows of

overnight loans and the flows of other payments separately. It is shown that

these two kinds of flows are not entirely independent. The flows of overnight

loans appear to counteract the imbalances in the bank’s reserve accounts cre-

ated by the flows of other payments. This is in accord with the dynamics

existing in the interbank money market, where the bank’s with the surplus of

reserves lend the surplus to the banks with depleted reserves.

Agent-based models are gaining popularity in the efforts to understand

economic and financial systems and their dynamics. The strength of these

models lies in the fact that they require the formulation of local rules of interac-

tion only without specifying the global constraints on the system’s behaviour,

which are often poorly understood. The simulations that use agent-based

models reveal the emergent behaviour of the economic and financial systems

that arises as a result of the collective actions of the agents following individ-

ual rules. In particular, agent-based models have been used to address the

problem of the wealth and income distribution in the economies. The work

conducted in the thesis examines one such model, referred to as the giver

scheme, where the rule of exchange stipulates that the transfer amount from

the giver to the receiver is equal to a fixed fraction of the giver’s wealth.

The giver model of asset exchanges is examined in the thesis by means

of multi-agent simulations. The system rapidly evolves to a steady state, in

which the distribution of wealth does not vary with time, even though agents

continue to exchange wealth. This model is amenable to the analysis based on

the master equation, which balances the influx and outflow of agents at every

wealth value. The thesis presents an investigation of the master equation by

means of the Laplace transform. A novel technique for calculating the wealth

distribution in the steady state is introduced and used to investigate wealth

inequality.

The giver model is shown to exhibit two distinct regimes in the steady

state. One of them is reminiscent of exchanges that occur in the economy

where the agents typically exchange a small fraction of their wealth. The other

regime is more akin to gambling and is characterised by exchanges where most

of the giver’s wealth is lost, so that fortunes are made and lost frequently. The

first regime is characterised by relatively low inequality, whereas the second

one is prone to exhibit very large inequality.

In addition to the study of inequality, the thesis investigates applicability

of the Boltzmann entropy as a measure of disorder in the giver model. The

giver model represent a closed system with no sources or sinks of agents or

wealth, i.e. it is conservative and in many respects is similar to an ideal gas.

However, numerical simulations reveal that the Boltzmann entropy does not

evolve monotonically in the giver model and, therefore, is not a faithful mea-

sure of disorder. This paradox is resolved by observing that the exchange rules

of the giver model are not time reversible, i.e. in order to reverse the dynamics

of exchanges a quantitatively different rule of exchange is required.

The foreign exchange market is a complex dynamical system that provides

prodigious amount of financial data, which makes it an attractive subject of

research. One of the approaches to modeling it relies on the lattice gauge

theory, where the lattice is constructed by considering two or more currencies

and discretising time and the gauge refers to the arbitrage on the lattice. A

brief investigation of one such model is undertaken in the thesis. It is shown

that the model is unstable in most realistic situations and thus cannot be used

to model the market behaviour.

Despite wealth of data, the foreign exchange market is difficult to study,

since the internal structure of the market is not well known. The Abelian

sandpile model, a cellular automaton that exhibits self-organised criticality,

is used in the thesis as a toy model that captures some of the features of the

foreign exchange market. The model is used to study temporal correlations in

the observed behaviour and their relation to the underlying internal structure.

A technique based on the site occupancy numbers is proposed in the thesis.

It reveals that the loss of memory in the sandpile model occurs in two distinct

stages, the fast stage characterised by rapid loss of memory and the subsequent

slow stage, during which memory of the initial state is lost at a much reduced

pace. Both stages are shown to be roughly exponential and the scaling of the

time decay is investigated.

The temporal correlations in Abelian sandpiles are also investigated in-

dependently by means of hidden Markov models, which show exceptional ca-

pabilities in detecting patterns in sequences of data. Hidden Markov models

have not been applied to Abelian sandpiles despite their popularity in a broad

range of applications ranging from bioinformatics to speech recognition. It is

demonstrated in the thesis that hidden Markov models do detect patterns in

the temporal variability of avalanche size, consistent with the results based on

the occupancy numbers. However, the connection between these patterns and

the internal structure of the sandpile has not been established. A number of

promising directions to address this problem are proposed.

Declaration

This is to certify that:

1. This thesis comprises only my original work, except where indicated in

the preface.

2. Due acknowledgement has been made in the text to all other material

used.

3. The thesis is less than 100,000 words in length, exclusive of tables, bib-

liographies, and appendices.

Andrey Sokolov

Preface

The majority of the work presented in this thesis is my own. The details of

specific contributions are listed below.

• Chapter 1 is an original review of the key research topics in econophysics,

based on a number of publications quoted in the text.

• Chapter 2 is based very closely on the following paper, co-authored with

Rachel Webster, AndrewMelatos, and Tien Kieu. It uses the data kindly

provided by the Reserve Bank of Australia. The work is original and my

own, including the code I wrote for data analysis and visualisation. The

idea to use the Anderson-Darling test was suggested by an anonymous

referee. Pip Pattison and Andre Gygax provided advice on the network

analysis.

Sokolov, Webster, Melatos, and Kieu (2012)

Loan and nonloan flows in the Australian interbank network.

• Chapter 3 is based very closely on the following publication, co-authored

with Andrew Melatos and Tien Kieu. The work is a result of a close

collaboration with Andrew Melatos who suggested the idea to use the

master equation.

Sokolov, Melatos, and Kieu (2010b)

Laplace transform analysis of a multiplicative asset transfer model.

• Chapter 4 is based very closely on the following publication, co-authored

with Tien Kieu and Andrew Melatos. Tien Kieu provided the gauge

theory expertise for this work.

Sokolov, Kieu, and Melatos (2010a)

A note on the theory of fast money flow dynamics.

• Chapter 5 is based on a paper submitted to Physica A and co-authored

with Andrew Melatos, Tien Kieu, and Rachel Webster. It uses an

Abelian sandpile simulator that I wrote. The idea to use hidden Markov

models can be traced to Hyam Rubinstein.

• Chapter 6 is original work.

Acknowledgements

In 2009, The School of Physics at the University of Melbourne offered a Ph.D.

scholarship in econophysics generously funded by the Portland House Foun-

dation. Tien Kieu, who was Director of Research at the The Portland House

Research Group at the time, played an instrumental role in making the fund-

ing available. The scholarship in econophysics was enthusiastically supported

by Rachel Webster and Andrew Melatos, who along with Tien Kieu became

my supervisors and collaborators. I am grateful to Tien, Rachel, and Andrew

for their support during my studies and research. I am eager to acknowledge

Andrew’s boundless enthusiasm and readiness to get involved, Rachel’s wise

advice, and Tien’s generosity with his time despite his pressing commitments

at The Portland House.

I thank the Portland House Foundation for funding this work.

The work on the interbank network would not be possible without Beth

Webster, who helped us get in touch with Chris Kent and Peter Gallagher

of the Reserve Bank of Australia. I thank the RBA and Peter Gallagher

in particular for making the interbank data available. During this project I

received valuable assistance from Andre Gygax and Pip Pattison.

I thank SIRCA for their timely assistance in obtaining financial data. I

gratefully acknowledge Hyam Rubenstein, Mark Joshi, Omar Foda, and Edda

Claus who advised me in the early stages of my studies. Thanks to David

Jamieson for his continued interest in my work. Many thanks to Sean and the

other members of the IT support group for dealing with my numerous requests.

And thanks to the students and faculty in the Astro group for having a lively

interest in my research.

Contents

Contents viii

List of Figures x

List of Tables xvi

1 Introduction 1

1.1 What is econophysics? . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Progress in econophysics . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Choosing the topics of research . . . . . . . . . . . . . . . . . . 10

1.4 Details of research projects . . . . . . . . . . . . . . . . . . . . 21

2 Loan and nonloan flows in the Australian interbank network 27

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.2 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.3 Overnight loans . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.4 Nonloans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.5 Loan and nonloan imbalances . . . . . . . . . . . . . . . . . . . 38

2.6 Flow variability . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.7 Net flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

2.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3 Laplace transform analysis of a multiplicative asset transfer

model 59

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.2 Giver scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.3 Laplace transform of the master equation . . . . . . . . . . . . 63

3.4 Steady-state wealth distribution by Laplace inversion . . . . . . 66

3.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

viii

3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

3.7 Iterative procedure . . . . . . . . . . . . . . . . . . . . . . . . . 79

4 A note on the theory of fast money flow dynamics 81

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.2 Lattice gauge theory and fast money flow dynamics . . . . . . . 82

4.3 Analysis of the Euler-Lagrange equations . . . . . . . . . . . . 86

4.4 Revisiting the action . . . . . . . . . . . . . . . . . . . . . . . . 91

4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

5 Memory on multiple time-scales in an Abelian sandpile 95

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.2 Abelian sandpiles . . . . . . . . . . . . . . . . . . . . . . . . . . 97

5.3 Site occupancy fraction distributions . . . . . . . . . . . . . . . 100

5.4 Short- and long-term memory in site occupancy . . . . . . . . . 101

5.5 Hidden Markov model . . . . . . . . . . . . . . . . . . . . . . . 104

5.6 HMM analysis of long-term memory . . . . . . . . . . . . . . . 106

5.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

6 Conclusion 111

6.1 The interbank network . . . . . . . . . . . . . . . . . . . . . . . 111

6.2 Wealth distributions . . . . . . . . . . . . . . . . . . . . . . . . 114

6.3 Critique of fast money flow theory . . . . . . . . . . . . . . . . 117

6.4 On memory in sandpiles . . . . . . . . . . . . . . . . . . . . . . 117

Bibliography 121

List of Figures

2.1 The distribution of transaction values v (in Australian dollars) on

a logarithmic scale, with bin size ∆ log10 v = 0.1; the vertical axis

is the number of transactions per bin. Components of the Gaussian

mixture model are indicated by the dashed curves; the solid curve

is the sum of the two components. The dotted histogram shows the

relative contribution of transactions at different values to the total

value (to compute the dotted histogram we multiply the number

of transactions in a bin by their value). . . . . . . . . . . . . . . . 35

2.2 Hypothetical interest rate rh versus value of the first leg of the

transaction pairs detected by our algorithm, with no restrictions

on value or interest rate. The dotted rectangle contains the trans-

actions that we identify as overnight loans. The least-squares fit is

shown with a solid red line. . . . . . . . . . . . . . . . . . . . . . 36

2.3 (a) The distribution of loan values v on a logarithmic scale. The

vertical axis is the number of loans per bin for bin size ∆ log10 v =

0.25. The dotted line is the same distribution multiplied by the

value corresponding to each bin (in arbitrary units). The date of

the first leg of the loans is indicated. (b) The distribution of loan

interest rates rh. The vertical axis is the number of loans per bin

for bin size ∆rh = 0.01. The date of the first leg of the loans is

indicated. The mean and standard deviation are 6.25% and 0.08%

on Monday (19-02-2007), and 6.26% and 0.07% on the other days. 37

x

2.4 The distribution of nonloan transaction values of the six largest

banks for Monday through Thursday (from left to right); the banks

are selected by the combined value of incoming and outgoing trans-

actions over the entire week. Black and red histograms correspond

to incoming (bank is the destination) and outgoing (bank is the

source) transactions; red histograms are filled in to improve visi-

bility. The banks’ anonymous labels, the combined daily value of

the incoming and outgoing transactions, and the daily imbalance

(incoming minus outgoing) are quoted at the top left of each panel

(in units of A$109). The horizontal axis is the logarithm of value

in A$. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.5 Left: loan imbalance ∆l vs nonloan imbalance ∆v for individual

banks and days of the week (in units of A$109). Right: the absolute

value of loan imbalance |∆l| vs nonloan total value (incoming plus

outgoing transactions) for individual banks and days of the week.

Thursday data are marked with crosses. . . . . . . . . . . . . . . 41

2.6 Nonloan flow value pairs on one day (horizontal axis) and the next

(vertical axis). Only flows present on both days are considered.

Flows that do not change lie on the diagonal (red dotted line).

The solid line is the weighted orthogonal least squares fit to the

scatter diagram; the weights have been defined to emphasize points

corresponding to large flows. . . . . . . . . . . . . . . . . . . . . . 43

2.7 As for Figure 2.6 but for loan flows. . . . . . . . . . . . . . . . . . 43

2.8 Loan flow values versus nonloan flow values combined over four

days. Triangles correspond to loan flows with three or more trans-

actions per flow. The solid line is the orthogonal least squares fit

to the scatter diagram; the weighting is the same as in Figure 2.6. 46

2.9 The distribution of values of net nonloan flows (black histogram)

on a logarithmic scale with bin size ∆ log10 v = 0.1. The compo-

nents of the Gaussian mixture model are indicated with the dashed

curves; the solid curve is the sum of the two components. Net loan

flows are overplotted in red. The vertical axis counts the number

of net flows per bin. . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2.10 (a) Degree distribution of the net nonloan flow networks (for conve-

nience, in-degrees are positive and out-degrees are negative). The

total value of the net flows corresponding to the specific degrees

is shown with red dots (the log of value in A$109 is indicated on

the right vertical axis). (b) Degree distribution of the net non-

loan flows when the degree data for all four days are aggregated

(in-degrees are circles; out-degrees are triangles). . . . . . . . . . . 49

2.11 (a) Same as Figure 2.10a, but for the net loan flow networks. (b)

Same as Figure 2.10b, but for the net loan flows. . . . . . . . . . . 50

2.12 Network of net nonloan flows on Tuesday, 20-02-2007. White (grey)

nodes represent negative (positive) imbalances. The bank labels

are indicated for each node. The size of the nodes and the thick-

ness of the edges are proportional to the logarithm of value of the

imbalances and the net flows respectively. . . . . . . . . . . . . . 52

2.13 Networks of daily net nonloan flows for D, AV, BP, T, W, BA, AH,

AF, U, AP, P, A, BG. All the other nodes and the flows to and

from them are combined in a single new node called “others”. The

size of the nodes and the thickness of the edges are proportional to

the logarithm of value of the imbalances and the net flows respec-

tively. The value of the flows and the imbalances can be gauged

by referencing a network shown in the middle, where the values of

the flows are indicated in units of A$1 billion. . . . . . . . . . . . 54

2.14 Networks of daily net loan flows. The same nodes as in Fig-

ures 2.13a–2.13d are used. The scale of the loan flows, the im-

balances, and the positions of the nodes are the same as those

used for the nonloan flows in Figures 2.13a–2.13d to simplify vi-

sual comparison. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.1 The Laplace transform g(z) for f = 0.1 obtained by solving (3.4) it-

eratively. (a) Re[g(z)] (top left panel), Im[g(z)] (bottom left panel),

and |g(z)| (right panel) versus r along the real axis (dashed curve),

the imaginary axis (solid curve), and the line inclined at θ = 60◦

to the real axis (dotted curve). The variables r and θ are defined

by z = reiθ. (b) A view of the real (top) and imaginary (bottom)

parts of g(z) (values above 1 and below −1 have been cut off). . . 65

3.2 The steady-state wealth probability distribution function ps(w) ob-

tained by inverting the Laplace transform g(z) for the following

values of the transfer fraction: f = 0.5 (bold solid curve), 0.25

(dash-dot curve), 0.1 (dotted curve), 0.05 (dashed curve), and 0.025

(thin solid curve). . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.3 The steady-state wealth probability distribution function ps(w) ob-

tained by inverting the Laplace transform g(z) for the following

values of the transfer fraction: f = 0.5 (bold solid curve), 0.6

(dash-dot curve), 0.7 (dotted curve), 0.8 (dashed curve), and 0.9

(thin solid curve). . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.4 The population distribution n(w), shown with crosses, as a function

of wealth w, measured in ficticious monetary units (m.u.) used in

the agent-based simulations. The distribution is computed as the

number of agents in every unit wealth interval after 100 steps in

the simulation of the giver scheme with total number of agents

N = 4 × 105 and transfer parameter (a) f = 0.95 and (b) f =

0.05. The initial distribution is uniform in the wealth interval (a)

[0, 100m.u.] and (b) [0, 500m.u.]. The corresponding solution of

the steady-state master equation for the same f is shown with a

solid curve, with ps(w) scaled to conform with the definition of

n(w) according to Nps(w/〈w〉)/〈w〉 where 〈w〉 is the mean wealth.

Both the agent-based simulations and the master equation predict

oscillations in the wealth distribution in (a) but not in (b). . . . . 69

3.5 (a) Boltzmann entropy Ss of the steady-state distribution as a

function of the variance σ2s = f/(1 − f). The critical values

σ2s = 0.062, 1, and 5.098, corresponding to f = 0.058, 0.5, and

0.836 respectively, are indicated with the dotted lines. (b) Entropy

as a function of time for the initial distribution given by (3.12) with

f = 0.058, computed from the multi-agent simulation of the giver

scheme. For the simulation, the distribution (3.12) was scaled up

to give N = 337123 agents in 0 ≤ w ≤ 1421. To compute the en-

tropy, the population distribution produced by the simulation was

normalized to a probability distribution with unit mean. . . . . . 72

3.6 (a) The limiting probability distribution ξ(w), shown with crosses,

obtained from one realization of the asymmetric random process (3.13)

for f = 0.05 after 106 iterations. The mean is 0.9998 and the vari-

ance is 0.0519 (cf. 0.0526 theoretically from the master equation).

The steady-state distribution ps(w) for the same f obtained by

Laplace inversion is shown as a solid curve for comparison. (b)

The first 1000 values of {wi}. . . . . . . . . . . . . . . . . . . . . . 74

3.7 Gini coefficient of the steady-state distribution ps(w) as a function

of the variance σ2s = f/(1− f). . . . . . . . . . . . . . . . . . . . 76

4.1 Re-creation of Ilinski’s solution of Eqs. (4.9–4.11) given on page 169

of Ilinski (2001) for α1 = 1.5, α2 = 10, C0 = 0, and the initial

conditions: η(0) = 0.2, υ(0) = 0, ρ(0) = 0.5. The factor α1 in

Eq. (4.10) is replaced with unity to match Ilinski’s Euler-Lagrange

equations. The displayed quantities are as follows: ρ− 1/2 (solid),

υ + η (dashed), η (dot-dashed), υ (dotted). . . . . . . . . . . . . . 84

4.2 The solution of Eqs. (4.9–4.11) for the same parameters and initial

conditions as in Fig. 4.1. The factor α1 in Eq. (4.10) is restored. . 85

4.3 The solution of Eqs. (4.9–4.11) for the same parameters and initial

conditions as in Fig. 4.2, except α1 = 0. . . . . . . . . . . . . . . . 87

4.4 The solution of Eqs. (4.9–4.11) for the same parameters and initial

conditions as in Fig. 4.2, except with C0 = 0.1 instead of C0 = 0.

The curves are coded as in Figs. 4.1 and 4.2. . . . . . . . . . . . . 89

4.5 As for Fig. 4.4, with C0 = −0.1. . . . . . . . . . . . . . . . . . . . . 90

4.6 The trading volume V (bold solid curve) and the return R (bold

dot-dashed curve) for the same parameters as in Fig. 4.2. For

comparison, we also display ρ− 1/2 (thin solid curve) and η (thin

dot-dashed curve). . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

4.7 Lattice diagram for the intra-day foreign exchange trading in two

currencies. Interest rates are ignored. . . . . . . . . . . . . . . . . 92

5.1 Probability density function p(fk) of the fraction fk of sites with

charge k = 0, 1, 2, 3 (left to right) for two sandpile simulations on

a square grid with N = 32 (blue) and N = 64 (red), based on

samples consisting of 103N2 steps in the recurrent regime. The

curves show the normal distributions for the corresponding values

of the mean and standard deviation; they agree well with the data.

The first 5N2 steps of the simulations are discarded to eliminate

transient configurations. . . . . . . . . . . . . . . . . . . . . . . . 100

5.2 Site occupancy analysis of an Abelian sandpile. (a) Ensemble-

averaged site occupancy fractions 〈δk〉 versus the time delay t− t0,where δk is the fraction of sites with charge k = 0, 1, 2, 3 in the

absolute difference matrix Dij = |zij(t) − zij(t0)|. The ensemble-

averaged curves are based on 104 trials (one representative trial

is shown in black). The standard error of the ensemble-averaged

values, which is ∼ 0.001, is not shown. The dashed lines represent

the expected values dk for t → ∞ and N = 32. (b) Approach of

〈δk〉 to the expected values dk shown on a logarithmic scale. . . . 103

5.3 Hidden Markov analysis of an Abelian sandpile. (a) Emission prob-

abilities Bij from hidden state i = 0 (red) and hidden state i = 1

(blue) to observed states j = 0, 1, 2 versus the averaging time-scale

Ta for the 32×32 sandpile. A discrete HMM with two hidden states

and three observed states is used. Observed states are defined by

binning the avalanche size into high, medium, and low bins, such

that there are equal number of samples in each bin for each aver-

aging interval. (b) Results of a control experiment where the same

input data is shuffled randomly before being fed into the HMM. . 107

List of Tables

2.1 The number of transactions (volume) and their total value (in units

of A$109) for each day. . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.2 Mean 〈u〉, variance σ2u, and mixing proportion P of the Gaussian

mixture components shown in Figure 2.1 (u = log10 v, where v is

value). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.3 Statistics of the overnight loans identified by our algorithm: the

number of loans (volume), the total value of the first leg of the loans

(in units of A$109), and the fraction of the total value of the loans

(first legs only) with respect to the total value of all transactions

on a given date. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.4 Loan and nonloan imbalances for the six largest banks (in units of

A$109). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.5 Mean 〈u〉, variance σ2u, and mixing proportion P of the Gaussian

mixture components appearing in Figure 2.9 (u = log10 v). . . . . . 46

5.1 Mean values and standard deviations for the samples shown in

Figure 5.1. The bottom row gives the analytical estimates of prob-

abilities pk, obtained in the limit of an infinitely large sandpile. . . 101

5.2 Transition probabilities Aij (left) between the hidden states i = 0, 1

(H0,H1) and emission probabilities Bij (right) from the hidden

states to observed states j = 0, 1, 2 (O0, O1, O2) for the averaging

time-scale Ta = 32. . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

xvi

Chapter 1

Introduction

1.1 What is econophysics?

Econophysics was established in 1995 as an interdisciplinary science with close

association to physics on one hand and economics/finance on the other. It is

conducted by physicists who apply data analysis and multi-agent simulations

to various problems in economics and finance.

The advent of econophysics coincides with availability of large arrays of

financial/economic data due to the implementation of various computer sys-

tems for tracking transactional data, which itself is part of “big data” ex-

plosion (Sagiroglu and Sinanc, 2013). For instance, such data is recorded on

various stock exchanges such as NYSE and NASDAQ, while Thomson Reuters

and Bloomberg provide a wealth of financial information on prices of stocks,

derivatives, foreign exchange, and other securities.

In 1995, a group of physicists gathered in Kolkata to discuss results of their

research on several topics traditionally associated with economics and finance

rather than physics. At this conference, Stanley coined the term econophysics

to refer to the research in economics and finance conducted by physicists us-

ing various techniques borrowed from statistical mechanics and other relevant

disciplines in physics. It is similar in etymology to geophysics and biophysics,

which are concerned with understanding geological and biological phenom-

ena from the physics-oriented point of view using the methods developed in

relevant fields of physics such as solid state physics, hydrodynamics, thermo-

dynamics, statistical mechanics, and quantum mechanics.

The difference between physics and economics is wider than between physics

and, say, biology, for one cannot apply the laws of physics to economic systems

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1. Introduction

directly. Therefore, the emphasis of the studies at present is empirical, where

many techniques developed in physics for analysing the data can be applied

to the economic systems directly. Econophysics is different from economet-

rics, which is a branch of economics also concerned with analysing data, since

physicists analyse data in order to attempt to uncover the underlying laws

governing the behaviour of the economic and financial systems, whereas the

econometrist views data in terms of already established economic theories, i.e.

economics is not empirically based and mostly interprets economic data in

terms of axiomatically postulated theories.

Economic and financial systems typically involve a large number of inter-

acting agents and are complex by nature. Even when the agents act according

to a set of simple rules, the systems demonstrate emergent properties char-

acteristic of various complex chaotic systems encountered in physics, such as

weather. It is natural to address such systems with multi-agent simulations,

which are frequently invoked in the studies of various complex systems. So,

econophysics emerges as a multi-disciplinary science that studies economic

and financial systems empirically and via multi-agent simulations by means

of various techniques developed by physicists for analysing natural systems.

Several factors contributed to the emergence of econophysics in the 1990s,

such as data availability, widespread use of computers in research, and poor

record of the standard economic theory in making accurate quantitative pre-

dictions. Arguably, the most important factor is the availability of economics

and financial data, which became possible thanks to the introduction of various

computer systems in stock (and other) exchanges, foreign exchange market,

private and central banking, and governments. At present, most financial

transactions are conducted using computers, which record and store trans-

actional data and make it easily available for analysis. Most of this data is

proprietary and is not publicly available, but some of it is available or can be

acquired upon request. Stock exchanges, where shares of public companies

are traded, provide the most comprehensive source of data, since each trans-

action must be reported (both its volume, number of shares, and the price

are thus made available). The price of various securities, such as bonds and

derivatives, and foreign exchange is recorded in real time and is available for

future analysis. Interbank transactions in many countries are conducted via

computer platforms provided by the central banks, where detailed records of

each high-value interbank transaction are stored (see for instance Gallagher

et al. (2010)). These data are not publicly available but have been made

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1. Introduction

available in some form to various research groups.

Another important reason that contributed to the advent of econophysics

is the inability of the traditional economic theory to make accurate quantita-

tive predictions. Most economic analysis at present is conducted on the basis

of a theory usually referred to as neoclassical economics, which is also domi-

nant in economics education as evidenced by most standard textbooks, while

other economic theories are marginalised (see Keen (2011) for a critical anal-

ysis). Neoclassical economics puts emphasis on exchange of goods between

various agents in the economy. It makes the assumption that the market is

in equilibrium, where the supply of goods matches the demand, and that the

economic agents act rationally and pursue the goal of maximising their util-

ity. The same assumptions are made in modeling the financial system, while

the banking system is usually assumed to play an auxiliary role in facilitating

exchanges. These assumptions are challenged by heterodox economic theories

such as Post-Keynesian economics and by econophysics (Keen, 2011), where

economic and financial systems are typically found in non-equilibrium states

and agents may act irrationally. The economy and finance are treated as dy-

namical systems and the actions of the economic agents are not necessarily

directed towards utility maximisation since their judgement is based at best

on incomplete data even if they act rationally.

Even though most research in econophysics occurred in the last twenty

years, some results, which are treated as part of modern econophysics, date

back to the turn of the twentieth century. In a pioneering study, economist

Vilfredo Pareto (1848–1923) examined data on wealth and income distribu-

tion from different countries and time periods and found that the number of

persons as a function of wealth/income follows the power-law distribution,

also referred to as Pareto distribution when applied to economic systems (see

(Richmond et al., 2006) for a historical account). The empirical fact that it

was not a Gaussian was unexpected; it suggested that the social laws may be

fundamentally different from physical laws known at the time, e.g. thermody-

namics, where bell-shaped curves predominated.

Another pioneering contribution was delivered by mathematician Louis

Bachelier (1870–1946) who in his PhD thesis made use of a Gaussian random

walk, which is a common stochastic process also known as Brownian walk,

to describe the evolution of stock option prices (Bachelier (2011)). Benoit

Mandelbrot (1924–2010) investigated price changes in some markets and found

that they do not follow a Gaussian distribution, with many more price changes

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1. Introduction

in excess of what might be expected from a Gaussian distribution (Mandelbrot,

1967). The distribution of price changes exhibits what is known as fat tails

characteristic of power-law distributions.

There is a methodological overlap between econophysics and some hetero-

dox economic theories and approaches that use dynamical models or multi-

agent simulations to model the economy. Heterodox economics encompasses a

large number of diverse economic theories such as Austrian economics, in-

stitutional economics, Marxian economics, Post-Keynesian economics, and

many others (see Keen (2011) for a brief review). Various branches of post-

Keynesian economics are of particular interest in the context of macroeco-

nomics, for they devote particular attention to monetary economics by mod-

eling money and banking and their effect on the economy explicitly.

The assumption frequently made by these theories is that money is en-

dogenous, i.e. it is produced by the banking system in response to prevailing

economic conditions in the market. In this view, money and credit become

important independent variables in describing the state of the economy via

a set of differential equations. An equilibrium is possible but may be unsta-

ble and there could be multiple equilibria, or the system could be in a state

of stable perpetual oscillations. Using dynamical equations for modeling the

economy may be described as the top-down approach. In contrast, multi-agent

simulations start with the micro level by establishing the rules that govern the

individual behaviour of the agents. Here, the agents can be persons, groups of

people, companies, etc. The emergent properties of the system on the macro

level arise as a result of action of the agents.

There are also certain connections between econophysics and financial en-

gineering. Financial engineering provides mathematical (analytical) and com-

putational support to the finance industry in the form of price assessment of

various securities, risk assessment, and technical analysis for predicting price

changes. In particular, the Black-Scholes option pricing model introduced in

Black and Scholes (1973) has gained significant prominence in finance indus-

try; similar statistical models have been applied to other securities (Joshi,

2003). Just as Bachelier’s model, Black-Scholes’ model relies on Brownian

motion to describe stochastic properties of the price changes. However, this

assumption is inconsistent with the observed statistical properties of price

changes where fat tails are typical, and therefore the model ignores rare event

that can nonetheless have a significant impact on the market due to large price

changes Nassim (2007). Despite strong criticism expressed in Haug and Taleb

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1. Introduction

(2011), Black-Scholes model is widely used in academic research.

Risk analysis is an important part of an investment strategy and also

plays a significant role in banking, where limited reserves make the banks

vulnerable to external shocks. A commonly used measure of financial risk is

the so-called value-at-risk measure, which is widely used in banking and has

been analysed by econophysicists (Bouchaud and Potters, 2003). Technical

analysis is used in financial trading of stocks and other securities and tries

to predict the changes in the price of a security based on the past behaviour

by utilising various techniques such as trend-following (momentum) and mean

reversion (Taylor and Allen, 1992), which are of interest to econophysicists,

e.g. Tanaka-Yamawaki and Tokuoka (2007).

1.2 Progress in econophysics

Econophysics celebrated its 15th anniversary in 2010 and a large number of

reviews of its recent progress has been written in the past few years. Some of

these reviews are outlined below.

In a recent interview (Gangopadhyay, 2013), Stanley tells that the name

econophysics first appeared at the STATPHYS–Kolkata II conference held in

Kolkata in 1995, which was probably the first conference with significant focus

on econophysics. Stanley coined the term econophysics to refer to research

based on the methodology adopted in physics and largely done by physicists

who address problems in economics and finance, in the same sense biophysics

or geophysics refer to the application of physics to problems in biology and

geology. Stanley highlights the difference in culture between physicists, who

adopt an empirical approach guided by the data, and economists, who follow

a mathematical (axiomatic) approach guided by theories.

In a thesis piece published in Nature (Buchanan, 2013), Buchanan lists

a number of lasting contributions made by econophysicists to economics and

finance. Econophysics is partially responsible for: 1) Precise empirical facts

about financial markets such as power-law distribution of large price move-

ments and its self-similar scaling properties (Borland et al., 2005). 2) Instruc-

tive links between markets and other natural phenomena such as Omori law for

earthquake aftershocks and market behaviour following a large crash (Weber

et al., 2007), which suggests that market dynamics may be governed by some

general dynamic principles not specific to financial markets. 3) More realistic

models of markets represented as an ecology of interactive adaptive agents,

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1. Introduction

giving rise to fat-tailed distributions (see Hommes (2002) for a review), 4) New

qualitative features of markets, one of which is that market dynamics depends

on the diversity of participants’ strategic behaviour (Johnson et al., 2012).

Markets operate smoothly when the agents use many diverse strategies but

break down if the strategies become similar. In view of recent international

financial crises, the focus of many econophysics studies on market instability

acquires new significance. Buchanan lists the following findings: 1) A market,

where participants compete with one another by increasing leverage, is inher-

ently unstable (Thurner et al., 2012). 2) Widespread risk sharing in a network

of financial institutions raises systemic risk since a contagion can spread too

easily through an over-connected network (Battiston et al., 2012a). 3) Market

efficiency as it becomes more complete in terms of the range and availability

of financial instruments brings with it inherent market instability (Caccioli

and Marsili, 2010). Buchanan concludes by mentioning DebtRank (Battiston

et al., 2012b; Thurner and Poledna, 2013), a network measure developed by

econophysicists that assesses the fragility of a particular institution embedded

in a complex network of mutual financial dependencies encompassing a large

number of financial institutions.

The importance of collaboration between hard sciences like physics and

social (soft) sciences, which includes sociology and economics, is emphasised

in the review by Chakrabarti and Chakraborti (2010), which introduces a

special issue of Science and Culture devoted to econophysics and published in

2010 (commemorating 15 years of econophysics). In particular, with regard

to fostering these collaborative links the review points out such centres of

interdisciplinary research as the Indian Statistical Institute (India), which fo-

cuses on application of statistics to natural and social sciences, and the Santa

Fe Institute (US), which focuses on the study of complex systems, including

physical, evolutionary, and social systems. The authors view econophysics as a

new research field, whose objective is to turn economics into a natural science.

According to the authors, important developments in econophysics include the

following: 1) Deviation from Gaussian statistics in financial markets (Stan-

ley, Mantegna, Bouchaud, Farmer) building on the earlier work of Mandelbrot

and Fama. 2) Correlations among different stocks/sectors (Mantegna, Marsili,

Kertesz, Kaski, Iori, Sinha). 3) Income and wealth distributions and the rel-

evant kinetic models (Redner, Souma, Yakovenko, Chakrabarti, Chakraborti,

Richmond, Patriarca, Toscani). 4) Behavioural models of market bubbles and

crashes (Bouchaud, Lux, Stauffer, Gallegati, Sornette, Kaizoji). 5) Learning

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1. Introduction

in multi-agent game models and minority games in particular (Zhang, Marsili,

Savit, Kaski).

Bouchaud says in an opinion essay (Bouchaud, 2008) that compared with

physics the quantitative success of economics has been disappointing as it

demonstrates recurrent inability to predict and avert crises. Economists have

no framework for understanding ‘wild’ markets, which are not efficient and

where judgement errors get amplified leading to crashes. These characteris-

tics of financial markets are better addressed in the framework of complex

chaotic systems, the study of which is largely neglected by economists who

instead adopt a number of axiomatic assumptions such as the rationality of

economic agents who maximise their profits, the ‘invisible hand’ of the market

that gives the best outcome for society as a whole even as agents pursue their

own interests, and market efficiency, which states that market prices perfectly

reflect all known information about the assets. According to Bouchaud, sta-

tistical regularities should emerge in the behaviour of large populations in the

same way statistical laws of ideal gases emerge as a result of chaotic motion

of molecules.

An interesting perspective on the progress of econophysics is given by

Roehner (2010) who emphasises the importance of studying social phenom-

ena and intereactions in general rather than pure economic phenomena. In

view of the recent trend in physics to establish and promote departments

focusing on complex systems (colloids, polymers, sandpiles, traffic jams, neu-

ral networks, colonies of social insects, stock markets, financial derivatives),

Roehner reminds that the success of physics and chemistry has been his-

torically predicated on the search for simplicity rather than complexity, by

breaking down and simplifying complex systems and phenomena to isolate

the fundamental laws. In physics, spurious exogenous effects are eliminated

or isolated as much as possible in order to get to the fundamental properties.

According to Roehner the same approach may prove fruitful when examining

economic data, where the big players (corporations, media, etc) that can affect

the system on a macro level need to be taken into account.

Mathematical economist Rossner who works in nonlinear economic dy-

namics (using chaos and complexity theory) provides another perspective on

econophysics (Rosser Jr, 2006). Even though classical physics has long been

influencing ideas and models used in economics, Rossner notes that the inter-

action between economics and biology must not be underestimated, especially

in the context of evolutionary theory, as is witnessed by the appearance of

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1. Introduction

ecological economics (Daly and Farley, 2010). Similar sentiment is echoed by

Carbone et al. (2007), who introduces a special issue of Physica A devoted to

papers from the conference on “Application of Physics in Financial Analysis”

(2006). It is noted that early developments in economics (utility maximi-

sation, equilibrium) were strongly influenced by Newtonian physics, but the

interaction between physics and economics ceased around the 1930s with the

development of non-equilibrium thermodynamics. Since the 1930s, scientists

were engaged in the study of many-body systems in far from equilibrium con-

ditions, which frequently demonstrate emergent behaviour characteristic of

complex adaptive systems, where nonlinear interactions between a large num-

ber of individual particles (or agents) engender a dynamic behaviour of the

system on the macroscopic level, i.e. macroscopic behaviour emerges from mi-

croscopic interactions. Traditional models used in economics are not suitable

for representing these dynamic features, which led to proliferation of new dis-

ciplines such as econophysics, behavioural economics, evolutionary economics,

and neuroeconomics.

A number of heterodox economists sympathetic to econophysics discuss

its strengths and weaknesses in (Gallegati et al., 2006). Among the most

important contributions of econophysics, the authors see 1) the work on fat

tails of asset price changes, which has been established as a universal feature of

financial markets, and 2) the discovery that the empirical correlation matrix of

price changes of different assets and classes of assets is poorly determined, i.e.

dominated by noise (Laloux et al., 1999). The latter finding undermines the

capital asset pricing model (Sharpe, 1964), which is still highly regarded by

many economists, while the former is in conflict with the assumptions of the

Black-Scholes model of option pricing (Black and Scholes, 1973), a popular tool

used frequently in financial trading. The critique of econophysics by Gallegati

et al. (2006) is mostly aimed at models of income and wealth distributions,

most of which assume that money is conserved (i.e. money is akin to energy

in physical systems). The authors points out that this assumption is not valid

in the modern economy where the amount of money grows over time and

production plays as important a role as exchange.

In response to the issues raised by Gallegati et al. (2006), McCauley (2006)

points out that conservation laws in physics follow from space-time symme-

tries, but only one such symmetry (related to arbitrage) has been identified

in empirical analysis of price distributions, while money is not a conserved

quantity in the modern economy. One of the main points McCauley makes is

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1. Introduction

that financial markets are non-stationary in the statistical sense (McCauley,

2008). Therefore any model that relies on stationarity, be it general equi-

librium theory or an agent-based model, is necessarily going to be invalid

over the relevant time scales. The difference between economics and econo-

physics from a philosophical and methodological perspective is considered by

Schinckus (2010), who argues in favour of the independence between eco-

nomics and econophysics. Economics adopts certain a priori postulates about

the real world, e.g. about the rationality of the economic agents or Gaussian

distribution of financial series returns, without adequate empirical verification.

A balanced view on the relationship between economics and econophysics

is taken in (Chen and Li, 2012), which introduces a number of papers from

Econophysics Colloquium 2010 held in Taipei and published in a special issue

of International Review of Financial Analysis. The authors document a long

history of connection between economics and classical physics, e.g. energy and

oscillations in physics versus utility and business cycles in economics. These

developments come under the rubric classical econophysics, which mostly con-

cerns 19th and early 20th centuries, or earlier. The authors trace the mul-

tidisciplinary character of modern econophysics, which they divide into four

main streams: (1) nonlinear dynamics (macroeconomic dynamics, non-linear

time series), (2) distributions of income, firm size, asset returns, etc, (3) so-

cial interactions (agent-based computational economics, Ising model, master

equation, cellular automata, kinetic and percolation model, minority games),

(4) complex networks and their statistical properties. The review sees econo-

physics as extending beyond statistical physics and finance, the focus of most

work to date.

A comprehensive review of the evolution of econophysics is presented in

(Ghosh and Chakrabarti, 2014). It provides succinct biographies of main

contributors to classical and modern econophysics, detailing their major con-

tributions. Both physicists and economists whose work can be classified as

econophysics are included; in all, over one hundred biographies. Besides in-

dividual scientists and research groups they lead, there are several institutes

that devote significant part of their activities to econophysics. These include:

Indian Statistical Institute, The Santa Fe Institute, and Institute for New Eco-

nomic Thinking. Other important centres of econophysics research include:

Boston University (Eugene Stanley), Saha Institute of Nuclear Physics (Bikas

Chakrabarti), Ecole Central Paris (Anirban Chakraborti, Damien Challet),

University of Maryland (Victor Yakovenko), University of Palermo (Rosario

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1. Introduction

Mantegna), Kyoto University (Hideaki Aoyama), Leiden University (Diego

Garlaschelli), University of Houston (Joseph McCauley), and several others.

The review lists eighteen books on econophysics and eleven special journal

issues devoted to econophysics.

Progressive institutionalisation of econophysics is documented in (Gingras

and Schinckus, 2012) which uses bibliometric methods. The authors identi-

fied 242 econophysics papers published in the period from 1980 to 2008; 147

papers are published in refereed journals, 90 of which appeared in Physica

A, 27 in European Physical Journal B, and the others in Physical Review E,

Quantitative Finance, and Journal of Economic Behaviour & Organization.

Unsurprisingly, most of these journals have at least one prominent econo-

physicist in their board of editors. Besides these journals, the authors find the

following two economics journals citing econophysics papers: Journal of Eco-

nomic Dynamics and Control and Macroeconomic Dynamics, which are both

macroeconomics journals. A related study (Ghosh and Chakrabarti, 2014)

uses Google Scholar to gather statistics of papers that mention econophysics

to illustrate graphically the growth of econophysics since the term’s inception

in 1995. The study reveals that the growth in the number of papers has been

roughly linear with only a few papers in 1998 and nearly 1000 papers in 2012.

A similar analysis of papers that mention sociophysics also reveals a roughly

linear growth from about 2000, with over 200 papers published in 2012.

The first dedicated conference in econophysics was held in 1998 in Bu-

dapest. Since then, a number of econophysics conferences have been held in

various locations. There are two prominent conferences on econophysics held

annually: Econophysics Colloquium (first held in 2005 at The Australian Na-

tional University) and Econophysics-Kolkata (first held also in 2005). Closely

related to econophysics is The Workshop on Economics with Heterogeneous

Interacting Agents (first held in 1996 at University of Ancona). The Inter-

national Conference on Statistical Physics, which also runs annually, includes

econophysics as one of its topics.

1.3 Choosing the topics of research

The economy is a complex phenomenon which involves agents from countries

and companies down to individuals who are engaged in social and economic

interactions via production and consumption of goods and services over an

intricate network. The great majority of exchanges in the economy are facili-

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1. Introduction

tated by the use of money. The origin and nature of money are controversial

and poorly understood despite its overwhelming presence in everyday life.

Money is usually described in economic texts through its properties as a unit

of account, a store of value, and a medium of exchange. Through credit, it

is also a driving force behind innovation and other entrepreneurial activity as

has been long recognised (Schumpeter, 1934). The origin of money is tradi-

tionally traced to primitive societies where money appears as some commonly

accepted commodity that simplifies direct exchange of goods (barter). This

widespread view is challenged by anthropological and historical studies that

find little evidence to corroborate it (Graeber, 2011). The view that equates

ancient money with precious metal coins emphasises the role of money as

the medium of exchange and store of value, whereas an alternative view that

money is based on credit and debt depends on the function of money as the

unit of account, which provides a record of mutual debt obligations between

members of an economic network. In modern economy, money is inextricably

linked with credit, whose role as a driver for innovation (and production) is

overshadowed by its more recent role as a driver of consumption and financial

speculation.

Money in modern economies is created by the central banks and the com-

mercial banking system via their credit facilities. Understanding its origin

and dynamics is essential for understanding modern economic and financial

systems. Unfortunately, a commonly accepted framework for understanding

and modeling money in econophysics has not been developed yet. Physical

concepts such as energy, which play crucial role in physics, fail to serve as a

viable theoretical representation of money that captures its endogenous char-

acter. Instead, research efforts have been concentrated on certain aspects and

manifestations of money via simplified modeling or data analysis of particular

segments of the economy where data is available. The data for such research

can come potentially from the banking system via tracking the majority (by

value) of monetary transfers in the economy and the records of credit is-

suance. Some indirect data are delivered by the foreign exchange market via

the exchange rate of various currencies, which is one of the richest sources of

data at present and can provide a valuable insight into the nature of money,

since exchange rates depend on the monetary policies of individual countries.

Dynamical models and multi-agent simulations of these systems provides ad-

ditional ways of addressing these issues.

Data availability determines in part the range of topics selected for re-

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1. Introduction

search in this thesis. The foreign exchange data is provided by SIRCA via

the Thomson Reuters Tick History database (TRTH). SIRCA was established

in 1997 by a number of Australian and New Zealand universities (including

the University of Melbourne) as a not-for-profit company to provide finan-

cial data for research at the member institutions. SIRCA provides access to

TRTH database, which is one of the premier sources of data used by most lead-

ing financial institutions. A sample of high-value transactional data between

Australian banks has been kindly provided by the Reserve Bank of Australia

(RBA), which is the central bank in Australia. RBA administers a real-time

gross settlement systems (called The Reserve Bank Information and Transfer

System or RITS) for settling high-value transactions between banks in Aus-

tralia. The data settled in RITS mostly originate from the SWIFT payment

delivery system and Austraclear securities settlement system and therefore it

mostly concerns financial transactions (purchase of various securities and for-

eign exchange). Small-value transactions of mostly non-financial nature are

settled by the banks on a daily basis and do not involve RITS. The dataset

provided by the Reserve Bank includes value, source, and destination of each

transfer and thus provides an exciting opportunity to investigate the dynam-

ics of monetary flows in the interbank network in Australia, as well as the

statistical and topological properties of the network itself.

Networks

The study of networks in mathematics is known as graph theory. A graph

is a representation of a set of objects (nodes or vertices) some of which are

connected by links (edges). This subject has a long history and originates

with a paper written by Leonhard Euler (1707–1783) on the Seven Bridges

of Konigsberg (1736). Networks are frequently employed in computer science

to represent communication routes, data structures, flow of computations,

and so on. They are also used extensively in linguistics, chemistry, physics,

ecology, and many other disciplines. The significance of networks in social and

economic contexts is well recognised through early examples of case studies in

sociology prior to the nineties and subsequent modeling of network formation

and games on networks in the last twenty years (Jackson and Zenou, 2013). In

an economic context, important questions concerning networks that have been

addressed in research are 1) network formation and growth by using random

or strategic assembly rules and 2) games on a network, which study the effects

12

1. Introduction

of network structure on the interaction between the nodes (Jackson, 2008a;

Goyal, 2012).

The properties of networks can be characterised by the average distance

between the nodes (the distance is the minimal number of edges that connect

two nodes), and maximum distance, which is also called the diameter of the

network, the degree distribution, where the degree of a node gives the number

of its neighbours, the assortativity, which measures the similarity (in terms

of the degree) between the neighbouring nodes, and a range of other charac-

teristics such as node clustering and centrality. Random networks, where the

degree of most nodes is close to the average degree of the network and the

overall degree distribution is binomial, played a significant role in the devel-

opment of graph theory (Bollobas, 1998). However, many real-world networks

frequently possess a small number of nodes with elevated degree. Many such

networks have been found to have a scale-free degree distribution (Caldarelli,

2007), i.e. the distribution is described by a power law without a characteristic

scale besides the upper cut-off. One of the earliest examples of such a network

was provided by the World Wide Web (Albert et al., 1999). Financial and

economic networks and in particular networks of transactional flows between

banks demonstrate similar characteristics. In these cases, the distribution of

distances is frequently peaked around small values, a phenomenon known as

the small-world effect.

If the links between the nodes of the network possess some directional

information (e.g. a web page has a hyperlink to another page, or one paper

cites another), the network is known as a directed network. In the context

of the banking networks, the edges correspond to transactional flows between

the banks, so that two edges pointing in the opposite directions are possible

between a single pair of banks. The banking networks are flow networks

with edges serving as conduits for payments (another important example of

a flow network is a transportation network). Flow networks are dynamic in

character; the flow characteristics are variable. Moreover, the structure of the

flow network can change with time when old nodes and edges are removed or

new ones are added.

In countries where the central bank operates a real-time gross settlement

(RTGS) system for settling large-value transactions between banks, the pay-

ments entered into RTGS are settled by the central bank by adjusting the

accounts of the participating banks held by the central bank. The member

banks are required to maintain a certain amount of funds in these accounts to

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1. Introduction

ensure real-time gross settlement. This contrasts with the deferred net settle-

ment applied to small payments made by individuals, which are tallied at the

end of a business day. Payment flows between banks cause the reserves of some

banks to increase, while those of the other banks decrease. Since RTGS oper-

ate in real time, these changes can be monitored by the central bank in real

time, which enables it if necessary to extend credit to banks whose reserves

are low, thereby improving the overall stability of the banking system.

Smooth functioning of RTGS is vital for the bank’s liquidity and systemic

stability of the banking system. It also has an effect on the conduct of mone-

tary policy by the central bank, since problems with liquidity in RTGS can lead

to rising interest rates in the money market and greater volatility and, on the

other hand, increased volatility on the money market may cause some banks

to stop submitting their payments through RTGS due to concerns over credit

risk. There has been a significant effort to analyse the properties of RTGS sys-

tems by means of standard neoclassical theoretical tools and also by means of

agent-based simulations. The latter have certain advantages over the former

as pointed out in (Alexandrova-Kabadjova et al., 2012), since 1) payment flows

do not follow any nice theoretical distribution, 2) each bank’s decision affects

the other banks so that a bank cannot solve its liquidity demand problem in

isolation, 3) the system design, which is becoming increasingly sophisticated,

strongly influences the participant’s incentives, 4) there is a high degree of

heterogeneity between the participants. Alexandrova-Kabadjova et al. (2012)

notes the lack of comprehensive empirical studies of these systems and the

participant’s behaviour in particular.

Banks whose reserves increase as a results of the flow of payments can

loan the excess in the overnight market to banks with depleted reserves at

the interest rate which is more favourable than the rate attached to their

accounts with the central bank. On the other hand, banks with depleted

reserves are anxious to restore them to a more comfortable level and therefore

seek funds in the overnight market. Thanks to overnight loans, the balance

of reserves in the banking system is restored. The flow of overnight loans

forms a network complementary to the flow network of other payments. This

creates an interesting causal relationship between the flows of loans and other

payments.

The topology of the interbank network is a significant factor in the overall

stability of the banking system, with important policy implications (Haldane

and May, 2011b). For instance, one of the recommendations is that capital

14

1. Introduction

and liquid asset requirements take into account the systemic importance of the

bank, characterised by its position in the network, with higher limits for banks

that are more interconnected and therefore can contribute to fast spread of a

financial contagion due to bad loans.

Agent-based simulations

A simulation is an approximate representation of a natural process or phe-

nomenon by means of a specific physical or computational model. Thanks

to computers, simulations have become an important part of the scientific

method, complementing deductive and inductive reasoning. Inductive reason-

ing uses experiments and observations to infer simple reproducible behaviours,

i.e. natural laws. Deductive reasoning starts with axioms (to be tested) or laws

(empirically verified) and uses logic to work out the physical consequences.

Simulations are employed when deductive reasoning alone is not sufficient to

derive the consequences that follow from the axioms and laws. Agent-based

simulations rely on the laws that govern the actions of individual autonomous

agents as they interact with one another and their environment. A cellular

automaton is a simplified example of an agent-based model, where the agents

have fixed positions on the grid and can only interact with their adjacent

neighbours, but even such a simple system can exhibit extremely complex

behaviour as exemplified by, for example, Conway’s game of life (Gardner,

1970).

Since the laws that govern the dynamics of social systems may be difficult

to infer from data, which are often sparse, agent-based simulations can play

an important role in sociology, psychology, politics, and economics. The pi-

oneering agent-based models used simple assumptions about the agents such

as zero-intelligence agents (Gode and Sunder, 1993) and swarm intelligence,

e.g. boids introduced by Reynolds (1987) to model bird flocks. Sugarscape

(Epstein, 1996) (see Gilbert and Troitzsch (2005) for an elementary introduc-

tion), which is used for simulating growing artificial societies, initiated another

set of agent-based models, where the agents demonstrate some goal-directed

behaviour. In the modern models, the agents can possess learning and plan-

ning and other advanced capabilities. Cooperation (e.g., in the prisoner’s

dilemma), reciprocity, prejudice and social influence have been investigated

via agent-based simulations (Axelrod, 1997). In sociophysics, agent-based

simulations have been used to address the universal features of group decision

15

1. Introduction

making, including democratic voting (Galam, 2012).

One of the well known applications of agent-based techniques in econo-

physics is the so-called minority game (Challet and Zhang, 1997), which is

based on the El Farol Bar problem (Arthur, 1994). Agents choose at each

step between two options, which can serve as a proxy for buying and selling

in a financial market, and those agents who adopted a less common option

(the minority view) gain, while the other agents lose. The agents have limited

memory that extends several steps into the past and they use adaptive strate-

gies from a limited pool of available strategies. The strategies are assessed

and selected based on their past performance. Under certain conditions, i.e.

when the number of agents is large enough compared to the number of avail-

able strategies, the asymptotic behaviour of the system depends strongly on

the initial conditions (Moro, 2004). The agents keep adapting their strate-

gies without ever settling on a particular strategy and no detailed balance is

observed. The minority game model is simple enough to address analytically

using techniques of statistical physics, but it does not explain some of the ob-

served facts about financial markets such as fat tails and volatility clustering

(the so-called stylised facts). The minority game model has become popular

in econophysics due to the fact that it shares many basic characteristics, such

as phase transitions, with other physical models.

The minority game in its simple form does not account for the stylised

facts mentioned previously, which requires more realistic agent-based models

of financial markets. For instance, Lux and Marchesi (1999) use two kinds

of agents: fundamentalists, who have a stabilising effect on the market, and

chartists, or noise traders, which can be either optimists or pessimists and

who are responsible for creating bubbles and crashes. The noise traders are

susceptible to herding behaviour. The agents can switch between these two

kinds depending on the market conditions; a random fluctuation of the mar-

ket price can cause many agents to become chartists, which increases volatility

thanks to an increased number of random buy and sell orders. Another no-

table agent-based model that simulates herding (Cont and Bouchaud, 2000)

is inspired by percolation theory, with clusters representing groups of agents

making the same decisions. This model and its subsequent variations explain

many features of financial markets; see Samanidou et al. (2007) for a review.

Other notable examples include the Santa Fe Artificial Stock Market model

(LeBaron, 2002), which uses heterogeneous agents who adapt their strategy

according to certain classifiers based on fundamentalist and chartist rules; the

16

1. Introduction

agent-based model for the stock market crash of October 19, 1987 (Kim and

Markowitz, 1989), which finds the cause of the crash in homogenisation of

algorithmic hedging strategies.

Asset exchange simulations

An asset exchange simulation is an agent-based simulation, where agents are

involved in the exchange of a certain asset, which can be identified with money,

for instance. The asset exchange simulations are of interest as they address

the question of income and wealth distribution and in particular wealth in-

equality characteristic of many human societies. Even though specific money

transfers among people occur for a variety of complex reasons, the aggregate

effect of these exchanges on the wealth distribution is simple enough, so that

simple exchange rules may be sufficient. Different rules of exchange can be

distinguished by their effect on the shape of the wealth distribution and the

degree of inequality.

The first multi-agent model of asset exchanges was developed in the 1980s

(Angle, 1986, 2002) in order to provide an explanation for the observed wealth

distributions, which can be closely approximated by the gamma distribution

(with the exception of the Pareto tail). In this model, the exchange amount

depends multiplicatively on the wealths of the agents. Various additive and

multiplicative models of asset exchanges were developed independently in (Is-

polatov et al., 1998); in the additive models, the amount exchanged does not

depend on the wealth of the agents. Besides multi-agent simulations, (Ispola-

tov et al., 1998) study the system analytically using a kinetic approach based

on the master equation, which balances the influx and outflow of agents at

specific wealths.

The model of Dragulescu and Yakovenko (2000) is conceptually the closest

to statistical mechanics of ideal gases. In this model, the exchange amount is

either fixed or equal to a random fraction of the total amount of money which

is conserved, the number of agents is fixed, and the pairing of agents in the

exchanges is completely random, provided that the agents have enough money

for an exchange. The behaviour of the agents in this model is reminiscent of

the behaviour of atoms in the ideal gas studied in the microcanonical ensem-

ble. Dragulescu and Yakovenko (2000) argue that the distribution of wealth

resulting after many additive exchanges must be given by the exponential

Boltzmann-Gibbs function, where the temperature parameter is related to

17

1. Introduction

the total wealth and the number of agents. Multi-agent simulations confirm

this supposition and also show that the Boltzmann entropy increases steadily

as expected in the ideal gas.

Chakraborti and Chakrabarti (2000) describe a multi-agent model of asset

exchanges in which agents are characterised by a propensity to save a certain

fraction of their wealth, while the rest of their wealth is made available for

random exchanges. This model is similar to the other multiplicative models

of asset exchange and the simulations based on this model yield in the steady-

state a distribution of wealth closely approximated by a gamma distribution.

Chatterjee et al. (2004) extended this model by using agents whose propensity

to save is assigned randomly from a uniform distribution. Once assigned, the

saving propensity of a given agent is fixed; agents with low propensity make

most of their wealth available for random exchanges, whereas agents with high

propensity save most of their wealth. Multi-agents simulations based on this

model yield a distribution of wealth which exhibits a power-law tail at high

wealths populated by agents with very high propensity to save who release

only a small fraction of their wealth for exchanges.

Foreign exchange

Foreign exchange markets are rich sources of data. However, the dynamics

of exchange rates is not well understood. The available data consist of the

indicative buy and sell prices offered by the so-called market makers (banks

and other financial institutions), while the records of actual transactions (their

value, source, and destination) are not available. Moreover, the structure of

the market is difficult to discover, since there is no central exchange where

transactions are processed and recorded. The total value of transactions on

the foreign exchange market is significant and plays a significant role in the

global economy.

Some insight into the structure and significance of the foreign exchange

market is provided by the Triennial Central Bank Survey prepared by the

Bank for International Settlement (BIS) every three years since 1989, with the

latest survey conducted in April 2013. According to the survey, the turnover

in foreign exchange markets averaged $5.3 trillion per day, of which swaps (for-

eign exchange transactions that reverse after a period of time) accounted for

$2.2 trillion per day and spots, which unlike swaps do not reverse, $2.0 trillion

per day. Reporting dealers accounted for 39% of turnover, smaller banks that

18

1. Introduction

are not participating in the survey as reporting dealers accounted for 24% of

turnover, pension funds, insurance companies, and the like contributed 11%,

while hedge funds and proprietary trading firms accounted for 11%. Trading

with non-financial customers (e.g. industrial firms) accounted for only 9% of

the turnover. The US dollar remains the dominant traded currency, followed

by the euro and Japanese yen. Most trading is done in the primary financial

centres such as London, New York, Singapore, and Tokyo. The survey found

that voice execution accounted for 43% of foreign exchange transactions, the

rest taken up by electronic execution, which includes single banks proprietary

trading systems, Thomson Reuters Matching and EBS platforms, as well as

other electronic communication networks such as Currenex, FX Connect, FX-

all, and Hotspot FX.

Patterns in foreign exchange series can be analysed in a number of ways by

means of the standard techniques of technical analysis routinely employed by

traders and by more exotic means such as artificial neural networks (ANN),

hidden Markov models (HMM), or dynamical models. One of the dynamical

model approaches was developed by Ilinski in (Ilinski, 2001) and relies on the

lattice gauge theory where the gauge is associated with arbitrage. On the

other hand, ANN and HMM do not rely on any underlying theory.

The model developed by Ilinski (2001) is based on the lattice-gauge theory,

where the gauge transformation is associated with the rescaling of currencies.

Indeed, whether the currency is expressed as a certain amount of dollars or

a certain amount of cents should not have any effect on the dynamics of the

exchange, i.e. invariance with respect to price dilation. The lattice represents

investors in 1) the two currencies in a given currency pair and 2) discrete time

steps. The curvature on the lattice depends on the arbitrage along elemen-

tary circular plaquettes (closed loops involving two consecutive time steps).

A similar model based on the arbitrage as a curvature of the gauge connection

was developed in (Young, 1999). The model can be extended to multiple cur-

rencies, or financial securities in general. The lattice geometry introduced by

Ilinski was used in (Dupoyet et al., 2010) to develop and perform numerical

simulations based on the Markov chain Monte Carlo technique, which yield the

distribution of price increments in agreement with the NASDAQ one-minute

price data over four orders of magnitude. Kostanjcar et al. (2011) derived the

discrete dynamics of asset price relations from the minimum arbitrage princi-

ple using Ilinki’s lattice geometry. Zhou and Xiao (2010) rewrite the model in

the form of a partial differential equation on a fibre bundle in covariant form.

19

1. Introduction

Sandpiles

The idea that the foreign exchange markets are hierarchically organised with

different traders contributing to the market dynamics on different time scales

(and possibly on different scale of trade value) leads to the notion of informa-

tion cascade in the market akin to the energy cascade observed in turbulence

Voit (2003). Similarly, the fact that local activity can precipitate a global

movement in the market in the right circumstances suggests that the behaviour

of avalanches in stochastic systems such as sandpiles may approximate some

of the processes occurring in the foreign exchange markets.

A sandpile is a cellular automaton that models some of the properties of

a pile of sand onto which grains of sand are dropped at random locations.

As grains accumulate, the local slope of the pile may become too steep to

support the grains, in which case an avalanche occurs, the grains in unstable

locations are redistributed in the pile until a new equilibrium position is found.

The sandpile is usually assumed to be located on a square grid, such that

the unstable grains at the edges of the pile can be assumed to drop off the

edges. Over the long term, the influx of grains due to random grain drops

is counterbalanced by the efflux of grains falling off the edges, so that the

sandpile enters stochastic equilibrium as soon as a certain number of grains is

deposited in the pile. Some drops that cause small avalanches do not result in

any efflux of grains from the pile, because the avalanches are contained inside

the pile. In this case, generally the internal stress of the sandpile configuration

increases, which means that a large avalanche is more likely to happen.

In terms of the relationship between the sandpiles and the foreign exchange

market, one can think of the grain drops as individual trades while avalanches

can be associated with the movements of the exchange rate. Then, a small

avalanche may be associated with a small random movement, while a large

avalanche is more indicative of a large systematic change in the exchange rate.

A sandpile simulation on a Sierpinski gasket has been used to represent market

behaviour (Ausloos et al., 2002; Ausloos, 2006).

A sandpile is a simplistic representation of the foreign exchange market,

but it is valuable in the following sense. The foreign exchange data available

for analysis consists of the time series of the exchange rate, which can be

easily converted into a series of the exchange rate differences, increments and

decrements. The time series of the exchange rate differences can be analysed in

order to discover any patterns in the series, for instance by applying a hidden

20

1. Introduction

Markov model to the series, which probabilistically associates an unobserved

(hidden) Markov chain to the observed series. However, whatever pattern is

discovered it is difficult to relate it to any structural properties of the market

since its structure is not available for analysis. On the other hand, the same

technique applied to a time series of avalanche power in a sandpile offers a

better chance of relating any pattern discovered by the HMM to the internal

structure of the sandpile, which is known precisely at all times.

1.4 Details of research projects

The chapters of the thesis are based on the four papers accepted (or submit-

ted) for publication in Physica A and The European Physical Journal B. The

following sections provide a brief introduction to the papers.

Australian interbank network

The research on the Australian interbank network is described in Chapter 2.

The data for the analysis of the interbank network originates from the real-

time gross settlement system operated by the Reserve Bank of Australia. It

covers five consecutive days in February 2007, when the financial markets were

relatively stable, and consists of records of all interbank transactions settled

through the RTGS (note that low-value transactions are settled on a net basis

daily and therefore are not present in the sample) including the amount, the

bank of the origin (the payer), and the bank of destination (the payee). The

dataset contains 123078 transactions with the combined value in excess of

A$848 billion.

Some of these transactions constitute overnight loans, which are payments

that are returned on the next day with the same amount plus some interest.

The interest can be expected to be close to the target rate set by the RBA,

which suggests a procedure to detect overnight loans and separate them from

the other payments. This procedure yields 897 overnight loans over the four

day period with the combined value of over A$42 billion (Friday is excluded

since the following business day is not present in the sample and, therefore,

the overnight loan detection procedure cannot be applied). Once loans and

other payments (nonloans) are separated, a detailed statistical analysis of the

two sets can be conducted. Furthermore, as the source and destination of

each loan and nonloan transaction are known, statistical properties of sent

21

1. Introduction

and received loans and nonloans can be analysed for each bank in the sample.

In particular, loan and nonloan daily imbalance, the difference between sent

and received payments, for each bank can be determined and the interplay

between loan and nonloan imbalances can be investigated.

The payments between the banks can be thought of as flows in the net-

work, whose vertices are banks. Typically, there are two flows going in the

opposite directions between large banks. The difference between them yields

the net flow between the banks. The flows can be measured by the number of

transactions or the total amount per day (in principle, intraday flows can be

similarly estimated, but precise time of transactions is not given in our sam-

ple). Naturally, the largest flows occur between big banks, whereas there are

no transactions and consequently no flows between many small banks. The

total number of nonloan flows is about 800 on a given day, out of 2970 possible

flows for the 55 banks present in the sample (there are about 470 net nonloan

flows per day). On the other hand, the number of loan flows is about 75 per

day (about 60 net loan flows per day). Daily variability of loan and nonloan

flows is investigated.

The properties of the network of net flows, both loan and nonloan, are

investigated by computing the degree distribution for in- and out-degree sepa-

rately, and the assortativity of the networks is estimated. Furthermore, visual

analysis of the topology of the net flows between the largest 12 banks is pre-

sented and the daily variability of the networks is discussed, with the view to

provide visual clues to the structure and stability of the banking network on

the basis of a limited data sample.

Asset transfers

In Section 3, a “giver scheme” of asset transfers is analysed. A large number

of agents exchange assets according to a rule which stipulates that the transfer

amount is proportional to the total amount of assets in possession of the giver

and is independent of the receiver. A numerical simulation of the giver scheme

reveals that the distribution of assets, i.e. the number of agents in a given asset

(or wealth) interval, quickly approaches a stable distribution regardless of the

initial distribution. Unfortunately, agent-based simulations are not sufficient

to accurately constrain the properties of the distribution of wealth, especially

in the tail of the distribution where the number of agents is small.

A novel procedure for computing the distribution of wealth in the steady

22

1. Introduction

state is presented in Chapter 3. In the steady state, the number of agents

arriving in each wealth interval is balanced by the number of agents departing

the interval. The distribution in the steady state can thus be analysed by

means of the master equation. The master equation for the giver scheme has

not been solved analytically, except for several special cases. In Chapter 3 a

new method to solve it numerically is presented. It relies on the Laplace trans-

form of the master equation, which gives a functional equation in the complex

plane that can be solved numerically by iterations. To obtain the wealth dis-

tribution, the numerical inverse Laplace transform has to be applied to the

solution of the functional equation. This technique is investigated for usabil-

ity for a wide range of the giver schemes and is applied to two representative

cases, which correspond to small and large constants of proportionality in the

exchange formula. The small value of the constant yields behaviour reminis-

cent of monetary exchanges, which correspond to buying and selling goods and

services in the economy. On the other hand, the large value is more appropri-

ate for monetary exchanges in gambling, where large sums of money change

hands frequently. Accurate numerical estimate of the steady-state distribu-

tion of wealth makes it possible to address the question of wealth inequality

in the giver scheme more precisely than it is possible to do with numerical

simulations.

The total amount of assets is conserved in the giver scheme and there are

no sources of sinks of assets, i.e. the system is conservative and the transfers

are similar to the exchange of energy in collisions between atoms in an ideal

gas. By analogy with the physical system, one expects the entropy, which

measures the level of disorder in the system, to increase until it reaches the

maximum when the agent-based simulation of asset transfer reaches the steady

state. The actual behaviour of the Boltzmann entropy in the giver scheme is

investigated by conducting numerical simulations of the asset transfer system.

In addition, a simple Markov chain random process that mimics some of

the features of the giver scheme is introduced. The Markov chain is discrete in

time and continuous in the state variable. It is asymmetric in the sense that

the increment is additive when it is positive and multiplicative otherwise. This

asymmetry corresponds to the asymmetry in the giver scheme where the loss

of the asset is proportional to the giver’s wealth while the gain can originate

from any agent in the whole population.

23

1. Introduction

Money flow dynamics

Chapter 4 is devoted to the critique of the gauge theory of arbitrage which

was introduced by Ilinski (2001) and applied to fast money flow dynamics.

The theory of fast money flow dynamics attempts to model the evolution of

currency exchange rates and stock prices on short, e.g. intraday, time scales.

It has been used to explain some of the heuristic trading rules, known as

technical analysis, that are used by professional traders in the equity and

foreign exchange markets. The study presented in chapter 4 demonstrates that

the choice of the input parameters used in Ilinski (2001) results in sinusoidal

oscillations of the exchange rate, in conflict with the results presented in Ilinski

(2001), because of an algebraic error in the derivation of the Euler-Lagrange

equations. It is also found that the dynamics predicted by the theory are

generally unstable in most realistic situations, with the exchange rate tending

to zero or infinity exponentially. Critical analysis of the action that governs

the dynamics of fast money flows is also given in chapter 4.

Temporal patterns in Abelian sandpiles

Abelian sandpile simulations on square two-dimensional grids of various sizes

are used to investigate memory effects in a sandpile. The objective is to see

if the observable pattern of “avalanches” in a slowly driven system with local

interactions like a sandpile can be related to its internal states. The sandpile

as a dynamical system can then provide a template, in the context of a toy

model, for analysing internal states and their time evolution in other complex

dynamical systems like the FX market where the internal states cannot be

observed.

A sandpile configuration (or state) at a given moment is represented as

a two-dimensional matrix of heights (or charges), which can also be thought

of as the number of grains in a given location in the sandpile. A grain drop

at a random location in the pile and a possible subsequent avalanche change

the configuration of the pile. However, most of the heights in the pile remain

unchanged after a single drop. As the number of drops and the avalanches

increases, the sandpile’s configuration gradually evolves away from the initial

configuration, which is assumed to be a recurrent one. In other words, the

sandpile is expected to gradually lose memory of the initial configuration.

Memory loss can be described by computing the difference between two

configurations, the initial configuration and a configuration at some later time,

24

1. Introduction

i.e. after a certain number of grain drops. The difference matrix represents

a detailed imprint of memory loss, which depends not only on the initial

configuration but also on the history of random grain drops. Since the latter

has nothing to do with the internal structure of the sandpile, the effect of

random grain drops can be diminished by computing the mean characteristics

of the difference matrix, such as the total number of sites with a given value

of the difference. The time dependence of such mean characteristics can then

be used as a convenient measure of memory loss. The effect of random drops

can be completely eliminated by combining a large number of simulations and

computing ensemble averaged mean quantities as a function of time delay from

the initial configuration in each simulation.

A complementary approach to the memory loss problem is to make use of

the hidden Markov modeling, which is effective at capturing temporal patterns

in the succession of observed states. In the case of Abelian sandpiles, many

properties can serve as observed states, e.g. the avalanche size (or power).

As the avalanche size at a particular moment in time depends on where a

randomly dropped grain lands in the pile, the randomness can be diminished

by using time averaged characteristics as the input for the HMM analysis. For

instance, the sequence of avalanche sizes can be averaged by computing the

mean of all avalanches that occurred during a contiguous sequence of grain

drops of certain fixed length, the averaging period. By comparing the results

of the HMM analysis based on the sequences obtained for different averaging

periods one can assess the effects of memory loss at different time-scales. These

results are described in chapter 5.

25

Chapter 2

Loan and nonloan flows in the

Australian interbank network

High-value transactions between banks in Australia are settled in the Reserve

Bank Information and Transfer System (RITS) administered by the Reserve

Bank of Australia. RITS operates on a real-time gross settlement (RTGS)

basis and settles payments and transfers sourced from the SWIFT payment

delivery system, the Austraclear securities settlement system, and the inter-

bank transactions entered directly into RITS. In this paper, we analyse a

dataset received from the Reserve Bank of Australia that includes all inter-

bank transactions settled in RITS on an RTGS basis during five consecutive

weekdays from 19 February 2007 inclusive, a week of relatively quiescent mar-

ket conditions. The source, destination, and value of each transaction are

known, which allows us to separate overnight loans from other transactions

(nonloans) and reconstruct monetary flows between banks for every day in

our sample. We conduct a novel analysis of the flow stability and examine

the connection between loan and nonloan flows. Our aim is to understand

the underlying causal mechanism connecting loan and nonloan flows. We find

that the imbalances in the banks’ exchange settlement funds resulting from

the daily flows of nonloan transactions are almost exactly counterbalanced by

the flows of overnight loans. The correlation coefficient between loan and non-

loan imbalances is about −0.9 on most days. Some flows that persist over two

consecutive days can be highly variable, but overall the flows are moderately

stable in value. The nonloan network is characterised by a large fraction of

persistent flows, whereas only half of the flows persist over any two consec-

utive days in the loan network. Moreover, we observe an unusual degree of

27

2. Loan and nonloan flows in the Australian interbank network

coherence between persistent loan flow values on Tuesday and Wednesday. We

probe static topological properties of the Australian interbank network and

find them consistent with those observed in other countries.

2.1 Introduction

Financial systems are characterised by a complex and dynamic network of

relationships between multiple agents. Network analysis offers a powerful way

to describe and understand the structure and evolution of these relationships;

background information can be found in Kolaczyk (2009), Jackson (2008b),

and Caldarelli (2007). The network structure plays an important role in de-

termining system stability in response to the spread of contagion, such as

epidemics in populations or liquidity stress in financial systems. The im-

portance of network studies in assessing stability and systemic risk has been

emphasised in Schweitzer et al. (2009) in the context of integrating economic

theory and complex systems research. Liquidity stress is of special interest in

banking networks. The topology of a banking network is recognised as one of

the key factors in system stability against external shocks and systemic risks

Haldane and May (2011a). In this respect, financial networks resemble eco-

logical networks. Ecological networks demonstrate robustness against shocks

by virtue of their continued survival and their network properties are thought

to make them more resilient against disturbances May et al. (2008). Often

they are disassortative in the sense that highly connected nodes tend to have

most of their connections with weakly connected nodes (see Newman (2003)

for details). Disassortativity and other network properties are often used to

judge stability of financial networks.

There has been an explosion in empirical interbank network studies in

the last years thanks largely to the introduction of electronic settlement sys-

tems. One of the first, reported in Boss et al. (2004), examines the Austrian

interbank market, which involves about 900 participating banks. The data

are drawn from the Austrian bank balance sheet database (MAUS) and the

major loan register (GKE) containing all high-value interbank loans above

e0.36× 106; smaller loans are estimated by means of local entropy maximisa-

tion. The authors construct a network representation of interbank payments

for ten quarterly periods from 1999 to 2003. They find that the network ex-

hibits small-world properties and is characterised by a power-law distribution

of degrees. Specifically, the degree distribution is approximated by a power

28

2. Loan and nonloan flows in the Australian interbank network

law with the exponent −2.01 for degrees & 40. This result, albeit with differ-

ent exponents, holds for the in- and out-degree distributions too (the exponent

is −3.1 for out-degrees and −1.7 for in-degrees). A recent study of transac-

tional data from the Austrian real-time interbank settlement system (ARTIS)

reported in Kyriakopoulos et al. (2009) demonstrates a strong dependence

of network topology on the time-scales of observation, with power-law tails

exhibiting steeper slopes when long time-scales are considered.

The network structure of transactions between Japanese banks, logged

by the Bank of Japan Financial Network system (BOJ-NET), is analysed in

Inaoka et al. (2004). The authors consider several monthly intervals of data

from June 2001 to December 2002 and construct monthly networks of inter-

bank links corresponding to 21 transactions or more, i.e. one or more trans-

action per business day on average. Truncating in this way eliminates about

200 out of 546 banks from the network. The resulting monthly networks have

a low connectivity of 3% and a scale-free cumulative distribution of degrees

with the exponent −1.1.

More than half a million overnight loans from the Italian electronic broker

market for interbank deposits (e-MID), covering the period from 1999 to 2002,

are analysed in De Masi et al. (2006). There are about 140 banks in the

network, connected by about 200 links. The degree distribution is found to

exhibit fat tails with power-law exponent 2.3 (2.7 for in-degrees and 2.15

for out-degrees), the network is disassortative, with smaller banks staying

on its periphery. In a related paper Iori et al. (2007), the authors make

use of the same dataset to uncover liquidity management strategies of the

participating banks, given the reserve requirement of 2% on the 23rd of each

month imposed by the central bank. Signed trading volumes are used as a

proxy for the liquidity strategies and their correlations are analysed. Two

distinct communities supporting the dichotomy in strategy are identified by

plotting the correlation matrix as a graph. The two communities are mainly

composed of large and small banks respectively. On average, small banks serve

as lenders and large banks as borrowers, but the strategies reversed in July

2001, when target interest rates in the Euro area stopped rising and started

to decrease. The authors also note that some mostly small banks tend to

maintain their reserves through the maintenance period. The evolution of the

network structure over the monthly maintenance period is examined in Iori

et al. (2008).

A study of the topology of the Fedwire network, a real-time gross settle-

29

2. Loan and nonloan flows in the Australian interbank network

ment (RTGS) system operated by the Federal Reserve System in the USA, is

reported in Soramaki et al. (2007). The study covers 62 days in the 1st quarter

of 2004, during which time Fedwire comprised more than 7500 participants

and settled 3.45× 105 payments daily with total value $1.3 trillion. It reveals

that Fedwire is a small-world network with low connectivity (0.3%), moderate

reciprocity (22%), and a densely connected sub-network of 25 banks responsi-

ble for the majority of payments. Both in- and out-degree distributions follow

a power law for degrees & 10 (exponent 2.15 for in-degrees and 2.11 for out-

degrees). The network is disassortative, with the correlation of out-degrees

equal to −0.31. The topology of overnight loans in the federal funds market

in the USA is examined in Bech and Atalay (2010), using a large dataset

spanning 2415 days from 1999 to 2006. It is revealed that the overnight loans

form a small-world network, which is sparse (connectivity 0.7%), disassorta-

tive (assortativity ranging from −0.06 to −0.28), and has low reciprocity of

6%. The reciprocity changes slowly with time and appears to follow the target

interest rate over the period of several years. A power law is the best fit for the

in-degree distribution, but the fit is only good for a limited range of degrees.

A negative binomial distribution, which requires two parameters rather than

one for a power law, fits the out-degree distribution best.

A comprehensive survey of studies of interbank networks is given in Imakubo

and Soejima (2010). The number of interbank markets being analysed con-

tinues to increase. For example, a study of the interbank exposures in Brazil

for the period from 2004 to 2006 was reported in Cajueiro and Tabak (2008).

A topological analysis of money market flows logged in the Danish large-value

payment system (Kronos) in 2006 was reported in Rørdam and Bech (2008),

where customer-driven transactions are compared with the bank-driven ones.

Empirical network studies have been used to guide the development of a net-

work model of the interbank market based on the interbank credit lending

relationships Li et al. (2010).

Establishing basic topological features of interbank networks is essential

for understanding these complex systems. Fundamentally, however, interbank

money markets are flow networks, in which links between the nodes correspond

to monetary flows. The dynamics of such flows has not been examined in depth

in previous studies, which mostly viewed interbank networks as static or slowly

varying. But the underlying flows are highly dynamic and complex. Moreover,

monetary flows are inhomogeneous; loan flows are fundamentally different

from the flows of other payments. Payments by the banks’ customers and the

30

2. Loan and nonloan flows in the Australian interbank network

banks themselves cause imbalances in the exchange settlement accounts of the

banks. For some banks, the incoming flows exceed the outgoing flows on any

given day; for other banks, the reverse is true. Banks with excess reserves lend

them in the overnight money market to banks with depleted reserves. This

creates interesting dynamics: payment flows cause imbalances, which in turn

drive compensating flows of loans. Understanding this dynamic relationship

is needed for advancing our ability to model interbank markets effectively.

In this paper, our objective is to define empirically the dynamics of inter-

bank monetary flows. Unlike most studies cited above, we aim to uncover the

fundamental causal relationship between the flows of overnight loans and other

payments. We choose to specialise in the Australian interbank market, where

we have privileged access to a high-quality dataset provided by the Reserve

Bank of Australia (RBA). Our dataset consists of transactions settled in the

period from 19 to 23 February 2007 in the Australian interbank market. We

separate overnight loans and other payments (which we call nonloans) using a

standard matching procedure. The loan and nonloan transactions settled on a

given day form the flow networks, which are the main target of our statistical

analysis. We compare the topology and variation of the loan and nonloan

networks and reveal the causal mechanism that ties them together. We inves-

tigate the dynamical stability of the system by testing how individual flows

vary from day to day. Basic network properties such as the degree distribution

and assortativity are examined as well.

2.2 Data

High-value transactions between Australian banks are settled via the Reserve

Bank Information and Transfer System (RITS) operated by the RBA since

1998 on an RTGS basis (Gallagher et al., 2010). The transactions are settled

continuously throughout the day by crediting and debiting the exchange set-

tlement accounts held by the RBA on behalf of the participating banks. The

banks’ exchange settlement accounts at the RBA are continuously monitored

to ensure liquidity, with provisions for intra-day borrowing via the intra-day

liquidity facility provided to the qualifying banks by the RBA. This obviates

the need for a monthly reserve cycle of the sort maintained by Italian banks

as discussed in Iori et al. (2008). The RITS is used as a feeder system for

31

2. Loan and nonloan flows in the Australian interbank network

Date Volume Value (A$109)

19-02-2007 19425 82.250620-02-2007 27164 206.102321-02-2007 24436 161.973322-02-2007 25721 212.135023-02-2007 26332 184.9202

Table 2.1: The number of transactions (volume) and their total value (in unitsof A$109) for each day.

transactions originating from SWIFT1 and Austraclear for executing foreign

exchange and securities transactions respectively. The member banks can also

enter transactions directly into RITS. The switch to real-time settlement in

1998 was an important reform which protects the payment system against

systemic risk, since transactions can only be settled if the paying banks pos-

sess sufficient funds in their exchange settlement accounts. At present, about

3.2 × 104 transactions are settled per day, with total value around A$168

billion.

The data comprise all interbank transfers processed on an RTGS basis by

the RBA during the week of 19 February 2007. During this period, 55 banks

participated in the RITS including the RBA. The dataset includes transfers

between the banks and the RBA, such as RBA’s intra-day repurchase agree-

ments and money market operations. The real bank names are obfuscated

(replaced with labels from A to BP) for privacy reasons, but the obfuscated

labels are consistent over the week. The transactions are grouped into separate

days, but the time stamp of each transaction is removed.

During the week in question, around 2.5 × 104 transactions were settled

per day, with the total value of all transactions rising above A$2 × 1011 on

Tuesday and Thursday. The number of transactions (volume2) and the total

value (the combined dollar amount of all transactions) for each day are given

in Table 2.1. Figure 2.1 shows the distribution of transaction values on a

logarithmic scale. Local peaks in the distribution correspond to round values.

The most pronounced peak occurs at A$106.

In terms of the number of transactions, the distribution consists of two

1Society for Worldwide Interbank Financial Telecommunication2The term “volume” is sometimes used to refer to the combined dollar amount of trans-

actions. In this paper, we only use the term “volume” to refer to the number of transactionsand “total value” to refer to the combined dollar amount. This usage follows the one adoptedby the RBA Gallagher et al. (2010).

32

2. Loan and nonloan flows in the Australian interbank network

Date Component 1 Component 2〈u〉 σ2u P 〈u〉 σ2u P

19-02-2007 4.00 1.12 0.81 6.68 0.68 0.1920-02-2007 3.55 0.72 0.43 5.73 1.49 0.5721-02-2007 3.66 0.86 0.55 5.86 1.43 0.4522-02-2007 3.87 1.01 0.68 6.42 1.07 0.3223-02-2007 3.82 0.87 0.61 6.12 1.19 0.39

Table 2.2: Mean 〈u〉, variance σ2u, and mixing proportion P of the Gaussianmixture components shown in Figure 2.1 (u = log10 v, where v is value).

approximately log-normal components, with lower-value transactions being

slightly more numerous. The standard entropy maximisation algorithm for a

Gaussian mixture model with two components (McLachlan and Peel, 2000)

produces a satisfactory fit with the parameters indicated in Table 2.2. The

lower- and higher-value components are typically centred around A$104 and

A$106 respectively. The high-value component is small on Monday (19-02-

2007) but increases noticeably on subsequent days, while the low-value com-

ponent diminishes. By value, however, the distribution is clearly dominated

by transactions above A$106, with the highest contribution from around A$2×108.

2.3 Overnight loans

The target interest rate of the RBA during the week of our sample was rt =

6.25% per annum. If the target rate is known, it is easy to extract the overnight

loans from the data by identifying reversing transactions on consecutive days.

A hypothetical interest rate can be computed for each reversing transaction

and compared with the target rate. For instance, suppose a transaction of

value v1 from bank A to bank B on day 1 reverses with value v2, from bank

B to bank A, on day 2. These transactions are candidates for the first and

second legs of an overnight loan from A to B. The hypothetical interest rate

for this pair of transactions is given by rh = 100% × 365 × (v2 − v1)/v1; note

that the quoted target rate is per annum. Since large banks participate in

many reversing transactions that can qualify as loans, we consider all possible

hypothetical pairs and prefer the one that gives rh closest to the target rate.

The algorithm for loan extraction is applied from Monday to Thursday; loans

issued on Friday cannot be processed since the next day is not available. A

33

2. Loan and nonloan flows in the Australian interbank network

Date Volume Value (A$109) Loan fraction

19-02-2007 185 7.50 9.12%20-02-2007 221 9.18 4.45%21-02-2007 226 11.08 6.84%22-02-2007 265 14.93 7.04%

Table 2.3: Statistics of the overnight loans identified by our algorithm: thenumber of loans (volume), the total value of the first leg of the loans (in unitsof A$109), and the fraction of the total value of the loans (first legs only) withrespect to the total value of all transactions on a given date.

similar procedure was pioneered by Furfine Furfine (2003); see also Ashcraft

and Duffie (2007).

The application of the above algorithm results in the scatter diagram

shown in Figure 2.2. There is a clearly visible concentration of the revers-

ing transaction pairs in the region v > 2 × 105 and |rt − rh| < 0.5% (red

box). We identify these pairs as overnight loans. Contamination from non-

loan transaction pairs that accidentally give a hypothetical rate close to the

target rate is insignificant. By examining the adjacent regions of the diagram,

i.e. v > 2× 105 and rh outside of the red box, we estimate the contamination

to be less than 2% (corresponding to ≤ 5 erroneous identifications per day).

It is also possible that some genuine loans fall outside our selection criteria.

However, it is unlikely that overnight interest rates are very different from the

target rate; and the lower-value transactions (below A$104), even if they are

real loans, contribute negligibly to the total value.

We identify 897 overnight loans over the four days. A daily breakdown is

given in Table 2.3. Here and below, we refer to the first leg of the overnight

loans as simply loans and to all other transactions as nonloans. The loans

constitute less than 1% of all transactions by number and up to 9% by value

(cf. Tables 2.1 and 2.3). The distribution of loan values and interest rates is

shown in Figures 2.3a and 2.3b. The interest rate distribution peaks at the

target rate 6.25%. The mean rate is within one basis point (0.01%) of the

target rate, while the standard deviation is about 0.07%. The average interest

rate increases slightly with increasing value of the loan; a least-squares fit

yields rh = 6.248 + 0.010 log10(v/A$106).

The same technique can be used to extract two-day and longer-term loans

(up to four-day loans for our sample of five consecutive days). Using the

same selection criteria as for the overnight loans, our algorithm detects 27,

34

2. Loan and nonloan flows in the Australian interbank network

log10v

23-02-2007

log10v

22-02-2007

21-02-2007

20-02-2007

19-02-2007

01

23

45

67

89

10

01

23

45

67

89

10

01

23

45

67

89

10

01

23

45

67

89

10

01

23

45

67

89

10

0

500

1000

15000

500

1000

15000

500

1000

1500

Figure

2.1:

Thedistribution

oftran

sactionvaluesv(inAustralian

dollars)on

alogarithmic

scale,

withbin

size

∆log10v=

0.1;

thevertical

axisisthenumber

oftran

sactionsper

bin.Com

pon

ents

oftheGau

ssianmixture

model

areindicated

bythedashed

curves;thesolidcu

rveisthesum

ofthetw

ocompon

ents.Thedottedhistogram

show

stherelative

contribution

oftran

sactions

atdifferentvalues

tothetotalvalue(tocompute

thedottedhistogram

wemultiply

thenumber

oftran

sactionsin

abin

bytheir

value).

35

2. Loan and nonloan flows in the Australian interbank network

r h

log10 v

2 3 4 5 6 7 8 9

4.5

5

5.5

6

6.5

7

7.5

8

Figure 2.2: Hypothetical interest rate rh versus value of the first leg of thetransaction pairs detected by our algorithm, with no restrictions on value orinterest rate. The dotted rectangle contains the transactions that we identifyas overnight loans. The least-squares fit is shown with a solid red line.

67, and 24 two-day loans, with total values A$1.3, A$2.2, and A$1.4 billion,

on Monday, Tuesday, and Wednesday, respectively. The total value of the

two-day loans is 1.5%, 1.0%, and 0.9% of the total transaction values on these

days respectively.

2.4 Nonloans

We display the distributions of the incoming and outgoing nonloan transac-

tions, for which the bank is the destination and the source respectively, for

the six largest banks in Figure 2.4. The distributions are similar to the to-

tal distribution shown in Figure 2.1, with the notable exception of BA (see

below). There is also an unusually large number of A$106 and A$400 transac-

tions from W to T on Monday. Note that the daily imbalance for each bank

is mostly determined by the highest value transactions; large discrepancies

between incoming and outgoing transactions at lower values are less relevant.

The distribution for BA is clearly bimodal; it contains an unusually high

proportion of transactions greater than A$106. Moreover, below A$106, in-

coming transactions typically outnumber outgoing ones by a large amount.

BA is also involved in many high value transactions that reverse on the same

day. These transactions probably correspond to the central bank’s repurchase

agreements, which facilitate intra-day liquidity of the banks (Campbell, 1998).

The banks shown in Figure 2.4 are also the largest in term of the number

36

2. Loan and nonloan flows in the Australian interbank network

log10v

22-02-2007

21-02-2007

20-02-2007

19-02-2007

45

67

89

10

45

67

89

10

45

67

89

10

45

67

89

10

0

10

20

30

400

10

20

30

400

10

20

30

400

10

20

30

40

(a)

r h

66.25

6.5

66.25

6.5

66.25

6.5

66.25

6.5

0

20

40

600

20

40

600

20

40

600

20

40

60

(b)

Figure

2.3:

(a)Thedistribution

ofloan

valuesvon

alogarithmic

scale.

Thevertical

axisisthenumber

ofloan

sper

bin

forbin

size

∆log10v=

0.25.Thedottedlineis

thesamedistribution

multiplied

bythevaluecorrespon

dingto

each

bin

(inarbitrary

units).Thedateof

thefirstlegof

theloan

sis

indicated

.(b)Thedistribution

ofloan

interest

ratesr h.Thevertical

axis

isthe

number

ofloan

sper

bin

forbin

size

∆r h

=0.01.Thedateof

thefirstlegof

theloan

sis

indicated

.Themeanan

dstan

dard

deviation

are6.25%

and0.08%

onMon

day

(19-02-2007),an

d6.26%

and0.07%

ontheother

days.

37

2. Loan and nonloan flows in the Australian interbank network

of transactions, with the exception of BA. The rank order by the number of

transactions matches that by value. For D, which is the largest, the number

of nonloan transactions reaches 48043 over the week. By the number of trans-

actions, the order of the top twelve banks is D, BP, AV, T, W, AH, AF, U,

AP, BI, BA, P. By value, the order is D, BP, AV, BA, T, W, BG, U, A, AH,

AB, BM. The situation is similar when considering the overnight loans. By

value, AV, D, BP, and T dominate. For these four banks, weekly total loans

range from A$11.5 to A$18 billion and number from 254 to 399. For the other

banks the total loan value is less than A$3 billion.

In view of the discussion above, it is noteworthy that Australia’s retail

banking system is dominated by four big banks (ANZ, CBA, NAB, andWBC)3

that in February 2007 accounted for 65% of total resident assets, according to

statistics published by Australian Prudential Regulation Authority (APRA);

see http://www.apra.gov.au for details. The resident assets of the big four

exceeded A$225 billion each, well above the next largest retail bank, St George

Bank Limited4 (A$93 billion). The distinction between the big four and the

rest of the banks in terms of cash and liquid assets at the time was less clear,

with Macquarie Bank Limited in third position with A$8 billion. According

to APRA, cash and liquid assets of the big four and Macquarie Bank Limited

accounted for 56% of the total.

2.5 Loan and nonloan imbalances

In order to maintain liquidity in their exchange settlement accounts, banks

ensure that incoming and outgoing transactions roughly balance. However,

they do not control most routine transfers, which are initiated by account

holders. Therefore, the imbalances arise. On any given day, the nonloan

imbalance of bank i is given by

∆vi = −∑

j

∑

k

vk(i, j) +∑

j

∑

k

vk(j, i), (2.1)

where {vk(i, j)}k is a list of values of individual nonloan transaction from bank

i to bank j, settled on the day. The nonloan imbalances are subsequently

compensated by overnight loans traded on the interbank money market. The

3Australia and New Zealand Banking Group, Commonwealth Bank of Australia, Na-tional Australia Bank, and Westpac Banking Corporation.

4In December 2008, St George Bank became a subsidiary of Westpac Banking Corpora-tion.

38

2. Loan and nonloan flows in the Australian interbank network

19-02-2007 20-02-2007 21-02-2007 22-02-2007nonloans loans nonloans loans nonloans loans nonloans loans

D −0.51 +0.12 −0.28 +0.12 −0.76 +0.45 −1.25 +1.44BP +2.08 −1.64 +0.80 −0.59 +1.38 −0.85 +0.16 +0.82AV −0.32 −0.17 +1.39 −0.79 +0.55 −0.65 +1.08 +0.25BA +0.03 −0.19 −0.10 −0.31 −0.32 −0.05 −1.53 −0.64T −0.76 +1.10 −0.75 +0.68 −0.62 +0.64 −0.21 +0.27W −0.09 +0.07 +0.08 +0.26 −0.36 +0.41 +0.36 −0.37

Table 2.4: Loan and nonloan imbalances for the six largest banks (in units ofA$109).

loan imbalances are defined in the same way using transactions corresponding

to the first leg of the overnight loans. Note that we do not distinguish between

the loans initiated by the banks themselves and those initiated by various

institutional and corporate customers. For instance, if the funds of a corporate

customer are depleted, this customer may borrow overnight to replenish the

funds. In this case, the overnight loan is initiated by an account holder,

who generally has no knowledge of the bank’s net position. Nevertheless, the

actions of this account holder in acquiring a loan reduce the bank’s imbalance,

provided that the customer deposits the loan in an account with the same

bank.

The loan and nonloan imbalances for the six largest banks are given in

Table 2.4. The data generally comply with our assumption that the overnight

loans compensate the daily imbalances of the nonloan transactions. The most

obvious exception is for BA on Thursday (22-02-2007), where a large negative

nonloan imbalance is accompanied by a sizable loan imbalance that is also

negative. Taking all the banks together, there is a strong anti-correlation

between loan and nonloan imbalances on most days. We see this clearly in

Figure 2.5. The Pearson correlation coefficients for Monday through Thursday

are −0.93, −0.88, −0.95, −0.36. It is striking to observe that many points

fall close to the perfect anti-correlation line. The anti-correlation is weaker on

Thursday (crosses in Figure 2.5), mostly due to BA and AV.

A correlation also exists between the absolute values of loan imbalances

and the nonloan total values (incoming plus outgoing nonloan transactions);

the Pearson coefficients are 0.74, 0.75, 0.66, 0.77 for Monday through Thurs-

day. This confirms the intuitive expectation that larger banks tolerate larger

loan imbalances.

39

2. Loan and nonloan flows in the Australian interbank network

W/34/+

0.3

6W

/20/-0

.36

W/27/+

0.0

8W

/7/-0

.09

T/30/-0

.21

T/21/-0

.62

T/29/-0

.75

T/15/-0

.76

BA/30/-1

.53

BA/30/-0

.32

BA/33/-0

.10

BA/23/+

0.0

3

AV/51/+

1.0

8AV/37/+

0.5

5AV/51/+

1.3

9AV/20/-0

.32

BP/57/+

0.1

6BP/41/+

1.3

8BP/57/+

0.8

0BP/17/+

2.0

8

D/66/-1

.25

D/51/-0

.76

D/60/-0

.28

D/15/-0

.51

01

23

45

67

89

01

23

45

67

89

01

23

45

67

89

01

23

45

67

89

0

50

100

150 0

100

200 0

20

40 0

100

200 0

100

200 0

100

200

300

Figu

re2.4:

Thedistrib

ution

ofnon

loantran

sactionvalu

esof

thesix

largestban

ksfor

Mon

day

throu

ghThursd

ay(from

leftto

right);

theban

ksare

selectedbythecom

bined

valueof

incom

ingan

dou

tgoingtran

sactionsover

theentire

week

.Black

andred

histogram

scorresp

ondto

incom

ing(ban

kisthedestin

ation)an

dou

tgoing(ban

kisthesou

rce)tran

sactions;red

histogram

sare

filled

into

improve

visib

ility.Theban

ks’an

onymou

slab

els,thecom

bined

daily

valueof

theincom

ingan

dou

tgoingtran

sactions,

andthedaily

imbalan

ce(in

comingminusou

tgoing)

arequoted

atthetop

leftof

eachpan

el(in

units

ofA

$109).

Thehorizon

talax

isis

thelogarith

mof

valuein

A$.

40

2. Loan and nonloan flows in the Australian interbank network

log10|∆

l|

log10 v

∆l

∆v

6 7 8 9 10 11−2 −1 0 1 25.5

6

6.5

7

7.5

8

8.5

9

9.5

−2

−1

0

1

2

Figure 2.5: Left: loan imbalance ∆l vs nonloan imbalance ∆v for individualbanks and days of the week (in units of A$109). Right: the absolute value ofloan imbalance |∆l| vs nonloan total value (incoming plus outgoing transac-tions) for individual banks and days of the week. Thursday data are markedwith crosses.

2.6 Flow variability

For each individual source and destination, we define the nonloan flow as

the totality of all nonloan transactions from the given source to the given

destination on any given day. The value of the flow is the sum of the nonloan

transaction values and the direction is from the source to the destination. On

any given day, the value of the flow from bank i to bank j is defined by

vflow(i, j) =∑

k

vk(i, j), (2.2)

where {vk(i, j)}k is a list of values of individual nonloan transaction from i to

j on the day. For example, all nonloan transactions from D to AV on Monday

form a nonloan flow from D to AV on that day. The nonloan transactions

in the opposite direction, from AV to D, form another flow. A flow has zero

value if the number of transactions is zero. Typically, for any two large banks

there are two nonloan flows between them. The loan flows are computed in a

similar fashion.

Nonloan flows

There are 55 banks in the network, resulting in Nflow = 2970 possible flows.

The actual number of flows is much smaller. The typical number of nonloan

flows is ∼ 800 on each day (the actual numbers are 804, 791, 784, 797). Even

though the number of nonloan flows does not change significantly from day to

day, we find that only about 80% of these flows persist for two days or more.

The other 20% are replaced by different flows, i.e. with a different source

41

2. Loan and nonloan flows in the Australian interbank network

and/or destination, on the following day. Structurally speaking, the network

of nonloan flows changes by 20% from day to day. However, persistent flows

carry more than 96% of the total value.

Even when the flow is present on both days, its value is rarely the same.

Given that 80% of the network is structurally stable from day to day, we as-

sess variability of the network by considering persistent flows and their values

on consecutive days. Figure 2.6 shows the pairs of persistent flow values for

Monday and Tuesday, Tuesday and Wednesday, and Wednesday and Thurs-

day. If the flow values were the same, the points in Figure 2.6 would lie on

the diagonals. We observe that the values of some flows vary significantly, es-

pecially when comparing Monday and Tuesday. Moreover, there is a notable

systematic increase in value of the flows from Monday to Tuesday by a factor

of several, which is not observed on the other days. For each pair of days

shown in Figure 2.6, we compute the Pearson correlation coefficient, which

gives 0.53 for Monday and Tuesday, 0.70 for Tuesday and Wednesday, and

0.68 for Wednesday and Thursday.

To characterize the difference between the flows on different days more

precisely, we compute the Euclidean distance between normalised flows on

consecutive days. We reorder the adjacency matrix {vflow(i, j)}ij of the flow

network on day d as an Nflow-dimensional vector vd representing a list of all

flows on day d (d = 1, 2, . . . , 5). For each pair of consecutive days we compute

the Euclidean distance between normalized vectors vd/|vd| and vd+1/|vd+1|,which gives 0.62, 0.50, 0.50 for all flows and 0.61, 0.49, 0.49 for persistent

flows (the latter are computed by setting non-persistent flows to zero on both

days). Since the flow vectors are normalized, these quantities measure random

flow discrepancies while systematic deviation such as between the flows on

Monday and Tuesday are ignored. For two vectors of random values uniformly

distributed in interval (0, 1), the expected Euclidean distance is 0.71 and the

standard deviation is 0.02 for the estimated number of persistent nonloan flows

of 640. So the observed variability of the nonloan flows is smaller than what

one might expect if the flow values were random.

Loan flows

Variability of the loan flows is equally strong. The number of loan flows varies

from 69 to 83 (actual numbers are 69, 75, 77, 83). Only about 50% of these

flows are common for any two consecutive days. Moreover, persistent flows

42

2. Loan and nonloan flows in the Australian interbank network

log10von22-02-2007

log10von21-02-2007

log10von21-02-2007log10von20-02-2007

log10von20-02-2007

log10von19-02-2007

01

23

45

67

89

10

01

23

45

67

89

10

01

23

45

67

89

10

0123456789

10

0123456789

10

0123456789

10

Figure

2.6:

Non

loan

flow

valuepairs

onon

eday

(horizon

talax

is)an

dthenext(verticalax

is).

Only

flow

spresenton

bothdays

areconsidered

.Flowsthat

donot

chan

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thediagonal

(red

dottedline).Thesolidlineis

theweigh

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alleast

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fitto

thescatterdiagram

;theweigh

tshavebeendefi

ned

toem

phasizepoints

correspon

dingto

largeflow

s.

log10von22-02-2007

log10von21-02-2007

log10von21-02-2007

log10von20-02-2007

log10von20-02-2007

log10von19-02-2007

56

78

910

56

78

910

56

78

910

56789

10

56789

10

56789

10

Figure

2.7:

AsforFigure

2.6butforloan

flow

s.

43

2. Loan and nonloan flows in the Australian interbank network

carry only about 65% of the total value of the loan flows on any given day,

cf. 80% of nonloan flows. For persistent loan flows, the Pearson correlation

coefficients are 0.63, 0.90, and 0.76 for the consecutive pairs of days starting

with Monday and Tuesday. The correlation is generally similar to that of the

nonloan flows, with the notable exception of the loan flows on Tuesday and

Wednesday, when the sub-network of persistent loan flows appears to be more

stable.

The Euclidean distances between the normalized loan flows for each pair

of consecutive days are 0.85, 0.68, 0.73 for all flows and 0.63, 0.44, and 0.44

for persistent flows. For two vectors of random values uniformly distributed

in interval (0, 1), the expected Euclidean distance is 0.7 and the standard

deviation is 0.1 for the estimated number of persistent loan flows of 40. So

the observed variability of the persistent loan flows is much smaller than what

one might expect if the flow values were random.

Relation between nonloan and loan flows

Some loan flows do not have corresponding nonloan flows between the same

nodes on the same day. These flows carry about 14% of loan value on Mon-

day, and about 7% on Tuesday through Thursday. Nonloan flows that have

corresponding loan flows account for 35% to 48% of all nonloan flows by value,

even though the number of these flows is less than 10% of the total.

To improve the statistics, we aggregate the flows on all four days. Fig-

ure 2.8 shows nonloan and corresponding loan flow values. We fail to find any

significant correlation between loan and nonloan flows (Pearson coefficient is

0.3). The correlation improves if we restrict the loan flows to those consisting

of three transactions or more; such flows mostly correspond to large persis-

tent flows. In this case the Pearson coefficient increases to 0.6; banks that

sustain large nonloan flows can also sustain large loan flows, even though the

loan flows on average are an order of magnitude lower than the corresponding

nonloan flows. The lack of correlation when all loans are aggregated is due to

the presence of many large loans that are not accompanied by large nonloan

transactions, and vice versa.

44

2. Loan and nonloan flows in the Australian interbank network

2.7 Net flows

The net flow between any two banks is defined as the difference of the opposing

flows between these banks. The value of the net flow equals the absolute value

of the difference between the values of the opposing flows. The direction of the

net flow is determined by the sign of the difference. If vflow(i, j) > vflow(j, i),

the net flow value from i to j is given by

vnet(i, j) = vflow(i, j) − vflow(j, i). (2.3)

For instance, if the flow from D to AV is larger than the flow in the opposite

direction, then the net flow is from D to AV.

General properties

The distributions of net loan and nonloan flow values are presented in Fig-

ure 2.9. The parameters of the associated Gaussian mixture models are quoted

in Table 2.5. The distribution of net nonloan flow values has the same general

features as the distribution of the individual transactions. However, unlike in-

dividual transactions, net flow values below A$104 are rare; net flows around

A$108 are more prominent.

There are on average around 470 net nonloan flows each day. Among these,

roughly 110 consist of a single transaction and 50 consist of two transactions,

mostly between small banks. At the other extreme, net flows between the

largest four banks (D, BP, AV, T) typically have more than 103 transactions

per day each. Overall, the distribution of the number of transactions per net

flow is approximated well by a power law with exponent α = −1.0± 0.2:

Nnet(n) ∝ nα, (2.4)

whereNnet(n) is the number of net nonloan flows that consist of n transactions

(n ranges from 1 to more than 1000). This is consistent with the findings for

Fedwire reported in Bech and Atalay (2010) (see right panel of Fig. 14 in Bech

and Atalay (2010)).

There are roughly 60 net loan flows each day. As many as 40 consist of

only one transaction. On the other hand, a single net loan flow between two

large banks may comprise more than 30 individual loans. The distribution

of the number of transactions per net loan flow is difficult to infer due to

poor statistics, but it is consistent with a power law with a steeper exponent,

45

2. Loan and nonloan flows in the Australian interbank network

log10l

log10 v

5 6 7 8 9 105

6

7

8

9

10

Figure 2.8: Loan flow values versus nonloan flow values combined over fourdays. Triangles correspond to loan flows with three or more transactions perflow. The solid line is the orthogonal least squares fit to the scatter diagram;the weighting is the same as in Figure 2.6.

Date Component 1 Component 2〈u〉 σ2u P 〈u〉 σ2u P

19-02-2007 5.14 1.88 0.60 7.51 0.36 0.4020-02-2007 5.70 2.17 0.51 7.82 0.50 0.4921-02-2007 5.73 1.97 0.52 7.72 0.44 0.4822-02-2007 5.78 2.06 0.57 7.86 0.45 0.43

Table 2.5: Mean 〈u〉, variance σ2u, and mixing proportion P of the Gaussianmixture components appearing in Figure 2.9 (u = log10 v).

−1.4± 0.2, than that of the nonloan distribution. There are no net loan flows

below A$105 or above A$109. Comparing net loan and nonloan flows, it is

obvious that net loan flows cannot compensate each and every net nonloan

flow. Not only are there fewer net loan flows than nonloan flows, but the total

value of the former is much less than the total value of the latter.

Net loan and net nonloan flows are not correlated; the correlation coeffi-

cient is 0.3. Restricting net loan flows to those that have three transactions or

more does not improve the correlation. If a net loan flow between two banks

46

2. Loan and nonloan flows in the Australian interbank network

log10v

22-02-2007

log10v

21-02-2007

20-02-2007

19-02-2007

01

23

45

67

89

10

01

23

45

67

89

10

05

10

15

20

25

3005

10

15

20

25

30

Figure

2.9:

Thedistribution

ofvalues

ofnet

non

loan

flow

s(black

histogram

)on

alogarithmic

scalewithbin

size

∆log10v=

0.1.

Thecompon

ents

oftheGau

ssianmixture

model

areindicated

withthedashed

curves;thesolidcu

rveis

thesum

ofthetw

ocompon

ents.Net

loan

flow

sareoverplotted

inred.Thevertical

axis

counts

thenumber

ofnet

flow

sper

bin.

47

2. Loan and nonloan flows in the Australian interbank network

was triggered to a significant degree by the magnitude and the direction of

net nonloan flow between these bank, one expects a correlation between net

loan and nonloan flows. Our examination shows that in this respect loan flows

are decoupled from nonloan flows. The connection between them is indirect.

Namely, nonloan flows cause an imbalance in the account of each bank, which

is subsequently compensated by loan flows, which are largely unrelated to the

nonloan flows that caused the imbalance.

Degree distribution and assortativity

We define the in-degree of node i as the number of net flows that terminate

at i, i.e. the number of net flows with destination i, and the out-degree as

the number of net flows that originate from i, i.e. the number of net flows

with source i. The degree distribution of the nonloan networks is shown in

Figure 2.10a. Node BA has the highest in-degree of 37 on Monday, but on the

other days it drops to 15 on average, while the out-degree is 11.75 on average

for this node. The highest in-degrees are usually found among the four largest

banks (D, BP, AV, T); the only exception is Monday, when AF’s in-degree of

22 is greater than AV’s 21, and BA has the highest in-degree. The highest

out-degrees are usually achieved by D, BP, AV, T, W, and AH; the exceptions

are Monday, when D’s out-degree of 17 is less than AR’s and AP’s 18, and

Thursday, when AV’s out-degree of 16 is less than P’s 18.

It is difficult to infer the shape of the degree distribution for individual

days due to poor statistics. The two-sample Kolmogorov-Smirnov (KS) test

does not distinguish between the distributions on different days at the 5%

significance level. With this in mind, we combine the in- and out-degree data

for all four days and graph the resulting distributions in Figure 2.10b. We

find that a power law distribution does not provides a good fit for either in- or

out-degrees. Visually, the distribution is closer to an exponential. However,

the exponential distribution is rejected by the Anderson-Darling test.

The degree distribution conceals the fact that flows originating or termi-

nating in nodes of various degrees have different values and therefore provide

different contributions to the total value of the net flows. Nodes with lower

degrees are numerous, but the flows they sustain are typically smaller than

those carried by a few high-degree nodes. In particular, for the nonloan flows,

nodes with in-degree d ≤ 10 are numerous, ranging from 35 to 37, but their

outgoing net flows carry about 20% of the value on average. On the other

48

2. Loan and nonloan flows in the Australian interbank network

d

22-02-2007

21-02-2007

20-02-2007

19-02-2007

05

10

15

20

25

30

35

40

95059950599505995059

−9

−5059

−9

−5059

−9

−5059

−9

−5059

(a)

d

100

101

100

101

(b)

Figure

2.10:(a)Degreedistribution

ofthenet

non

loan

flow

networks(for

convenience,in-degrees

arepositivean

dou

t-degrees

arenegative).Thetotalvalueof

thenet

flow

scorrespon

dingto

thespecificdegrees

isshow

nwithreddots(thelogof

valuein

A$10

9isindicated

ontherigh

tvertical

axis).

(b)Degreedistribution

ofthenet

non

loan

flow

swhen

thedegreedataforallfour

daysareaggregated

(in-degrees

arecircles;

out-degrees

aretriangles).

49

2. Loan and nonloan flows in the Australian interbank network

d

05

10

15

9 5 0 5 9 9 5 0 5 9 9 5 0 5 9 9 5 0 5 9

−9

−5 0 5 9

−9

−5 0 5 9

−9

−5 0 5 9

−9

−5 0 5 9

(a)

d

100

101

100

101

(b)

Figu

re2.11:

(a)Sam

eas

Figu

re2.10a,

butfor

thenet

loanflow

netw

orks.

(b)Sam

eas

Figu

re2.10b

,butfor

thenet

loanflow

s.

50

2. Loan and nonloan flows in the Australian interbank network

hand, nodes with d ≥ 17 are rare, but their flows carry 50% of the value. The

same effect is observed for the out-degrees.

The degree distribution of the network of net loan flows is shown in Fig-

ure 2.11a (we ignore the nodes that have zero in- and out- degrees over four

days). Similarly to nonloan flows, the KS test does not distinguish between

the distributions on different days at the 5% significance level. The combined

distribution is shown in Figure 2.11b.

To probe assortativity of the net flow networks, we compute the in-assortativity

defined in Piraveenan et al. (2010) as the Pearson correlation coefficient be-

tween the in-degrees of sources and destinations of the net flows (out-assortativity

is computed similarly using the out-degrees). The net nonloan flow network

is disassortative, with in-assortativity of −0.39, −0.37, −0.38, −0.37 and out-

assortativity of −0.35, −0.38, −0.39, −0.37 on Monday, Tuesday, Wednesday,

and Thursday, respectively. The net loan flow network is less disassortative;

the in-assortativity is −0.16, −0.26, −0.18, −0.19 and the out-assortativity

is −0.03, −0.10, 0.02, −0.20 for the same sequence of days. In biological

networks, the tendency of out-assortativity to be more assortative than in-

assortativity has been noted in Piraveenan et al. (2010).

Topology of the net flows

Given the source and destination of each net flow, we can construct a network

representation of the net flows. An example of the network of net nonloan flows

is shown in Figure 2.12. The size of the nodes and the thickness of the edges

are proportional to the net imbalances and net flow values respectively (on a

logarithmic scale). We use the Fruchterman-Reingold algorithm to position

the nodes Fruchterman and Reingold (1991); the most connected nodes are

placed in the centre, and the least connected nodes are moved to the periphery.

The core of the network is dominated by the four banks with the largest total

value and the largest number of transactions: D, BP, AV, and T. The other

big banks, such as AF, AH, and W, also sit near the core. It is interesting to

note the presence of several poorly connected nodes (Q, V, BF, and especially

X) that participate in large incoming and outgoing flows, which produce only

negligible imbalances in the banks themselves.

The sub-network consisting of D, BP, AV, BA, T, W, U, A, AH, AF, AP,

and P is fully connected on all five days, i.e. every node is connected to every

other node. The sub-network of D, AV, and BP is fully connected, even if we

51

2. Loan and nonloan flows in the Australian interbank network

Figure 2.12: Network of net nonloan flows on Tuesday, 20-02-2007. White(grey) nodes represent negative (positive) imbalances. The bank labels areindicated for each node. The size of the nodes and the thickness of the edgesare proportional to the logarithm of value of the imbalances and the net flowsrespectively.

restrict the net flows to values above A$108.

In Figure 2.12, the flows between the largest nodes are difficult to discern

visually, because the nodes are placed too close to each other in the image.

We therefore employ the following procedure to simplify the network. We

consider the fully connected sub-network of twelve nodes, plus node BG, and

combine all other nodes into a new node called “others” in such a way that

the net flows are preserved (BG is included because it usually participates

in large flows and is connected to almost every node in the complete sub-

52

2. Loan and nonloan flows in the Australian interbank network

network). The result of this procedure applied to the daily nonloan networks

is presented in Figures 2.13a–2.13d. For these plots, we employ the weighted

Fruchterman-Reingold algorithm, which positions the nodes with large flows

between them close to each other. The imbalances shown in Figure 2.13b are

the same as those of the full network in Figure 2.12. The daily networks of

net loan flows for the same nodes are shown in Figures 2.14a–2.14d.

We observe that the largest flows on Monday (19-02-2007) were signifi-

cantly lower than the flows on the subsequent days. The largest nodes (D,

BP, AV, T, W) are always placed close to the center of the network, because

they participate in the largest flows. The topology of the flows is complex and

difficult to disentangle, even if one concentrates on the largest flows (above

A$5 × 108). For instance, on Monday, probably the simplest day, the flow of

nonloans is generally from BG to “others” to D to BP. There are also siz-

able flows from T to AV and from AV to “others” and BP. However, lower

value flows (below A$5 × 108) cannot be neglected completely because they

are numerous and may contribute significantly to the imbalance of a given

node.

Nodes D, T, BP, AV, and W form a complete sub-network of net loan flows

on Monday, Tuesday, and Wednesday. This sub-network is almost complete on

Thursday too, except for the missing link between BP and W. The appearance

of the net loan network is different from that of the nonloan network, since the

same nodes participate in only a few loan flows. Therefore, the position of a

node in the network image is strongly influenced by the number of connections

of that node. Some of the poorly connected nodes are placed at the periphery

despite the fact that they possess large flows. The four largest nodes (D, T,

BP, AV) are always positioned at the center of the network.

Network variability

The net nonloan flow network is extremely volatile in terms of flow value and

direction. For example, a A$109 flow from D to BP on Monday transforms

into a A$3.2×109 flow in the same direction on Tuesday, only to be replaced by

a A$6.3 × 108 flow in the opposite direction on Wednesday, which diminishes

further to A$2.5×109 on Thursday. Nodes T and BP display a similar pattern

of reversing flows between Tuesday and Wednesday. On the other hand, the

net flow between T and AV maintains the same direction, but the flow value

is strongly fluctuating. In particular, a moderate A$4.8×108 flow on Monday

53

2. Loan and nonloan flows in the Australian interbank network

(a) 19-02-2007 (b) 20-02-2007

(c) 21-02-2007 (d) 22-02-2007

Figure 2.13: Networks of daily net nonloan flows for D, AV, BP, T, W, BA,AH, AF, U, AP, P, A, BG. All the other nodes and the flows to and from themare combined in a single new node called “others”. The size of the nodes andthe thickness of the edges are proportional to the logarithm of value of theimbalances and the net flows respectively. The value of the flows and theimbalances can be gauged by referencing a network shown in the middle,where the values of the flows are indicated in units of A$1 billion.

54

2. Loan and nonloan flows in the Australian interbank network

(a) 19-02-2007 (b) 20-02-2007

(c) 21-02-2007 (d) 22-02-2007

Figure 2.14: Networks of daily net loan flows. The same nodes as in Fig-ures 2.13a–2.13d are used. The scale of the loan flows, the imbalances, andthe positions of the nodes are the same as those used for the nonloan flows inFigures 2.13a–2.13d to simplify visual comparison.

55

2. Loan and nonloan flows in the Australian interbank network

rises to A$1.9×109 on Tuesday, then falls sharply to A$2×108 on Wednesday

and again rises to A$2.2× 109 on Thursday.

Considering any three nodes, we observe that circular and transitive flows

are present on most days, the latter being more common. The most obvious

example is a circular flow between D, T, and BP on Thursday and a transitive

flow involving BG, T, and AV on the same day. The circular flows are unstable

in the sense that they do not persist over two days or more.

The net loan flow network exhibits similar characteristics. Few net loan

flows persist over the four days. For example, the flow from AV to T has the

same direction and is similar in value on all four days. Circular loan flows are

also present, as the flow between AV, T, and BP on Thursday demonstrates.

2.8 Conclusions

In this paper, we study the properties of the transactional flows between Aus-

tralian banks participating in RITS. The value distribution of transactions

is approximated well by a mixture of two log-normal components, possibly

reflecting the different nature of transactions originating from SWIFT and

Austraclear. For the largest banks, the value distributions of incoming and

outgoing transactions are similar. On the other hand, the central bank dis-

plays a high asymmetry between the incoming and outgoing transactions, with

the former clearly dominating the latter for transactions below A$106.

Using a matching algorithm for reversing transactions, we successfully sep-

arate transactions into loans and nonloans. For overnight loans, we estimate

the identification rate at 98%. The mean derived interest rate is within 0.01%

of the central banks’ target rate of 6.25%, while the standard deviation is

about 0.07%. We find a strong anti-correlation between loan and nonloan im-

balances (Pearson coefficient is about 0.9 on most days). A likely explanation

is that nonloan flows create surpluses in some banks. The banks lend the

surplus to banks in deficit, creating loan flows that counteract the imbalances

due to the nonloan flows. Hence, loan and nonloan imbalances of individual

banks are roughly equal in value and opposite in sign on any given day.

The flow networks are structurally variable, with 20% of nonloan flows

and 50% of loan flows replaced every day. Values of persistent flows, which

maintain the same source and destination over at least two consecutive days,

vary significantly from day to day. Some flow values change by several orders

of magnitude. Persistent flows increase in value several-fold between Monday

56

2. Loan and nonloan flows in the Australian interbank network

and Tuesday. Individual flow values can change by several orders of magnitude

on the following day. Overall, there is a reasonable correlation between the

flow values on consecutive days (Pearson coefficient is 0.65 for nonloans and

0.76 for loans on average). We also find that larger banks tend to sustain

larger loan flows, in accord with the intuitive expectations. However, there is

no correlation between loan and nonloan flows.

We examine visually the topology of the net loan and nonloan flow net-

works. The centre of both networks is dominated by the big four banks.

Twelve banks form a complete nonloan sub-network, in which each bank is

connected to every other bank in the sub-network. The three largest banks

form a complete sub-network even if the net flows are restricted to values

above A$108. Our examination reveals that the network topology of net flows

is complicated, with even the largest flows varying greatly in value and direc-

tion on different days.

Our findings suggest a number of avenues for future research on interbank

networks. Firstly, the relationships we uncovered can be used to constrain

analytical models and numerical simulations of interbank flows in financial

networks. In particular, our explanation of the link between the loan and

nonloan imbalances needs to be tested in numerical simulations. Secondly, it

is necessary to analyse interbank markets in other countries to establish what

elements of our results are signatures of general dynamics and what aspects

are specific to the epoch and location of this study. Even when high qual-

ity data are available, most previous studies concentrate on analysing static

topological properties of the networks or their slow change over time. The

internal dynamics of monetary flows in interbank networks has been largely

ignored. Importantly, one must ask whether the strong anti-correlation be-

tween loan and nonloan imbalances is characteristic of RTGS systems whose

institutional setup resembles the Australian one or whether it is a general

feature. For instance, in Italy a reserve requirement of 2% must be observed

on the 23rd of each month, which may encourage strong deviations between

loan and nonloan imbalances on the other days.

57

Chapter 3

Laplace transform analysis of

a multiplicative asset transfer

model

We analyze a simple asset transfer model in which the transfer amount is

a fixed fraction f of the giver’s wealth. The model is analyzed in a new

way by Laplace transforming the master equation, solving it analytically and

numerically for the steady-state distribution, and exploring the solutions for

various values of f ∈ (0, 1). The Laplace transform analysis is superior to

agent-based simulations as it does not depend on the number of agents, en-

abling us to study entropy and inequality in regimes that are costly to address

with simulations. We demonstrate that Boltzmann entropy is not a suitable

(e.g. non-monotonic) measure of disorder in a multiplicative asset transfer sys-

tem and suggest an asymmetric stochastic process that is equivalent to the

asset transfer model.

3.1 Introduction

A vibrant research theme in econophysics is the analysis of asset exchange

models. In these models, a large number of agents iteratively exchange assets,

typically representing monetary amounts. In the simplest model that has been

considered, the transfer amount is constant and independent of the agent’s

wealth, producing an exponential wealth distribution in the steady state (see

Yakovenko and Rosser Jr. (2009) for a review). More complicated fractional

59

3. Laplace transform analysis of a multiplicative asset transfer model

exchange models have also been considered by several authors Chatterjee and

Chakrabarti (2007); Hayes (2002), in which the size of each transfer is a linear

function of the wealths of the agents involved in the exchange.

It has been found both analytically and numerically that the steady-state

wealth probability distribution function ps(w) in fractional exchange models

depends strongly on the parameters that characterize the exchange Matthes

and Toscani (2008). Certain parameter values or exchange rules yield a

strongly peaked distribution with an exponential tail, while other values yield

a broad distribution with Pareto-like qualities. The dichotomy is exemplified

by two simple models. If the transfer amount is a fixed fraction f of the

giver’s wealth (the giver is the agent who surrenders the asset in the transfer),

then the resulting steady-state distribution is strongly peaked and decays ex-

ponentially in the tail. If, on the other hand, the transfer amount is a fixed

fraction f of the poorer agent’s wealth, then one finds a broad steady-state

distribution, which can be fitted well by a power law with exponent −1 across

a broad interval of wealths. In this paper, we refer to these two models as the

giver scheme and the poorer scheme respectively.1

In some of the asset exchange models considered in Chakraborti and

Chakrabarti (2000) and several other studies Chatterjee and Chakrabarti

(2007); Chatterjee et al. (2005), the fractional exchange amount is a random

linear combination of the wealths of the participating agents. The controlling

parameter is the saving propensity, λ, which determines the fraction of the

agents’ wealths that they do not offer to exchange. Comparing the output of

simulations for the giver scheme and the exchange schemes based on the sav-

ing propensity, one observes that the schemes are closely related, with f ≈ 0

corresponding to λ ≈ 1. If the saving propensity is the same for all the agents,

then the resulting steady-state distributions are similar to those obtained for

the giver scheme. On the other hand, if the saving propensity is uniformly

distributed, the steady-state distribution is a power law, ps(w) ∝ w−2. In

a recent study based on numerical simulations Saif and Gade (2007), it was

found that a combination of the poorer and giver schemes in one simulation

results in a power-law wealth distribution whose exponent depends on the

relative contributions of the two schemes. The more agents follow the giver

scheme, the greater the exponent.

Asset exchange models can be treated analytically via a kinetic or mas-

1They are also known as the theft-and-fraud and yard-sale models respectively (see Hayes(2002)).

60

3. Laplace transform analysis of a multiplicative asset transfer model

ter equation, which tracks the rate of change of the number of agents at any

given wealth. In particular, the master equation for the giver scheme has been

derived in Ispolatov et al. (1998). These authors found an expression for the

second moment in the steady state, which agrees with the expression found

in Angle (2006) for a similar model by assuming the gamma distribution of

wealth. The standard deviation converges to its steady-state value exponen-

tially on a time scale ∼ [f(1− f)]−1. The authors also found the asymptotic

behaviour of the wealth distribution at small values of wealth. The master

equation for the poorer scheme was derived recently Moukarzel et al. (2007),

but its solutions have not yet been studied. In Chatterjee et al. (2005), the au-

thors derived the kinetic equation for the case of uniformly distributed saving

propensity. They demonstrated that the solution follows a power law with the

same exponent as in the simulations. The kinetic equation approach was also

used in Slanina (2004) to analyze self-similar solutions of a non-conservative

asset exchange system. The author found a closed-form solution in the limit

of continuous trading by means of the Laplace transform and observed that

the distribution exhibits power-law behaviour at large wealths.

The dependence of the relaxation time on the exchange parameters has

been investigated numerically in Patriarca et al. (2007) for the models con-

sidered in Chatterjee and Chakrabarti (2007); Chatterjee et al. (2005). The

relaxation time-scale was found numerically to scale as ∼ (1 − λ)−1. This is

consistent with the values found analytically in the giver scheme for the stan-

dard deviation. The authors also considered how the relaxation time depends

on the number of agents but failed to find any significant trend.

Recently, much effort has been directed profitably at developing more so-

phisticated and realistic multi-agent models to be analyzed by means of numer-

ical simulations. In the present paper we take the opposite tack and return

instead to one of the simplest multiplicative models, the giver scheme. We

show that its master equation can be solved efficiently by a Laplace transform

technique. Armed with this new tool backed by multi-agent simulations, we

identify the following new properties of the system. (1) We get precise val-

ues of various quantities such as the steady-state entropy as a function of the

model parameter f , independently of the number of agents. (2) We explore

the thinly studied regime 1/2 < f < 1 and identify its unusual properties,

e.g. oscillations in ps(w). (3) Using multi-agent simulations, we investigate

how the Boltzmann entropy evolves with time as the system approaches equi-

librium and argue that the Boltzmann entropy is not a suitable entropy for

61

3. Laplace transform analysis of a multiplicative asset transfer model

the giver scheme, even though the system is closed and conservative. (4) We

propose a simple asymmetric stochastic process that is equivalent to the giver

scheme. (5) We investigate how the degree of inequality, characterized by

the Gini coefficient, depends on f . (6) Finally, we apply phase-space tech-

niques from statistical mechanics to the giver scheme in order to illuminate

the difficulties and opportunities that this asset transfer model presents.

3.2 Giver scheme

We consider a simple asset transfer model, in which the transfer amount is

equal to a fixed fraction of the giver’s wealth.2 If wg is the giver’s wealth and

wr is the receiver’s wealth prior to the transfer, then their wealths after the

transfer are given by wg−∆w and wr+∆w respectively, with ∆w = fwg and

f ∈ (0, 1). The model comprises a large number of agents, who are assigned

wealths initially according to some distribution. The transfers are assumed to

take place over a fixed time interval ∆t. At each discrete time ti, the agents

are divided randomly into pairs and the transfer formula is applied to each

pair. The transfers are complete by the time ti+1 = ti + ∆t and the process

repeats at the time ti+1. The probability of drawing any pair is the same.

In each pair, the giver is assigned randomly regardless of the wealths of the

agents.

The master equation for this system was derived in Ispolatov et al. (1998)

and is given by

∂p(w, t)

∂t= −p(w, t)+ 1

2(1 − f)p

(

w

1− f, t

)

+1

2f

∫ w

0dw′ p

(

w − w′

f, t

)

p(w′, t).

(3.1)

It is easy to verify that the mean of the distribution, µ1 =∫∞0 dw wp(w, t),

does not depend on time. Upon integrating by parts, one arrives at the evo-

lution equationdµ2(t)

dt= −f(1− f)µ2 + fµ21 (3.2)

for the second moment, µ2(t) =∫∞0 dw w2p(w, t), first reported in Ispolatov

et al. (1998). This equation can be solved for the variance

σ2(t) = µ2(t)− µ21 =

(

µ2(0) −µ21

1− f

)

e−f(1−f)t +fµ211− f

. (3.3)

2The giver is also called the payer or the loser in the literature.

62

3. Laplace transform analysis of a multiplicative asset transfer model

In the steady state, one has σs = σ(t→ ∞) = µ1[f/(1−f)]1/2. For simplicity,

we assume henceforth that the mean of p(w, t) equals unity.3 In the following

sections, we are mostly concerned with the steady-state distribution ps(w) =

p(w, t → ∞).

3.3 Laplace transform of the master equation

The Laplace transform of the master equation in the steady state is given by

g(z) =1

2g(z − fz) +

1

2g(z)g(fz), (3.4)

with g(z) =∫∞0 dw e−zwps(w). Note that the functional equation (3.4) applies

to any integral transform whose kernel depends only on the product of the

arguments of the function and its transform. For f = 1/2, the functional

equation has a closed form solution

g(z) =1

1 + Cz, (3.5)

where C is a complex-valued constant. Using the definition of the transform,

we have g(0) = 1 from the normalization of ps(w) and g′(0) = −1 from the

assumption that ps(w) has unit mean, which gives C = 1. Applying the

inverse Laplace transform to this solution gives the exponential distribution,

ps(w) = e−w, which was obtained in Ispolatov et al. (1998) by substituting

simple “test” functions into the master equation. No closed-form solutions

have been found for other values of f ∈ (0, 1).

The Taylor expansion of g(z) at z = 0 can be derived by substituting the

expansion in (3.4) and using g(0) = 1 and g′(0) = −1. For the first four terms

of the expansion, this procedure gives

g(z) = 1− z +1

2(1− f)z2 − 1 + f

6(1− f)2z3 +O(z4). (3.6)

In general, for g(z) =∑∞

n=0 an(−z)n/n!, a0 = 1, and a1 = 1, we obtain

an =

n−1∑

k=1

(

n

k

)

fkakan−k

1− fn − (1− f)nfor n > 1. (3.7)

Since g(−z) is the moment-generating function for the distribution ps(w),

the n-th moment of the distribution µn equal an, i.e. all moments of the

3If the mean µ1 6= 1, one can consider the function q(x) = µ1p(µ1x), which is normalizedand has unit mean.

63

3. Laplace transform analysis of a multiplicative asset transfer model

steady-state wealth distribution can be computed for any f using the recursive

formula (3.7).

Using the Taylor expansion, one has an → 1 as f → 0 and hence g(z) →e−z. Note that the functional equation (3.4) becomes an identity for f = 0

and g(0) = 1. Taking the inverse Laplace transform of g(z) = e−z, formally

one gets ps(w) = δ(w− 1). However, this wealth distribution is never reached

because the relaxation time scale tr = [f(1 − f)]−1 determined from (3.3)

tends to infinity in this limit. Indeed, ps(w) is equal to the initial distribution

if f = 0. On the other hand, the Taylor expansion does not have a limit as

f → 1. The functional equation (3.4) has the solution g(z) = 1 when f = 1,

but it does not satisfy the condition g′(0) = −1. It appears that ps(w) does

not have a proper limit as f → 1. Note also that the relaxation time tends to

infinity as f approaches unity as well.

The asymptotic behaviour of g(z) at infinity can also be deduced readily

from the functional equation. In the directions in the complex plane for which

one has g(z) → 0 as |z| → ∞, the equation

g(z − fz) = 2g(z) (3.8)

must be approximately true for large enough |z|. Assuming a power-law shape

|g(z)| ∝ |z|−α as |z| → ∞, equation (3.8) gives

α =−1

log2(1− f). (3.9)

By Watson’s lemma Davies (2002), this is consistent with the asymptotic

behaviour p(w) ∝ wα−1 as w → 0 that was found in Ispolatov et al. (1998) by

the method of dominant balance.

The functional equation (3.4) can be solved iteratively when it is cast in

the form

gi+1(z) =gi(z − fz)

2− gi(fz), (3.10)

where gi(z) is the i-th iteration. Experimentation shows that the choice

g0(z) = 1/(1 + z) works well for all f . A detailed description of the com-

putational procedure is given in 3.7; there are some subtleties involved in the

choice of grid and interpolation method. An example of the numerical solution

for f = 0.1 is presented in Figures 3.1a and 3.1b. The power-law behaviour

at large |z|, with the exponent given by (3.9), is confirmed numerically for

f = 0.1 (right panel of Figure 3.1a) and a range of other values. The itera-

tions converge in the negative half-plane, Re(z) < 0, despite the complicated

64

3. Laplace transform analysis of a multiplicative asset transfer model

θ=

90◦

θ=

60◦

θ=

0◦

log10|g(z)|

log10

r

Im[g(z)]

log10

r

Re[g(z)]

log10

r

−2

−1

01

23

−2−1

01

23

−2−1

01

23

−16

−14

−12

−10

−8

−6

−4

−20

−1

−0.50

0.51

−1

−0.50

0.51

(a)

Im

(z)

Re(z)

Im

[g(z)]

Im

(z)

Re(z)

Re[g

(z)]

−5

05

10

−5

05

10

−10

−5

05

10

−10

−5

05

10

−1

−0.50

0.51

−1

−0.50

0.51

(b)

Figure

3.1:

TheLap

lace

tran

sformg(z)forf=

0.1ob

tained

bysolving(3.4)iteratively.

(a)Re[g(z)]

(top

left

pan

el),

Im[g(z)]

(bottom

left

pan

el),

and|g(z)|

(rightpan

el)versusralon

gthereal

axis

(dashed

curve),theim

aginaryax

is(solid

curve),an

dthelineinclined

atθ=

60◦to

thereal

axis

(dottedcu

rve).Thevariab

lesran

dθaredefi

ned

byz=re

iθ.(b)A

view

ofthe

real

(top

)an

dim

aginary(bottom)parts

ofg(z)(values

above1an

dbelow

−1havebeencu

toff

).

65

3. Laplace transform analysis of a multiplicative asset transfer model

structure of g(z), illustrated in Figure 3.1b, as it gradually approaches e−z

for decreasing values of f . The convergence does not depend on the initial

function g0(z); e.g. g0(z) = e−z works just as well for small values of f .

3.4 Steady-state wealth distribution by Laplace

inversion

The steady-state probability distribution function ps(w) can be obtained by

inverting its Laplace transform g(z) numerically. A number of inversion al-

gorithms were reviewed recently in Hassanzadeh and Pooladi-Darvish (2007)

and Abate and Whitt (2006). The reviewers advised that at least two different

algorithms should be used as a cross-check, because different algorithms work

well for specific classes of functions and none of the algorithms is universally

accurate. Fortunately, the algorithms are easy to implement. We test four (re-

ferred to as the Euler, Talbot, Stehfest, and Zakian algorithms in Hassanzadeh

and Pooladi-Darvish (2007); Abate and Whitt (2006)) and find that the first

two give accurate results over a wider range of w. Euler has an additional ad-

vantage over Talbot: it samples g(z) in the positive half-plane only, where the

function g(z) has a simpler structure, as one sees in Figure 3.1b. The results

of the inversion are presented in Figures 3.2a and 3.2b for 0.025 ≤ f ≤ 0.5

and Figures 3.3a and 3.3b for 0.5 ≤ f ≤ 0.9. The exponential analytic solu-

tion is recovered numerically for f = 0.5. From the output of the inversion

algorithms, we compute the moments of the distribution (µ0, µ1, and µ2)

and find agreement with the analytical results to 8 significant digits. In Fig-

ures 3.4a and 3.4b for f = 0.95 and f = 0.05 respectively, we compare the

wealth distributions obtained from the Laplace transform (curves) and from

the agent-based simulations (crosses). We find excellent agreement between

these two methods for all values of f that we consider.

The algorithms that perform the inverse Laplace transform suffer from

truncation errors. Maximum precision is achieved near the peak of ps(w). In

the tail, the precision decreases until the results are completely dominated

by the truncation errors below a threshold value of ps(w). For example, we

perform all computations with 16 significant digits and achieve ∼ 8 significant

digits of precision at the peak of ps(w), but the algorithms break down at

ps(w) . 10−8.

The wealth distributions that we find for f < 1/2 are characterized by

66

3. Laplace transform analysis of a multiplicative asset transfer model

f=

0.025

f=

0.05

f=

0.1

f=

0.25

f=

0.5

ps(w)

w

00.5

11.5

22.5

30

0.51

1.52

2.53

(a)linearscale

f=

0.025

f=

0.05

f=

0.1

f=

0.25

f=

0.5

log10ps(w)

w

01

23

45

67

8−

7

−6

−5

−4

−3

−2

−101

(b)log-linearscale

Figure

3.2:

Thesteady-state

wealthprobab

ilitydistribution

functionps(w

)ob

tained

byinvertingtheLap

lace

tran

sformg(z)

forthefollow

ingvalues

ofthetran

sfer

fraction

:f

=0.5(boldsolidcu

rve),0.25

(dash-dot

curve),0.1(dottedcu

rve),0.05

(dashed

curve),an

d0.025(thin

solidcu

rve).

67

3. Laplace transform analysis of a multiplicative asset transfer model

f=

0.9

f=

0.8

f=

0.7

f=

0.6

f=

0.5

log10 ps(w)

log10w

−3

−2.5

−2

−1.5

−1

−0.5

00.5

11.5

2−

5

−4

−3

−2

−1 0 1 2

(a)log-lo

gsca

le

f=

0.9

f=

0.8

f=

0.7

f=

0.6

f=

0.5

log10 ps(w)

w

05

10

15

20

25

30

35

40

45

50

−5

−4

−3

−2

−1 0 1

(b)log-lin

earsca

le

Figu

re3.3:

Thestead

y-state

wealth

prob

ability

distrib

ution

function

ps (w

)ob

tained

byinvertin

gtheLap

lacetran

sformg(z)

forthefollow

ingvalu

esof

thetran

sferfraction

:f=

0.5(bold

solidcu

rve),0.6

(dash

-dot

curve),

0.7(dotted

curve),

0.8(dash

edcu

rve),an

d0.9

(thin

solidcu

rve).

68

3. Laplace transform analysis of a multiplicative asset transfer model

log10n(w)

log10w

(m.u.)

−1

−0.5

00.5

11.5

22.5

33.5

4−

10123456

(a)f=

0.95

log10n(w)w

(m.u.)

0100

200

300

400

500

600

700

−1

−0.50

0.51

1.52

2.53

3.54

(b)f=

0.05

Figure

3.4:

Thepop

ulation

distribution

n(w

),show

nwithcrosses,

asafunctionof

wealthw,measuredin

ficticiousmon

etary

units(m

.u.)

usedin

theagent-based

simulation

s.Thedistribution

iscomputedas

thenumber

ofagents

ineveryunit

wealth

interval

after100step

sin

thesimulation

ofthegiverschem

ewithtotalnumber

ofagentsN

=4×

105an

dtran

sfer

param

eter

(a)f=

0.95

and(b)f=

0.05.Theinitialdistribution

isuniform

inthewealthinterval

(a)[0,100

m.u.]an

d(b)[0,500

m.u.].

Thecorrespon

dingsolution

ofthesteady-state

masterequationforthesamefis

show

nwithasolidcu

rve,

withps(w

)scaled

toconform

withthedefi

nitionofn(w

)accordingto

Nps(w/〈w〉)/〈w〉where〈w

〉is

themeanwealth.Boththeagent-based

simulation

san

dthemasterequationpredictoscillationsin

thewealthdistribution

in(a)butnot

in(b).

69

3. Laplace transform analysis of a multiplicative asset transfer model

power-law behaviour at w ≪ 1, in accord with the analytical results, and

approximately exponential tails at large w. A careful examination of the tails

confirms that the asymptotic behaviour at large wealths is not exactly ex-

ponential. However, we have not been able to find a closed-form expression

for it. The distribution becomes tightly concentrated around its peak as f

decreases; the peak of the distribution gradually shifts towards w = 1. On

the other hand, the peak shifts towards w = 0 as f increases; the distribu-

tions eventually turns into an exponential function for f = 1/2. This overall

behaviour is similar to that observed in the asset exchange models based on

the saving propensity with 0 < λ < 1 Chatterjee and Chakrabarti (2007).

The structure of the steady-state solutions for f > 1/2 is very different.

The asymptotic approximation ps(w) ∝ wα−1, with α = −1/ log2(1 − f), is

valid for f > 1/2 as well and indicates that the distribution diverges at w = 0

(as α < 1). For values of f sufficiently close to 1, the wealth distribution

acquires a shape that is akin to a power law, ps(w) ∝ w−1, with overlaid

oscillations that become more prominent as f increases. This power-law be-

haviour cuts off exponentially at some critical wealth that increases slowly as

f approaches 1. At the same time the exponential drop-off at large wealths

becomes shallower as evident from Figure 3.3b. The oscillations of ps(w) ap-

pear to be periodic on a logarithmic scale, with the period depending on f .

For example, the periods for f = 0.9 and f = 0.99 are roughly one and two

decades respectively. This is directly related to the fact that all givers retain

1% of their wealth for f = 0.99 and 10% for f = 0.9.

3.5 Discussion

We now use the Laplace transform tools developed in Section 3.4 to address

two questions that are costly to explore with agent-based simulations: the

nature of disorder (entropy) and its evolution in the giver scheme, and the

degree of inequality in the steady state.

Entropy and the approach to equilibrium

According to Boltzmann, states with higher entropy are more probable be-

cause they correspond to a larger number of microscopic configurations of the

system. A closed system evolves to a state of maximum entropy, i.e. maxi-

mum disorder, which is characterized by the Boltzmann-Gibbs distribution.

70

3. Laplace transform analysis of a multiplicative asset transfer model

The exponential distribution observed in models where the transfer amount

is fixed and constant (see section II.C in Yakovenko and Rosser Jr. (2009))

is thus consistent with entropy maximization ideas. On the other hand, it is

argued in Yakovenko and Rosser Jr. (2009) that multiplicative asset exchanges

may lead to non-exponential distributions because of the broken time-reversal

symmetry, whereas, in models with fixed additive exchanges, the time-reversal

symmetry is preserved. Despite this, the entropy maximization technique has

been applied in Chakraborti and Patriarca (2008) to an asset exchange model

described by a Hamiltonian quadratic in wealth variables. It predicts a gamma

distribution of wealth, but ps(w) in multiplicative asset exchange models is

not a gamma distribution in general.

In this section, we explore the applicability of the Boltzmann entropy to

the giver scheme. Using the Laplace transformed solutions of the master

equation (3.1), we compute the steady-state Boltzmann entropy4

Ss = −∫ ∞

0dw ps(w) log[ps(w)] (3.11)

for several values of f in the range 0.01 ≤ f ≤ 0.9 and plot the results

in Figure 3.5a. Entropy maximization arguments Kapur (1989) imply that

the entropy defined by (3.11) leads uniquely to the exponential distribution,

p(w) = e−w with Ss = 1, if the only condition is that the mean of the dis-

tribution is fixed to µ1 = 1. In our model, however, the transfer fraction f

places additional constraints on how the distribution of wealth evolves with

time. Therefore, it is not surprising that we observe a range of steady-state

entropies Ss 6= 1 corresponding to different values of f . The exponential distri-

bution for f = 1/2 appears to be the most disordered state of the system with

the highest entropy Ss = 1. For all other values of f , the entropy is smaller

and it becomes negative as f approaches 0 or 1. Reading off the graph, we

find that Ss is negative for 0 < f < 0.058 and 0.836 < f < 1. The entropy

tends to negative infinity as f → 0 or f → 1, which is in accord with the

behaviour of ps(w) in these limits.

4We stress that this definition applies to a normalized distribution with unit mean. Un-like the case of discrete probability distributions, continuous entropy can be negative andit is not invariant with respect to the change of variable. It may be more appropriate toconsider the Kullback-Leibner divergence Ds =

∫∞

0dw ps(w) log[ps(w)/m(w)], which is a

measure of the divergence between ps(w) and the reference distribution m(w). It is conve-nient to take m(w) = e−w, in which case the divergence essentially reverts to Boltzmannentropy because Ds = 1− Ss.

71

3. Laplace transform analysis of a multiplicative asset transfer model

Ss

σ2s

10−

210−

1100

101

−1

−0.8

−0.6

−0.4

−0.2 0

0.2

0.4

0.6

0.8 1

(a)stea

dy-sta

teen

tropy

S(t)

t(step

s)

010

20

30

40

50

60

70

80

90

100

0

0.0

5

0.1

0.1

5

0.2

0.2

5

(b)en

tropyevolutio

n

Figu

re3.5:

(a)Boltzm

ann

entrop

ySsof

thestead

y-state

distrib

ution

asafunction

ofthevarian

ceσ2s=f/(1

−f).

The

criticalvalu

esσ2s=

0.062,1,

and5.098,

correspon

dingtof=

0.058,0.5,

and0.836

respectively,

areindicated

with

thedotted

lines.

(b)Entrop

yas

afunction

oftim

efor

theinitial

distrib

ution

givenby(3.12)

with

f=

0.058,com

puted

fromthemulti-

agentsim

ulation

ofthegiver

schem

e.For

thesim

ulation

,thedistrib

ution

(3.12)was

scaledupto

giveN

=337123

agents

in0≤w

≤1421.

Tocom

pute

theentrop

y,thepop

ulation

distrib

ution

produced

bythesim

ulation

was

norm

alizedto

aprob

ability

distrib

ution

with

unitmean

.

72

3. Laplace transform analysis of a multiplicative asset transfer model

Negative values of Ss are already a warning that the Boltzmann entropy

may not be a faithful measure of disorder in a multiplicative asset transfer

system like the giver scheme. However, the situation worsens when we look

at how S(t) = −∫∞0 dw p(w, t) log[p(w, t)] evolves with time by conducting

multi-agent simulations. In many cases, it decreases instead of increasing. For

example, if we choose p(w, 0) = e−w initially, S(t) decreases with time for all

f 6= 1/2 and remains constant for f = 1/2. Moreover, we can easily find

realistic situations where S(t) does not change monotonically with time, as

the experiment described below shows. Consider an initial distribution of the

form

p(w, 0) =

p1, 0 ≤ w ≤ 1,

p2, 1 < w ≤ w2,

0, otherwise,

(3.12)

with the parameters p1, p2, and w2 chosen to give S(0) = 0 (p1 ≈ 0.296, p2 ≈1.669, and w2 ≈ 1.421). The evolution of entropy for this initial distribution

with f = 0.058 is plotted in Figure 3.5b. The entropy grows initially but

after about ten steps in the simulation it turns over and begins to decrease,

eventually reaching Ss = 0 as expected for f = 0.058. This is in marked

contrast to the behaviour of S(t) in an ideal gas, where one has dS(t)/dt ≥ 0

according to Boltzmann’s H-theorem.

The population distribution is determined by dividing the wealth axis into

small bins and computing the number of agents that fall in each bin. One can

define the multiplicityW as the number of permutations of the agents between

different wealth bins such that the occupation numbers of the bins do not

change. The definition of entropy, Ss = logW , leads to the expression (3.11) in

the continuous limit. Entropy maximization under the assumption that total

wealth is conserved gives an exponential distribution. However, this ignores

the global constraints on the probability distribution ps(w) imposed by the

transfer fraction f . The maximization procedure must take these constraints

into account to derive the steady-state wealth distribution appropriate to the

giver scheme. Unfortunately, at the time of writing, we have been unable to

derive these additional constraints, and they do not appear in the literature.

73

3. Laplace transform analysis of a multiplicative asset transfer model

log10 ξ(w)

w

00.5

11.5

22.5

3−

5

−4

−3

−2

−1 0 1

(a)

wi

i

0100

200

300

400

500

600

700

800

900

1000

0

0.2

0.4

0.6

0.8 1

1.2

1.4

1.6

1.8 2

(b)

Figu

re3.6:

(a)Thelim

itingprob

ability

distrib

ution

ξ(w),show

nwith

crosses,ob

tained

fromon

erealization

oftheasy

mmetric

random

process

(3.13)for

f=

0.05after

106iteration

s.Themean

is0.9998

andthevarian

ceis

0.0519(cf.

0.0526theoretically

fromthemaster

equation

).Thestead

y-state

distrib

ution

ps (w

)for

thesam

efob

tained

byLap

laceinversion

isshow

nas

asolid

curve

forcom

parison

.(b)Thefirst

1000valu

esof{w

i }.

74

3. Laplace transform analysis of a multiplicative asset transfer model

Random process

The evolution of wealth in the giver scheme can be analyzed in terms of a

random process defined by

wi+1 = wi +∆wi, (3.13)

where w1 = 1 and ∆wi = +f or ∆wi = −fwi with equal probability. This

process is asymmetric, i.e. multiplicative in the negative direction and addi-

tive in the positive. To illuminate the relationship between the giver scheme

and the random process we note that (1) an agent’s loss of wealth is always

proportional to his wealth, i.e. it is multiplicative, and (2) an agent’s gain

of wealth can originate from any other agent in the population and therefore

equals f〈w〉 on average, or simply f if we set 〈w〉 = 1. We compute the lim-

iting distribution ξ(w) of this process by applying (3.13) a sufficiently large

number of times and then constructing a histogram of all {wi}. By the ergodic

assumption, this is equivalent to computing a large number of realizations of

this random process and using the final values in each realization to find the

limiting distribution.

Figures 3.6a and 3.6b display one particular realization of the random

process (3.13) and the corresponding limiting distribution. Given the close

link between the transfer model and the random process, it is not surprising

that the limiting distribution ξ(w) of the random process (3.13) appears to be

identical to the steady-state distribution ps(w) of the giver scheme. We obtain

similar results for other values of f ≪ 1. Note that the slight discrepancy

between ξ(w) and ps(w) at w > 2 is due to insufficient sampling of large

wealths by the random process. The agreement improves as the number of

iterations increases.

Inequality of wealth

A traditional measure of inequality in economic systems is the Gini coefficient,

defined asG = 1−2∫ 10 L(X)dX, where L(X) is the Lorenz curve. For a contin-

uous distribution, we have L(w) =∫ w0 dw′ w′ps(w

′), X(w) =∫ w0 dw′ ps(w

′),5

and hence

Gs = 1− 2

∫ ∞

0dw ps(w)

∫ w

0dw′ w′ps(w

′), (3.14)

5 Note that we have L(X) ≤ X for all X because L(0) = 0, L(1) = 1, and dL/dX is amonotonically increasing function.

75

3. Laplace transform analysis of a multiplicative asset transfer model

Gs

σ2s

10−2 10−1 100 1010

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 3.7: Gini coefficient of the steady-state distribution ps(w) as a functionof the variance σ2s = f/(1− f).

such that Gs = 0 corresponds to perfect equality, and Gs = 1 to perfect

inequality.

In Figure 3.7 we plot Gs versus the steady-state variance σ2s = f/(1 − f)

for 0.01 ≤ f ≤ 0.9. As expected, Gs increases monotonically with σ2s , since

both quantities are measures of dispersion. Interestingly, however, there is

an inflection point in the Gs(σ2s) curve at σ2s = 1, Gs = 1/2, corresponding

to the exponential distribution (i.e. f = 1/2). For f → 0, we have ps(w) →δ(w− 1), which corresponds to perfect equality since all agents have the same

wealth. On the other hand, for f → 1, the distribution ps(w) becomes sharply

peaked near w = 0, while the standard deviation approaches infinity. This

corresponds to the situation where most agents have zero wealth, except for

one who has everything, i.e. perfect inequality.

We can understand the evolution towards inequality in terms of the state

vector of the system. Consider N agents whose wealths are characterized

by random variables wi, i = 1, 2, . . . , N . The state of the system can be

described by the phase-space vector w = (w1, w2, . . . , wN ). The constraints

that define the phase-space are (1) 0 ≤ wi ≤ 1 for all i, and (2)∑N

i=1 wi = 1

(we assume for convenience that the total wealth is unity). These constraints

76

3. Laplace transform analysis of a multiplicative asset transfer model

define a segment of the (N − 1)-dimensional hyperplane embedded in N -

dimensional space. Without any additional constraints, entropy maximization

Kapur (1989) gives gi(wi) = (N−1)(1−wi)N−2 for the probability distribution

of the wealth wi of the i-th agent, with mean 〈wi〉 = 1N and variance σ2wi

=N−1

N2(N+1). However, in our system, the asset transfer process and the value of

the parameter f place additional restrictions on the evolution of w. For the

increment of the phase-space vector w from time tk to time tk+1, i.e. after one

generation of asset transfers, we have

|∆w|2 =N∑

i=1

[wi(tk+1)− wi(tk)]2. (3.15)

The terms in the sum on the right hand side can be split into two groups,

associated with the givers and the receivers. Since the transfer amount is

proportional to the giver’s wealth, we get

|∆w|2 = 2f2∑

i∈givers

[wi(tk)]2. (3.16)

Therefore the following inequality must always be satisfied:

|∆w| ≤ 21/2f |w|. (3.17)

In addition, we have

N−1/2 ≤ |w| ≤ 1 (3.18)

due to the restrictions of the phase-space itself. Note that the state wi = 1/N

for all i is the nearest point to the origin.

When f is small, the norm of the increment |∆w| is also small compared

with the maximum linear extent of the phase space (which equals 21/2); the

evolution of w is gradual. Furthermore, |∆w| is also constrained by |w|, whichcan be very small if N is large and all agents are clustered as close to the origin

as possible. So, if the dispersion in wealth is modest, w moves slower through

the phase space than if there is great inequality. On the other hand, w changes

more rapidly on average if |w| is close to unity, which corresponds to large

inequality. The states of equality are therefore more probable, which explains

why the steady-state distribution tends towards a delta function for f → 0.

Even if the initial wealth distribution is very unequal, w drifts quickly towards

the states of near equality.

When f is close to 1, |∆w| can be comparable to the size of the phase

space. Since f is large, the gains in wealth of the individual agents can be

77

3. Laplace transform analysis of a multiplicative asset transfer model

large as well. This leads to the situation where a few agents own most of the

wealth. These agents retain their large wealth for a short time only (typically a

few time steps) before they become givers and pass their large wealth to other

agents. In the extreme case f = 1, one agent possesses all the wealth at any

instant, while all the other agents have zero wealth. This maximum wealth is

passed from agent to agent frequently. This corresponds to w jumping from

one corner of the phase space to another. For f ≈ 1, w evolves similarly, with

ps(w) peaking strongly at zero wealth.

3.6 Conclusions

We develop a new technique for computing the steady-state probability dis-

tribution of a multiplicative asset transfer model, which we call the giver

scheme, by Laplace transforming the associated master equation to give a

functional equation for the characteristic function of the distribution. In the

giver scheme, the transfer amount fwg is proportional to the giving agent’s

wealth wg, so the model depends on a single parameter f ∈ (0, 1). We develop

an efficient iterative method to solve the functional equation for any f , and

we employ several Laplace inversion algorithms to recover the steady-state

distribution ps(w).

We comprehensively explore the dependence of the wealth distribution on

the value of f , especially the thinly studied regime 1/2 ≤ f < 1. We find

a stark qualitative difference between the distributions for f ≈ 0 (sharply

peaked distribution centred around the mean wealth) and f ≈ 1 (broad dis-

tribution of approximately power-law shape with overlaid oscillations). These

two extremes correspond to near-perfect equality and inequality respectively,

as characterized by the Gini coefficient. Both extremes are also characterized

by negative Boltzmann entropy. While the regime f ≈ 0 is generally thought

to represent to some extent the exchange processes occuring in the real econ-

omy, the regime f ≈ 1 is probably less applicable to realistic economic systems,

except perhaps in situations involving extreme leverage. The regime f ≈ 1

may also be relevant to the analysis of gambling, where transitory fortunes

are made and lost frequently.

We show that the Boltzmann entropy is unlikely to be a faithful measure of

disorder in a multiplicative asset transfer system, since it does not vary mono-

tonically as a function of time, assuming the second law of thermodynamics.

This is an important and counterintuitive result, because the system in the

78

3. Laplace transform analysis of a multiplicative asset transfer model

giver scheme is closed and the microscopic transfer rules conserve wealth, in a

manner reminiscent of the microcanonical ensemble in statistical mechanics.

In a multiplicative transfer system, the correlations between various subsys-

tems (e.g. subclasses corresponding to a particular historical sequence of giving

and receiving) and the time-reversal asymmetry of the microscopic rules are

crucial to the system’s dynamics and, therefore, cannot be ignored.

3.7 Iterative procedure

We assume that the computations are carried out with 16 significant digits.

For a given complex argument z, define a uniform grid u = {uk}K1 that covers

the interval [−4, log10(|z|)]. The approximation (3.6) gives sufficient precision

for |z| < 10−4 for computations with 16 significant digits. Choose the num-

ber of points K such that there are a large number of points in every unit

interval, say, 103 logarithmic grid points per decade in [10−4, |z|]. Define two

auxiliary grids, u(f) = log10(f) + u and u(1−f) = log10(1 − f) + u, and ini-

tialize the iterations with g0(10uz/|z|) = 1/(1 + 10uz/|z|). For a given set of

values gi(10uz/|z|), defined on the grid u, find the corresponding values on

the auxiliary grids by performing a spline interpolation or using the approxi-

mation (3.6) where appropriate. Then use these values in equation (3.10) to

find gi+1(10uz/|z|). Continue iterating until the convergence criterion is met

(we find that the convergence spreads gradually from zero to |z|). Typically

the convergence requires a few dozens of iterations for |z| ∼ 100 in the posi-

tive half-plane. Once the convergence is reached, apply a spline interpolation

to find g(z′) for any z′ along the same direction in the complex plane as z,

provided that |z′| < |z|.The obvious disadvantage of the procedure outlined above is that it relies

on interpolation. Its precision is therefore limited by the number of points in

the grid u, i.e. the discretization of the interval [0, |z|]. It is possible, however,to avoid interpolation altogether by defining a special non-uniform grid that

is invariant with respect to division by f and (1 − f). This gives rise to an

alternative procedure for computing the iterations.

Define a grid rk,m = fk(1 − f)m with 0 ≤ k ≤ K and 0 ≤ m ≤ M ,

where K = ⌈log(10−4/|z|)/ log(f)⌉ and M = ⌈log(10−4/|z|)/ log(1 − f)⌉ are

defined such that |z|rK,0 < 10−4 and |z|r0,M < 10−4. The function g(z) on

the grid zk,m = rk,mz, defined according to gk,m = g(rk,mz), has the following

properties: g(fzk,m) = g(zk+1,m) = gk+1,m and g[(1 − f)zk,m] = g(zk,m+1) =

79

3. Laplace transform analysis of a multiplicative asset transfer model

gk,m+1. Therefore the iteration rule becomes

gk,m =gk,m+1

2− gk+1,m, (3.19)

for 0 ≤ k ≤ K − 1 and 0 ≤ m ≤ M − 1. For gK,M one can use the ap-

proximation (3.6). In fact, the Taylor expansion can be used for any point

|zk,m| < 10−4. Thus, no interpolation is required and the iterations can be

computed more efficiently. However, unlike the approach based on interpola-

tion, this procedure must be repeated for different arguments even if they lie

in the same direction in the complex plane.

80

Chapter 4

A note on the theory of fast

money flow dynamics

The gauge theory of arbitrage was introduced by Ilinski in Ilinski (1997) and

applied to fast money flows in Ilinskaia and Ilinski (1999); Ilinski (2001). The

theory of fast money flow dynamics attempts to model the evolution of cur-

rency exchange rates and stock prices on short, e.g. intra-day, time scales. It

has been used to explain some of the heuristic trading rules, known as tech-

nical analysis, that are used by professional traders in the equity and foreign

exchange markets. A critique of some of the underlying assumptions of the

gauge theory of arbitrage was presented by Sornette in Sornette (1998). In

this paper, we present a critique of the theory of fast money flow dynamics,

which was not examined by Sornette. We demonstrate that the choice of the

input parameters used in Ilinski (2001) results in sinusoidal oscillations of the

exchange rate, in conflict with the results presented in Ilinski (2001). We

also find that the dynamics predicted by the theory are generally unstable in

most realistic situations, with the exchange rate tending to zero or infinity

exponentially.

4.1 Introduction

Fast money flows are analyzed in Ilinskaia and Ilinski (1999); Ilinski (2001) in

terms of the lattice gauge theory of arbitrage developed in Ilinski (1997). The

main idea of the theory is that the dynamics should only depend on gauge

invariant quantities rather than the exchange rates themselves. Changing the

units in which stocks of currency are denominated obviously changes the nom-

81

4. A note on the theory of fast money flow dynamics

inal exchange rate. However, it is obvious that such changes of scale, i.e. gauge

transformations, should have no effect on its dynamics. Some assumptions of

the theory have been criticized in Sornette (1998); for example, the lack of

justification for the exponential form of the weight of a given market config-

uration. However, the results of the theory reported in Ilinskaia and Ilinski

(1999); Ilinski (2001) seem impressive, reproducing in particular some of the

phenomenological rules of technical trading employed by professional traders.

Hence the theory appears to be a promising tool for analyzing the markets.

In this note, we present our analysis of the theory of fast money flow dy-

namics and re-examine the results presented in Ilinskaia and Ilinski (1999);

Ilinski (2001). In Sect. 4.2, we present the derivation of the dynamical equa-

tions of the theory. In Sect. 4.3, we examine the dynamics predicted by the

theory for various initial conditions. We highlight certain inconsistencies in

the theory, the unstable dynamics for most realistic values of the parameters

and initial conditions, and the resulting problems in applying the theory to

technical trading. In Sect. 4.4, we revisit the action and demonstrate that the

expression used in Ilinskaia and Ilinski (1999); Ilinski (2001) is inconsistent

with the evolution operator resulting from the lattice formulation.

4.2 Lattice gauge theory and fast money flow

dynamics

In analogy with quantum electrodynamics, Ilinski identified the exchange rate

S between two currencies with the field and the trading agents with matter.

In general, the exchange rate dynamics depends on the interest rates of the

underlying currencies. However, since we are interested in intra-day dynamics

only, we consider the special case of zero interest rates. Ilinski tacitly assumed

that the interest rates of the two currencies are identical, i.e r1 = r2. In this

paper we set r1 = r2 = 0 and assume that transaction costs are zero.

The part of the action s1 that describes the dynamics of the field on its

own is formulated by identifying arbitrage on the lattice with the curvature,

which gives

s1 = − 1

2σ2

∫ T

0dt

(

dy

dt

)2

. (4.1)

In Eq. (4.1), T is the investment horizon and σ2 is the volatility (presumed

to be constant in the interval 0 ≤ t ≤ T ). This expression is equivalent to a

Gaussian random walk in y = lnS.

82

4. A note on the theory of fast money flow dynamics

The effect of the field y on “matter”, i.e. the trading agents, is described

by the Hamiltonian

H(ψ1, ψ+1 , ψ2, ψ

+2 ) = H21ψ

+1 ψ2 +H12ψ

+2 ψ1, (4.2)

where ψ+k and ψk are creation and annihilation operators for agents in currency

k (k = 1, 2), and the coefficients H21 and H12 depend on y. According to

Ilinski, H21 = heβy and H12 = he−βy, where h and β are constants (we

discuss the motivation behind these formulas in Sect. 4.4). Following the

standard treatment of a quantum harmonic oscillator (see, e.g. Slavnov and

Faddeev (1980)), Ilinski Ilinski (2001) derived a path-integral expression for

the evolution operator in terms of the coherent states ψ1 and ψ2, which are

the eigenstates of the annihilation operators ψ1 and ψ2 respectively. From the

evolution operator one can obtain the expression for the part of the action s2

that represents the field’s effect on matter:

s2 =

∫ T

0dt

[

ψ1dψ1

dt+ ψ2

dψ2

dt+H(ψ1, ψ1, ψ2, ψ2)

]

, (4.3)

where the overbar denotes complex conjugation.

Finally, departing from the electrodynamics analogy, Ilinski introduced

Farmer’s term F to describe the effect of matter on the field. As a result, the

action s1 is replaced by

s1F = − 1

2σ2

∫ T

0dt

[

d(y + F )

dt

]2

, (4.4)

where

F =f

M(ψ1ψ1 − ψ2ψ2), (4.5)

M is the total number of agents, and f is a constant (Ilinski (2001) uses α in

place of f).

The total action is given by

s = s1F + s2. (4.6)

Following Ilinski, we introduce new variables η = βy and τ = ht, and replace

complex-valued ψk with φk and ρk, defined by ψk = (Mρk)1/2e−iφk (k = 1, 2)

and ρ1 + ρ2 = 1. Ilinski identifies Mρk with the number of agents in currency

k; the total number of agents is conserved. The action can be written as

s =M

∫ hT

0dτ L, (4.7)

83

4. A note on the theory of fast money flow dynamics

0 1 2 3 4 5 6 7 8 9 10−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

Figure 4.1: Re-creation of Ilinski’s solution of Eqs. (4.9–4.11) given on page 169of Ilinski (2001) for α1 = 1.5, α2 = 10, C0 = 0, and the initial conditions:η(0) = 0.2, υ(0) = 0, ρ(0) = 0.5. The factor α1 in Eq. (4.10) is replaced withunity to match Ilinski’s Euler-Lagrange equations. The displayed quantitiesare as follows: ρ− 1/2 (solid), υ + η (dashed), η (dot-dashed), υ (dotted).

where the Lagrangian L is given by

L = −(2α2)−1(

η′ + α1ρ′)2

+ ρυ′ + φ′2+

+ 2[ρ(1 − ρ)]1/2 cosh(υ + η), (4.8)

with α1 = 2βf , α2 =Mβ2σ2/h, ρ = ρ1, υ = φ1−φ2. A prime denotes a deriva-

tive with respect to τ . Due to the unique structure of the Lagrangian (4.8),

the resulting Euler-Lagrange equations can be simplified to the following first

84

4. A note on the theory of fast money flow dynamics

0 1 2 3 4 5 6 7 8 9 10−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

Figure 4.2: The solution of Eqs. (4.9–4.11) for the same parameters and initialconditions as in Fig. 4.1. The factor α1 in Eq. (4.10) is restored.

order differential equations:

η′ = α2(1/2 − ρ)−− 2α1[ρ(1− ρ)]1/2 sinh(υ + η) + C0,

(4.9)

υ′ = 2ρ− 1)[ρ(1 − ρ)]−1/2 cosh(υ + η)+

+ 2α1[ρ(1− ρ)]1/2 sinh(υ + η),(4.10)

ρ′ = 2[ρ(1 − ρ)]1/2 sinh(υ + η). (4.11)

However, some of the second-order nature of the Euler-Lagrange equations is

retained in the constant C0 = η′(0) + α1ρ′(0) + α2[ρ(0) − 1/2], whose value

depends explicitly on the derivatives ρ′(0) and η′(0). The equation for φ2 is

85

4. A note on the theory of fast money flow dynamics

trivial and we omit it. To solve Eqs. (4.9–4.11), one needs to specify the initial

conditions η(0), υ(0), ρ(0), and η′(0), which uniquely determine C0 (note that

ρ′(0) is given by Eq. (4.11)). Alternatively, one can set η(0), υ(0), ρ(0), and

C0, which uniquely determine η′(0).

4.3 Analysis of the Euler-Lagrange equations

Missing coefficient

By introducing new variables, ρ = ρ − 1/2 and η = υ + η, and linearizing

(|ρ| ≪ 1, |η| ≪ 1), we obtain η = ρ′ and

ρ′′ + (α2 − 4)ρ = C0. (4.12)

For α2 > 4, the general solution is

ρ = A sin(2πνt+ θ) + C0(α2 − 4)−1, (4.13)

η = 2πνA cos(2πνt+ θ), (4.14)

with ν = (α2−4)1/2/2π (A and θ are found from the initial conditions). This is

inconsistent with the solutions presented in Ilinskaia and Ilinski (1999); Ilinski

(2001), which exhibit oscillations decaying slowly with time. The origin of this

inconsistency can be traced to a simple algebraic mistake in the derivation of

the equations of motion given in Ilinskaia and Ilinski (1999); Ilinski (2001).

On page 168 of Ilinski (2001), the second term on the right-hand side of the

equation for υ′ is missing a factor α1. The same coefficient is also missing in the

equations given in Ilinskaia and Ilinski (1999). This is essentially equivalent

to replacing α1 in our Eq. (4.10) with unity, while keeping α1 in our Eq. (4.9)

intact.

We verify the above by numerically solving Eqs. (4.9–4.11) in their incor-

rect form (with α1 missing from one of the equations as in Ilinskaia and Ilinski

(1999); Ilinski (2001)) and in their correct form derived in this paper. We are

able to perfectly reproduce1 the plots presented on page 169 of Ilinski (2001)

by solving the incorrect equations (see Fig. 4.1). Note that we have α1 = 1.5

and α2 = 10 for the parameters used in Ilinski (2001). Ilinski claimed to set

η′(0) = 0 (dy(0)/dt = 0 in his notation), but this is obviously incorrect; the

solutions he presented are obtained for C0 = 0, which gives η′(0) ≈ −0.3020.

1In the caption of figure 7.2 in Ilinski (2001), it is claimed that one of the quantitiesdisplayed is η (y in Ilinski’s notation), but actually η + υ is plotted.

86

4. A note on the theory of fast money flow dynamics

0 1 2 3 4 5 6 7 8 9 10−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

Figure 4.3: The solution of Eqs. (4.9–4.11) for the same parameters and initialconditions as in Fig. 4.2, except α1 = 0.

As anticipated by the linearized analysis, the correct nonlinear equations of

motion do not show any decay in the oscillation amplitude (see Fig. 4.2).

Furthermore, we do not observe any enhancement of oscillations for smaller

values of α1, as Farmer’s term becomes less important. In fact, the solutions

for α1 = 0 plotted in Fig. 4.3 are only slightly different from those for α1 = 1.5

(cf. the plots given on page 171 of Ilinski (2001)). After some exploration, we

conclude that Farmer’s term does not have any critical effect on the dynamics

of the system; it only affects the amplitude of oscillations of η and υ, and their

phase shift from ρ.

87

4. A note on the theory of fast money flow dynamics

Unstable solutions

In Sect. 4.3, we explored the dynamics of η = η + υ in the case C0 = 0.

However, there is no a priori reason why the initial conditions should conspire

to give C0 = 0. In this section, we briefly examine the dynamics of η = β lnS

in the more general case C0 6= 0.

Linearizing Eqs. (4.9) and (4.10) gives

η′ = −α2ρ− α1η + C0, (4.15)

υ′ = 4ρ+ α1η. (4.16)

We find that the solutions for η and υ are also harmonic oscillations plus an

extra term linear in time. The average value of η changes linearly with time

at a rate −4C0(α2−4)−1, while the average of υ changes at the same rate but

with the opposite sign. This behaviour is illustrated in Figs. 4.4 and 4.5 (note

that ρ and η remain small, so the linearization assumption is not broken).

Thus, for C0 > 0, the exchange rate S decays exponentially to zero, whereas

for C0 < 0, it grows exponentially. In both cases the exponential time-scale is

given by τc = 0.25(α2 − 4)|C0|−1.

Technical trading

Ilinski justified certain rules employed in technical trading (see Ilinskaia and

Ilinski (1999) and pages 170–173 of Ilinski (2001)), e.g., the use of positive

and negative volume indices (PVI and NVI respectively), by appealing to the

solutions of the equations of motion. The relevant figures are presented in

Ilinski (2001) on pages 170 (figure 7.3) and 172 (figure 7.7). We identify the

trading volume V with |ρ′| and the return R with η′/β = S′/S. In Ilinski

(2001), the derivative of η = υ+ η is used incorrectly instead of η to compute

the return (see also footnote 1). For comparison, we plot the volume and the

return curves in Fig. 4.6, computed using the correct equations of motion and

C0 = 0. The quantities plotted in figure 7.7 of Ilinski (2001) are not specified,

nor are the parameters and initial conditions, so we do not comment on that

figure’s validity.

Ilinski used the trading volume and the return curves to construct con-

tinuous2 versions of PVI and NVI. The details of the construction are left

unspecified. However, the PVI and NVI are usually computed from daily re-

turns, not from continuous intra-day variables. In any event, the resulting

2In technical trading, these quantities are discrete and defined by recursive formulas.

88

4. A note on the theory of fast money flow dynamics

0 1 2 3 4 5 6 7 8 9 10−1

−0.5

0

0.5

1

Figure 4.4: The solution of Eqs. (4.9–4.11) for the same parameters and initialconditions as in Fig. 4.2, except with C0 = 0.1 instead of C0 = 0. The curvesare coded as in Figs. 4.1 and 4.2.

construction must depend strongly on the time-scale that is chosen, since the

indices are defined recursively. Examining figure 7.7 in Ilinski (2001), one

observes that, for instance, the continuous PVI is constant if the trading vol-

ume V is decreasing with time and changes linearly if V is increasing, with a

slope of +1 where the return curve R is positive and −1 where R is negative.

However, this simple trend is inconsistent with the recursive definitions of the

PVI and NVI employed in technical trading.

Moreover, the constant-amplitude solutions we employed in this section

only exist for C0 = 0. In all other cases, the exchange rate converges to

89

4. A note on the theory of fast money flow dynamics

0 1 2 3 4 5 6 7 8 9 10−1

−0.5

0

0.5

1

Figure 4.5: As for Fig. 4.4, with C0 = −0.1.

zero or diverges to infinity exponentially on a short time scale. The condition

C0 = 0 requires precise alignment between the initial values ρ(0), v(0), η(0),

and η′(0). There is no reason to expect that such a precise alignment will

be observed at any time in the real market. Therefore, the lattice gauge

model predicts unrealistic behaviour (e.g., exponential divergence if C0 < 0)

of the exchange rate under most circumstances. Given the issues raised in

this section, it is premature to conclude that the technical trading schemes

employed by market participants can be justified by the lattice gauge model.

90

4. A note on the theory of fast money flow dynamics

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

Figure 4.6: The trading volume V (bold solid curve) and the return R (bolddot-dashed curve) for the same parameters as in Fig. 4.2. For comparison, wealso display ρ− 1/2 (thin solid curve) and η (thin dot-dashed curve).

4.4 Revisiting the action

We conclude by re-examining the derivation of the action s given by Eq. 4.6.

Consider two currencies, referred to as currency 1 and currency 2, linked by

an exchange rate S(t) that depends on time t, such that the amount C2 of

currency 2 at time t corresponds to the amount C1 = S(t)C2 of currency 1. We

assume that the currencies can only be exchanged at the discrete times tn =

n∆t (n = 0, . . . , N) and define Sn = S(tn). At any given time tn, an agent can

decide to either exchange his stock of currency for the counterpart currency

91

4. A note on the theory of fast money flow dynamics

or keep his position, in which case his stock of currency remains unchanged

(recall that we neglect interest rates completely since we are interested in the

intra-day dynamics). We display these possibilities in Fig. 4.7, showing part

of the lattice from time tn to time tn+1.

✲Sn

✛S−1n

✲Sn+1

✛S−1

n+1

✻

❄

✻

❄

Currency 2 Currency 1

tn

tn+1

Figure 4.7: Lattice diagram for the intra-day foreign exchange trading in twocurrencies. Interest rates are ignored.

The returns on arbitrage along the closed loops of the elementary plaquette

shown in Fig. 4.7 are given by S−1n Sn+1−1 for the clockwise loop and SnS

−1n+1−

1 for the counter-clockwise loop. The total return SnS−1n+1 + S−1

n Sn+1 − 2 is

identified in Ilinski (1997) with the curvature on the lattice and, therefore, the

corresponding discrete action is given by

A1 =N∑

n=0

an(SnS−1n+1 + S−1

n Sn+1 − 2). (4.17)

Assuming that for any n we have an∆t → 1/2σ2 in the limit ∆t → 0, we

obtain the continuous action s1 given by (4.1). No justification is given in

Ilinski (1997, 2001) for why the limit of an∆t must be finite. The expression

for Farmer’s term was derived in Ilinski (2001), but we omit it because its

inclusion has no critical effect on the dynamics (see Sect. 4.3).

In order to derive the Hamiltonian given by Eq. (4.2) and the expressions

for the coefficients H12 and H21, Ilinski considered the case of a single trader

first and then generalized to multiple traders by using creation and annihi-

lation operators. In the case of a single trader, Ilinski postulated that the

probability of a given path Q through the lattice from t0 to tN is exponen-

tially weighted with respect to s(Q) = ln(U1U2 . . . UJ), where {Uj} are the

parallel transport coefficients on the lattice (note that J > N for most paths).

Thus, for a given path Q, the probability is given by

P (Q) ∼ eβs(Q). (4.18)

92

4. A note on the theory of fast money flow dynamics

Depending on the path, a given Uj can be Sn, S−1n , or unity (note that Ilinski

introduces a new gauge, under which the exchange rates remain unchanged,

except at t0 and tN where they equal unity; see pages 131–132 of Ilinski (2001)

for more details). The state of the trader is characterized by the probabilities

p1 and p2 of being in currency 1 and currency 2 respectively. The evolution

of the state vector ( p1p2 ) can be described by the transition matrix

P (tn; tn−1) =

(

1 Sβn

S−βn 1

)

, (4.19)

which Ilinski essentially identifies3 with the discrete version of the continuous

evolution operator U(t, t′) that satisfies

∂U

∂t= HU, (4.20)

where H is the Hamiltonian and U(0, 0) is the identity matrix. Ilinski claim

that the expression for the transition matrix (4.19) and the formula (4.20)

result in

H =1

∆t

(

0 Sβ

S−β 0

)

. (4.21)

Finally, identifying the parameter h with 1/∆t, we obtain the expressions for

H12 and H21, the Hamiltonian H given by (4.2), and the action s2.

In deriving the action Ilinski considered a more general case of non-zero

interest rates, but this does not nullify the two issues pointed out below.

Firstly, we note that the Hamiltonian given by (4.21) becomes infinite in

the limit ∆t → 0. It is stated in Ilinski (2001) that ∆t in the continuous-

time calculations “stands for the smallest time-scale of the theory, the time

cut-off” (see page 133). However, if ∆t is retained in the finite form in the

Hamiltonian and, therefore, the action s2, it must also appear in the finite

form in the expression for the action s1 for consistency. Secondly, we observe

that the transition matrix P (tn; tn−1) is degenerate; its determinant is zero.

Therefore, it cannot possibly be identified with the evolution operator. We

conclude that the justification provided for the Hamiltonian (4.2) in Ilinski

(2001) is insufficient.

3 In the case of non-zero interest rates, P (tn; tn−1) is related to U(tn; tn−1) by a simplematrix transform (see page 132 of Ilinski (2001)); however, P (tn; tn−1) = U(tn; tn−1) ifr1 = r2 = 0 and the transaction costs are zero.

93

4. A note on the theory of fast money flow dynamics

4.5 Conclusions

We have examined the theory of fast money flow dynamics developed in Ilin-

skaia and Ilinski (1999); Ilinski (2001) and uncovered errors in 1) the derivation

and the analysis of the equations of motion based on the theory, and 2) the

justification of the action based on the lattice gauge formalism.

The equations of motion presented in Ilinskaia and Ilinski (1999); Ilin-

ski (2001) are missing the coefficient α1 in one term, crucially modifying the

dynamics of the system. We also find that most of the solutions of the equa-

tions of motion, in their correct form derived in this paper, are unstable with

respect to the initial conditions, resulting in unrealistic behaviour of the ex-

change rate. We show that the justification of the technical trading given in

Ilinski (2001) is based on an erroneous interpretation of the variables related

to the exchange rate and on the stability predicted by the incorrect equations

of motion.

The theory of fast money flows relies on a particular form of the Hamilto-

nian that describes the effect of the exchange rate on the actions of the agents.

We demonstrate that this form is not consistent with the lattice gauge formu-

lation and diverges in the continuum limit.

94

Chapter 5

Memory on multiple

time-scales in an Abelian

sandpile

We report results of a numerical analysis of the memory effects in two-dimen-

sional Abelian sandpiles. It is found that a sandpile forgets its instantaneous

configuration in two distinct stages: a fast stage and a slow stage, whose du-

rations roughly scale as N and N2 respectively, where N is the linear size

of the sandpile. We confirm the presence of the longer time-scale by an in-

dependent diagnostic based on analysing emission probabilities of a hidden

Markov model applied to a time-averaged sequence of avalanche sizes. The

application of hidden Markov modeling to the output of sandpiles is novel. It

discriminates effectively between a sandpile time series and a shuffled control

time series with the same time-averaged event statistics and hence deserves

further development as a pattern-recognition tool for Abelian sandpiles.

5.1 Introduction

The Abelian sandpile is a simple open dynamical system governed by deter-

ministic rules, which demonstrates interesting emergent behaviour. Its charac-

teristic feature is the presence of avalanches, whose scale-free size distribution

is bounded above by the size of the sandpile. Sandpiles have been studied ex-

tensively in the past two decades both computationally and analytically, but

many of their properties remain unexplained Dhar (2006); Pruessner (2012).

95

5. Memory on multiple time-scales in an Abelian sandpile

Most research has concentrated on the microstructure and properties of the

avalanches or on long-term average properties of the recurrent configurations.

The intermediate time-scale, that covers many avalanches but not so many as

to reach the long-term averages, has not been investigated as thoroughly due

to its complexity. In this paper we present the results of numerical experiments

on Abelian sandpiles on the intermediate time-scale.

Starting from an arbitrary configuration in the recurrent regime, subse-

quent grain drops and the avalanches they cause gradually modify the dis-

tribution of charges that defines the starting configuration. This results in a

gradual loss of the memory of the starting configuration. Some features are

lost quickly while others persist for a long time until eventually all memory of

the starting configuration is lost. Even though the basic features of memory

loss are known Dhar (2006), the details of this process are not well understood.

For instance, we do not know how the rate of memory loss changes with time,

nor do we know which features of the starting configuration are responsible

for maintaining memory on different time scales. These issues are important

for developing a deeper understanding of the dynamical properties of Abelian

sandpiles. We explore these issues by conducting numerical simulations of a

two-dimensional sandpile and analysing the output in two independent ways:

(i) the distribution of occupation numbers in absolute difference maps, and

(ii) emission probabilities in a hidden Markov model (HMM). Hidden Markov

models Rabiner (1989) owe their success to ease of implementation and ef-

fectiveness in capturing temporal patterns. They have become one of the

standard tools in such diverse areas as speech recognition Pieraccini (2012)

and bioinformatics Durbin (1998). Since hidden Markov models incorporate

Markov chains in their setup, they allow one to examine statistically the suc-

cession of events, or states, an important capability missing in standard tech-

niques of statistical analysis. To our knowledge, this technique has not been

applied to sandpiles. We show here by way of a control experiment that it

captures hidden patterns in the succession of time-averaged avalanche sizes.

We discuss the relevant properties of Abelian sandpiles and introduce site

occupancy fractions in Section 5.2; their statistical properties are briefly ex-

plored in Section 5.3. Section 5.4 defines absolute difference maps and presents

results on the dual-time-scale evolution of sandpile memory. Hidden Markov

models are introduced in Section 5.5, where we emphasize their pattern recog-

nition and classification capacities. Their application to time-averaged se-

quences of avalanche sizes as a memory diagnostic is described in Section 5.6.

96

5. Memory on multiple time-scales in an Abelian sandpile

5.2 Abelian sandpiles

A sandpile is a cellular automaton introduced in Bak et al. (1987) as an exam-

ple of a slowly driven system with dissipative boundaries that spontaneously

evolves to a critical state, which is characterised by scale-free distributions. It

does not require any parameter adjustment to achieve criticality, unlike (say)

the Ising model. Hence this system and its numerous derivatives demonstrate

what has become known as self-organised criticality Jensen (1998). The orig-

inal model described in Bak et al. (1987) is closely based on its prototype,

an actual pile of sand onto which grains of sand are dropped. The slope of

the pile increases, as grains are added, until an avalanche occurs, whereupon

the slope decreases. The time of the next avalanche is unpredictable and the

distribution of avalanche sizes is approximated by a power law. A sandpile

is an open dynamical system with random driving and deterministic rules for

toppling.

In Dhar (1990), a generalisation of the original toppling rules was proposed,

which has the property that the outcome of an avalanche does not depend on

the order in which the critical sites topple. Most research in the last two

decades has concentrated on this model known as the Abelian sandpile rather

than the original model where an avalanche does depend on the order of

the toppling. The properties of Abelian sandpiles have been investigated by

means of computer simulations and analytical approaches including branching

processes and spanning trees, Abelian operators, loop-removed random walks,

mean-field theory, renormalisation techniques and the logarithmic conformal

field theory (see Pruessner (2012) for a comprehensive review).

Toppling rules

In a typical simulation of a two-dimensional sandpile one considers a square

grid of sites described by a matrix zij , i, j = 1, . . . , N , where N is frequently

chosen to be a multiple of two. The values of zij are variously known as

height, slope, or charge. Initially the sandpile is assumed to be empty, with

zij = 0. At each step of the evolution, the charge of a randomly selected site

is increased by one, representing a grain drop at this site. That is, the charge

of a site can be thought as the number of grains located at the site. A site

is considered stable if zij < 4 and unstable or critical if zij ≥ 4. If a grain

drop creates an unstable site, an avalanche begins and further drops are halted

97

5. Memory on multiple time-scales in an Abelian sandpile

until the avalanche finishes. A critical site contains at least four grains and

it topples by distributing four of its grains into the four neighbouring sites,

i.e. if (i, j) is a critical site, then it topples according to the following rule:

zij → zij − 4, zi±1,j → zi±1,j + 1, and zi,j±1 → zi,j±1 + 1. If a critical site

is at the edge of the sandpile, it lacks one or two neighbours and the grains,

which would have gone to those neighbouring sites, are removed (i.e. they fall

off the edges of the pile). The avalanche continues until there are no critical

sites left, after which regular evolution of the sandpile consisting of random

grain drops resumes. The number of topples in an avalanche is referred to as

the size (or power) of the avalanche.

Site occupancy

If the sandpile starts from an empty configuration, initially there are no or few

critical sites and the charge increases steadily at most sites. The sandpile is in

the transient regime. After about 2.125N2 steps the mean charge reaches the

expected value of 2.125, after which the influx of grains is balanced on average

by the efflux over the edges due to avalanches. At this stage the sandpile is

in the recurrent regime. Once the sandpile is in a recurrent configuration, it

can only move to another recurrent configuration; given enough time any re-

current configuration is recreated eventually. The so-called burning algorithm

Dhar (1990) can be used to establish whether a configuration is recurrent or

transient. It recursively removes every site, whose charge is equal to or greater

than the number of its neighbours, starting from the edges of the sandpile.

The number of recurrent configurations is estimated to be ∼ 3.21N2

Dhar

(2006), which for a large sandpile is a vanishingly small fraction of the total

number of configurations 4N2

.

The problem of determining the probability pk that a randomly chosen

site has charge k in a recurrent configuration has been solved analytically in

the limit N → ∞ using various techniques such as uniform spanning trees

Priezzhev (1994); Caracciolo and Sportiello (2012), loop-erased random walks

Kenyon and Wilson (2011), and the dimer model Poghosyan et al. (2011). The

analytical expressions give p0 ≈ 0.0736, p1 ≈ 0.1739, p2 ≈ 0.3063, p3 ≈ 0.4462.

The number of sites nk with charge k = 0, 1, 2, 3 can be easily computed for

a given configuration zij. Then, fk = nk/N2 gives the fraction of sites with

charge k = 0, 1, 2, 3, and one has∑3

k=0 fk = 1. Considering recurrent configu-

rations only, the ensemble-averaged quantities 〈fk〉 approach the probabilities

98

5. Memory on multiple time-scales in an Abelian sandpile

pk as N → ∞. For finite sandpiles, the ensemble-averaged fractions 〈fk〉 devi-ate from pk, and the discrepancy increases as N decreases. The mean sandpile

charge can be estimated numerically by∑3

k=1 k〈fk〉 for finite N and is given

by∑3

k=1 kpk = 2.125 in the limit N → ∞.

Toppling waves

Avalanches in an Abelian sandpile can be viewed as waves of toppling Ivashke-

vich et al. (1994), where each wave is defined by allowing all critical sites, bar

the original critical site, to topple until there are no critical sites left, at which

point the original critical site is toppled again thereby launching the next

wave. The region encompassed by a wave consists of a set of contiguous sites

(without holes) each of which has toppled once. After a wave stops, the sites

whose charge differs from the original charge before the wave passes through

are located at the boundary of this region, while all internal sites regain their

original charge. The boundary itself consists of two thin layers; the wave

causes the charge of the sites at the outer layer to increase and the charge of

the sites at the inner layer to decrease. Avalanches that consist of multiple

waves can be thought of as a series of consecutive waves; subsequent waves

can be larger or smaller than preceding waves, and there is no restriction on

the number of waves in a given avalanche (note that the number of waves

is equal to the number of times the trigger site topples Fey et al. (2010)).

The structure of a multi-wave avalanche is therefore more complex than that

of a single-wave avalanche. Furthermore, the interaction of the waves in a

multi-wave avalanche is likely to be a factor in the observed deviation in two

dimensions of the avalanche size distribution from the power-law predicted by

the mean-field approximation Abdolvand and Montakhab (2010).

Temporal correlations

In this paper, we concentrate on the problem of sandpile memory, as mani-

fested in the temporal sequence of avalanche sizes. Detailed, time-averaged

statistics of output sequences have not received much attention in the litera-

ture. Most of the emphasis has been on the power spectral density of sandpile

“noise”. Indeed, the original motivation for studying sandpiles was to model

the long-range temporal correlations with a 1/f -type power spectrum exhib-

ited by some dynamical systems. Further studies revealed that the power

spectrum of the sandpile’s temporal activity has 1/f2 behaviour (see Jensen

99

5. Memory on multiple time-scales in an Abelian sandpile

0.0 0.1 0.2 0.3 0.4 0.5fk, k = 0, 1, 2, 3

100

101

102

p(f

k)

Figure 5.1: Probability density function p(fk) of the fraction fk of sites withcharge k = 0, 1, 2, 3 (left to right) for two sandpile simulations on a squaregrid with N = 32 (blue) and N = 64 (red), based on samples consisting of103N2 steps in the recurrent regime. The curves show the normal distributionsfor the corresponding values of the mean and standard deviation; they agreewell with the data. The first 5N2 steps of the simulations are discarded toeliminate transient configurations.

(1998) for details). Studies of temporal and spatial correlations in sandpiles

typically use some modification of the standard rules of the Abelian sandpile,

e.g. Davidsen and Paczuski (2002); Barrat et al. (1999), or confine the in-

vestigation to sandpiles consisting of narrow strips (a quasi-one-dimensional

geometry), e.g. Yadav et al. (2012); Maslov et al. (1999). As loss of memory in

sandpiles is related to relaxation due to grain movement caused by avalanches,

we also note the study of residence time of grains Dhar and Pradhan (2004)

where the scaling function of probability distribution of residence time is de-

rived explicitely for a simplified one dimensional model and a constraint is

obtained in the general case. In this paper, in contrast, we use the standard

Abelian sandpile model to investigate memory effects concerned with specific,

time-ordered changes in the global structure of a given sandpile configuration,

as the sandpile evolves away from it.

5.3 Site occupancy fraction distributions

The simulations we perform follow the standard recipe for a two-dimensional

Abelian sandpile on a square grid withN2 sites. We start each simulation from

the zero configuration zij = 0 (the empty sandpile). A single grain is added

at each step at a random position; no grains are added during an avalanche.

100

5. Memory on multiple time-scales in an Abelian sandpile

N µ1 µ2 µ3 µ4 σ1 σ2 σ3 σ432 0.0790 0.1844 0.3110 0.4256 0.0062 0.0102 0.0142 0.010564 0.0763 0.1793 0.3090 0.4354 0.0030 0.0050 0.0071 0.0051∞ 0.0736 0.1739 0.3063 0.4462

Table 5.1: Mean values and standard deviations for the samples shown inFigure 5.1. The bottom row gives the analytical estimates of probabilities pk,obtained in the limit of an infinitely large sandpile.

Dissipation happens at the edges of the sandpile, where grains fall out of the

system. The first 5N2 steps are excluded from the statistical analysis to avoid

transient configurations. Following each grain drop the sandpile is allowed to

relax, if the drop causes an avalanche. The sandpile configuration at time-step

t is given by the matrix zij after t grain drops and relaxations.

The first set of simulations explores the baseline behaviour of the fractions

fk introduced earlier. The probability density function p(fk) for each k is

presented in Figure 5.1 for a simulation consisting of 103N2 time-steps in

the recurrent regime. The numerical output can be fitted reasonably well

(see Figure 5.1) by Gaussian distributions with the parameters recorded in

Table 5.1. The standard deviations decrease by approximately a factor of

two as N doubles. As N increases the mean values µk = 〈fk〉 approach the

analytical estimates pk obtained in the limit N → ∞.

5.4 Short- and long-term memory in site

occupancy

One way to study memory in an Abelian sandpile is to measure the difference

between two configurations separated by a certain time interval. The details

of this approach are as follows. For a given t0 > 5N2 we compute the absolute

difference matrix Dij(t, t0) = |zij(t)−zij(t0)| between the initial configuration

zij(t0) at time-step t0 and a subsequent configuration zij(t) at t > t0; this

approach is somewhat reminiscent of how damage is measured in (Stapleton

et al., 2004) even though in our case no damage is introduced but two con-

figuration at different times are compared. Since for each position (i, j) we

have zij(t) ∈ {0, 1, 2, 3} at any time-step t, entries in the matrix Dij(t, t0) are

restricted to the same set of values, and we have Dij(t0, t0) = 0. Dij(t, t0)

can be used to describe the rate and dynamics of how the sandpile forgets

the configuration at t0 as t increases, but it contains too much information to

101

5. Memory on multiple time-scales in an Abelian sandpile

be practical. It is more manageable to consider the fractions δk of sites with

value k in the absolute difference matrix Dij . This is the same calculation

applied previously to the matrix zij to find fk but now applied to Dij to find

δk. The fractions δk are functions of the time delay t − t0 and can change

significantly after an avalanche. As we do not wish to be biased towards a

particular avalanche history, we consider the ensemble-averaged fractions 〈δk〉based on a large sample of trials.

As a control experiment, one can compute the fractions δk for the absolute

difference matrix of two sandpile configurations taken randomly from the set

of recurrent configurations. Given a large enough sample of such configura-

tions, one obtains distributions for δk and mean values 〈δk〉, which can also

be estimated analytically in the limit N → ∞ as

d0 = p20 + p21 + p22 + p23, d1 = 2(p0p1 + p1p2 + p2p3),

d2 = 2(p0p2 + p1p3), d3 = 2p0p3, (5.1)

with approximate values d0 ≈ 0.3285, d1 ≈ 0.4055, d2 ≈ 0.2003, d3 ≈ 0.0657.

For instance, a site (i, j) with Dij = 3 arises only if the corresponding location

in the two configurations used to compute the absolute difference has 0 in

one configuration and 3 in the other, or vice versa. Since the probabilities

to observe charge 0 and 3 at a given location are given by p0 and p3, the

probability to find 0 at a given location in one configuration and 3 at the same

location in the other configurations is given by p0p3, which implies d3 = 2p0p3.

The ensemble-averaged fractions 〈δk〉 converge to dk once enough time elapses,

such that the configuration at t loses all memory of the initial configuration

at t0 as t→ ∞. Note that for a relatively small sandpile such as 32× 32, it is

more relevant to use the mean values µk in place of pk to compute dk, since

the probabilities pk are determined in the limit N → ∞.

To test for this loss of memory, we conduct a second set of simulations,

where we compute δk as a function of t− t0 for a large number of trials. We

use several sizes ranging from N = 16 to N = 128 and run each trial for at

least N2 time-steps, after which the value of t0 is reset to the current time-

step t for the next trial. The number of trials varies from 105 for N = 16

to 100 for N = 128. The results for N = 32 with 104 trials are shown in

Figures 5.2a and 5.2b. We observe that the fractions 〈δk〉 converge to dk on

the time-scale ∼ N2, which is close to the time-scale over which a sandpile

starting with zij = 0 approaches the recurrent regime. Note that the ensemble-

averaged curves shown in Figure 5.2 eliminate most of the jitter characteristic

102

5. Memory on multiple time-scales in an Abelian sandpile

100

101

102

103

t−

t 0

0.0

0.2

0.4

0.6

0.8

1.0

〈δk〉

k=0

k=1

k=2

k=3

(a)

0200

400

600

800

1000

t−t 0

10−4

10−3

10−2

10−1

100

log10|〈δk〉−dk|

k=0

k=1

k=2

k=3

(b)

Figure

5.2:

Siteoccupan

cyan

alysisof

anAbeliansandpile.

(a)Ensemble-averagedsite

occupan

cyfraction

s〈δ

k〉v

ersusthetime

delay

t−t 0,whereδ k

isthefraction

ofsiteswithchargek=

0,1,2,3in

theab

solute

differen

cematrixD

ij=

|z ij(t)−z ij(t

0)|.

Theen

semble-averagedcu

rves

arebased

on10

4trials

(onerepresentative

trialis

show

nin

black).

Thestan

darderrorof

the

ensemble-averagedvalues,whichis

∼0.001,

isnot

show

n.Thedashed

lines

representtheexpectedvaluesdkfort→

∞an

dN

=32.(b)Approachof

〈δk〉t

otheexpectedvaluesdkshow

non

alogarithmic

scale.

103

5. Memory on multiple time-scales in an Abelian sandpile

of individual trials, the level of which (estimated visually to be ∼ 0.1) gives

∼ 0.1/√104 = 0.001 for the standard error of the ensemble-averaged curves.

Interestingly, Figure 5.2a reveals that the memory of a given configuration is

lost in two stages: a fast stage on the time-scale ∼ N , characterised by rapid

changes in 〈δk〉, and a slow stage on the long time-scale, where the fractions

〈δk〉 vary slowly. The transition from the fast stage to the slow stage is followed

by a small bump or depression in 〈δk〉 near t − t0 ≈ 100 superimposed over

approximately exponential curves. Moreover, 〈δ2〉 approaches d2 noticeably

quicker than the other fractions.

The loss of memory during the fast and slow stages is roughly exponential,

and fitting the sum of two exponentials (Koehler, 2012) to each memory curve

shown in Figure 5.2b yields the following decay times: τ0 = 21, τ1 = 20,

τ2 = 22, τ3 = 24 for the fast stage and τ0 = 190, τ1 = 200, τ2 = 100,

τ3 = 230 for the slow stage, where the subscript refers to the charge k. The

large uncertainty in the exponential fits does not allow us to infer accurate

scaling dependence of the decay times, but we find it broadly consistent with

the approximate scaling dependence of the fast and slow stages, i.e. ∼ N and

∼ N2.

At the beginning of the fast stage, each avalanche contributes fully (or

nearly fully) to the departure from the original configuration and therefore to

rapid memory loss. Even though some of the sites affected by an avalanche

may have already been changed by previous avalanches, they constitute a

small fraction of the affected sites. However, the fraction of such sites tends to

increase with each avalanche, until eventually the majority of sites affected by

a new avalanche have already been changed by previous avalanches. At this

stage, each avalanche contributes marginally to memory loss and therefore

the memory loss slows down. We hypothesise that this corresponds to the

transition from the fast stage to the slow stage.

5.5 Hidden Markov model

The short- and long-term memory loss displayed in Figure 5.2 pertains to

global quantities, such as the site occupancy fractions 〈δk〉. Another global

characteristic, which can be used to address the problem of memory loss, is the

sequence of avalanches and their associated sizes. By averaging avalanche size

over a fixed time interval we obtain a quantity describing the average activity

of the sandpile during that period. The patterns in the sequence of average

104

5. Memory on multiple time-scales in an Abelian sandpile

sizes contain information about sandpile memory. To access this information,

we now conduct an experiment using hidden Markov methods. The method

is described in Section 5.5 and the results of the experiments in Section 5.6.

Given a sequence of observed states characterising some system, a hidden

Markov model seeks to represent the observed sequence in terms of 1) an un-

derlying finite-state Markov chain that cannot be observed and 2) an emission

model, which gives the probability of each observed state when the Markov

chain is in a given hidden state (see Rabiner (1989)). In this paper we only

consider HMMs with discrete observed and hidden states. At each time-step

the system is described by one of the observed states and is assumed to be

in one of the hidden states. The meanings (i.e. the physical significance) of

the hidden states are not known a priori. The Markov chain is described by a

set of transition probabilities, which form an n × n matrix A, where n is the

number of hidden states. The emission model is described by a set of emission

probabilities, which form an n ×m matrix B, where m > n is the number of

observed states. To complete the HMM, one also needs to define a vector π of

initial probabilities of the hidden states. The set {A,B, π} defines the HMM.

The Baum-Welch forward-backward algorithm is a likelihood-maximisation

technique that recursively adjust the probabilities of the HMM in such a way

that the probability of the observed sequence increases Rabiner (1989). The

recursive adjustment, also referred to as training (or learning), continues until

the probability of the observed sequence ceases to increase, and the HMM pa-

rameters stabilise. To start training, one specifies some initial HMM, i.e. some

values for transition probabilities, emission probabilities, and initial probabili-

ties of the hidden states. The rate of convergence and the limit of convergence

depend strongly on the initial specification. There is no guarantee of conver-

gence to an optimal model, so one performs training for a range of initial

HMMs. The training algorithm is easy to implement and its description can

be found in e.g. Rabiner (1989); Li et al. (2000).

To illustrate the classification potential of an HMM consider the following

example Cave and Neuwirth (1980); Stamp (2004). A text in English can be

treated as a sequence of 27 states, consisting of 26 characters of the alphabet

and the symbol for space, after all punctuation marks and other symbols are

removed. An HMM with two hidden states, whose meanings are not known

a priori, can be trained on a given text. Remarkably, the resulting emission

probabilities reveal that one hidden state, say, H0, corresponds to vowels and

the other, say, H1, to consonants. Specifically, if the system is in the hidden

105

5. Memory on multiple time-scales in an Abelian sandpile

H0 H1

H0 0.32 0.68H1 0.65 0.35

O0 O1 O2

H0 0.00 0.32 0.68H1 0.65 0.35 0.00

Table 5.2: Transition probabilities Aij (left) between the hidden states i =0, 1 (H0,H1) and emission probabilities Bij (right) from the hidden states toobserved states j = 0, 1, 2 (O0, O1, O2) for the averaging time-scale Ta = 32.

state H0 at a certain position in the text, the probability of the observed

state in that position is high for vowels and low for consonants. The reverse

is true for the hidden state H1, i.e. the probability is high for consonants

and low for vowels. Since an HMM strives to capture the foremost statistical

properties of the succession of observed states, one concludes that the most

prominent statistical feature of an arbitrary English text in terms of the two-

state classification is the dichotomy between vowels and consonants.

5.6 HMM analysis of long-term memory

We harness the classifying power of HMMs to investigate memory loss in a

sandpile. At each step a sandpile can be characterised by the output size of

the released avalanche; if no avalanche is initiated, the size is zero. The power

released by an avalanche after each drop depends in an unpredictable way on

where exactly the dropped grain lands. Since the grains are dropped randomly,

this translates into shot noise in the observed avalanche size, which has little

to do with the intrinsic internal structure of the sandpile. To suppress the

shot noise, we average the avalanche size over an averaging interval Ta and

bin the average size into three bins (low, medium, high), such that the number

of samples in each bin is the same.

We apply an HMM to the average size time-series generated by the 32 ×32 sandpile (N = 32), specifically to a sequence of 103N3 time-steps in the

recurrent regime (the first 5N2 steps are discarded). We consider several

averaging time-scales spaced logarithmically in the range N ≤ Ta ≤ N3.

For each averaging time-scale we feed a sequence of 103 samples of average

avalanche size into the HMM; that is, the observed sequence contains 103

terms for each averaging time-scale Ta. The observed sequence consists of

three states. We assume there are two hidden states. For each Ta we use 20

random initial HMMs and train each model for several thousand steps, which

is sufficient for the best models to stabilise.

106

5. Memory on multiple time-scales in an Abelian sandpile

3226

2728

291024

211

212

213

214

215

Ta

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0.751

Bij

(a)sandpilesequen

ce

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211

212

213

214

215

Ta

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Bij

(b)shuffled

sandpilesequen

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Figure

5.3:

Hidden

Markovan

alysisof

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(a)Emission

probab

ilitiesB

ijfrom

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statei=

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Adiscrete

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ofacontrol

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entwherethesameinputdatais

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lybeforebeingfedinto

theHMM.

107

5. Memory on multiple time-scales in an Abelian sandpile

The transition and emission probabilities of the final HMM with the high-

est likelihood are shown in Table 5.2 for Ta = 32. We observe that the hidden

state i = 0 is biased towards the lower values of the averaged size and i = 1

is biased towards the higher values. In other words, when the sandpile is in

the hidden state i = 0 for the duration of the averaging period Ta = 32, the

avalanches have higher power on average (loud state) as compared to when

the sandpile is in the hidden state i = 1, when avalanches have lower power

on average (quiet state). We also observe that a transition from one to the

other hidden state is roughly twice as likely as the lack of a transition.

The emission probabilities Bij of the best HMM at the end of training

for other values of Ta are displayed in Figure 5.3a. The values Bij are fairly

stable in the range N ≤ Ta ≤ 2N2, where N = 32, and the same applies to the

transition probabilities. This indicates that there is no fundamental difference

in the statistics of the average size sequence on these time-scales. However,

for Ta > 2N2 the final HMMs are qualitatively different. Loud and quiet

states are no longer distinguishable, as can be seen in Figure 5.3a, and the

likelihood of a transition between the hidden states increases markedly. This

result is consistent with the memory horizon observed in the memory plots

discussed previously (Figure 5.2). It is unlikely that any memory will persist

over a time interval longer than ∼ N2 time steps. Indeed, a square sandpile

with N2 sites takes about 2.125N2 steps to reach the recurrent regime from

an empty configuration, where 2.125N2 is the expected mean number of sand

grains in the sandpile. In the recurrent regime, one might expect the sandpile

to migrate to a completely different configuration on the same time-scale.

The results of a control experiment are shown in Figure 5.3b, where we

apply the same procedure as above to randomly shuffled sequences of sizes, i.e.

the temporal pattern is shuffled without changing the probability distribution

of sizes. We find that the final HMM resembles that obtained in Figure 5.3a

for Ta > 2N2. That is, shuffling the sequence of sizes completely destroys the

classification pattern found by the HMM on the original unshuffled sequences

of average sizes.

5.7 Conclusion

Our simulations indicate that an Abelian sandpile forgets a given configuration

in two stages: a fast stage on the time-scale ∼ N and a slow stage on the

time-scale ∼ N2. The details of memory loss are embedded in the particular

108

5. Memory on multiple time-scales in an Abelian sandpile

sequence of grain drops and avalanches, which is reflected in the behaviour of

site occupancy fractions as functions of time. By taking the ensemble averages

of the site occupancy curves over many configurations, we eliminate the details

of a particular sequence of random drops and derive the smooth memory curves

characteristic of a given Abelian sandpile. We find that memory loss is roughly

exponential during the fast and slow stages, with the site occupancy fraction

for charge 2 decaying much faster than the other fractions. We also observe a

hint of an oscillation in the memory curves following the transition from the

fast to the slow stage.

An independent analysis based on hidden Markov modeling confirms that

memory extends up to the time scale ∼ N2. The analysis identifies the hidden

states with quiet and loud periods in the sandpile’s evolution, during which

the avalanche power is respectively low or high on average. We note a re-

markable consistency in the output of the hidden Markov models obtained

from avalanche size sequences averaged on different time-scales ranging from

N to N2. This indicates that the statistical properties of the succession of the

observed states, and by implication the hidden states too, are similar across a

broad range of time-scales. As the waiting time between individual avalanches

is exponential, the temporal patterns detected by the hidden Markov model

must be driven by longer term structural changes in the sandpile. The tem-

poral patterns disappear if the observed sequence is shuffled, even though the

distribution of avalanche sizes remains the same.

Our HMM analysis focuses on the avalanche size, since this is the simplest

and most prominent characteristic of the sandpile evolution. The patterns

detected by the HMM are interesting but difficult to interpret without further

insight into the underlying structure of the sandpile. Of course, the sandpile

possesses many other characteristics that can be fed, after averaging, into an

HMM. Future studies could address and exploit sandpile properties like the

waiting time between avalanches, number of waves per avalanche, the mean

charge of the sandpile, or the occupancy numbers, just to name a few. More-

over, one can combine several characteristics into a single observed quantity to

take advantage of correlations between variables. We do not know at present

which combination of parameters will be the most successful in detecting the

underlying patterns of the sandpile, but future numerical experiments should

be able to address these questions.

Other promising avenues for future studies include 1) improving statisti-

cal significance of the reported results by increasing the number of samples

109

5. Memory on multiple time-scales in an Abelian sandpile

used for computing the memory curves and in the hidden Markov analysis, 2)

extending the simulations to larger two-dimensional and higher-dimensional

sandpiles, as well as other sandpile models such as Manna and Oslo models

(Manna, 1991; Christensen et al., 1996), 3) developing a dynamical model of

memory loss, and possibly based on the mean-field approximation or other

analytical techniques, which explains the evolution of the ensemble-averaged

site occupancy fractions.

110

Chapter 6

Conclusion

The main objective of this work is to contribute to our understanding of

the economy and financial system by conducting a number of projects in the

emerging field of econophysics, which uses techniques developed in statistical

physics to model and analyse economic and financial systems. The thesis ad-

dresses the following issues: 1) empirical investigation of transactional flows

between commercial banks and their network properties, 2) analytical and

agent-based investigation of wealth distributions and income inequality, 3) an

analysis of lattice-gauge theories of fast money flows on the foreign exchange

market, 4) numerical study of memory effects in stochastic dynamical systems

and the application of hidden Markov models to discover temporal patterns

in such systems. An important component of this work was to create a foun-

dation for future research in the field of econophysics at the University of

Melbourne. This was achieved by exploring a broad range of diverse issues of

interest and methods employed in econophysics research such as multi-agent

simulations and network science. The over-arching theme that ties the issues

explored in the thesis is the nature and properties of monetary instruments

and their role in the modern economy.

6.1 The interbank network

Empirical investigation of the Australian interbank transactional flows and

their network properties described in chapter 2 contributes to the growing

body of literature on the interbank network properties in other countries,

their network topologies, and properties of the flows. It is based on the data

provided by the Reserve Bank of Australia through their real-time gross set-

111

6. Conclusion

tlement system that captures all high-value transactions between banks in

Australia. The study offers a unique view of the structure and variability of

daily monetary flows in the Australian banking system and is the first study

to report on the Australian interbank network and its variability. Another

unique feature of this study is simultaneous investigation of the transactional

flows due to overnight loans and the flows due to other (nonloan) payments.

The sample of interbank transactions provided by the RBA consists of all

payments including overnight loans; the loans have been found by following

Furfine (2003). The procedure involves comparing transactions on two con-

secutive days and detecting those transactions that reverse on the next day

with the same amount plus interest, which closely matches the central bank’s

target rate. Just under 900 overnight loans have been identified over the

course of four days (a week in February 2007) out of over 95000 transactions

over the same period. The overnight loans account for less than 1% of all

transactions and about 6.5% of value of all payments. The interest rate of the

overnight loans is found to lie within 0.1% per annum of the target rate set

by the RBA, which was 6.5% per annum during this period. The distribution

of the nonloan transactions is well represented by a mixture of two lognormal

components, which are likely to correspond to the transactions arising from

the SWIFT and Austraclear feeds of the gross settlement system.

A major finding of the study is strong anti-correlation (whose value is

about −0.9 on most days) between the daily imbalances of overnight loans

and nonloans. The daily imbalance is computed for each bank and represents

the cumulative change in the reserve account of the bank, i.e. it is equal to

the difference in value of all incoming and outgoing transactions on a given

day. The payments recorded by the RBA correspond to financial and other

transactions between the customers of commercial banks and to a smaller

extent between the banks themselves. The daily stream of such payments is

largely random and results in an unpredictable change in the reserve accounts

of the commercial banks, with some banks’ accounts increasing in value while

the others decreasing. The money held in the reserve accounts attracts smaller

interest than the target rate set by the RBA, which encourages the banks with

the positive imbalances to lend the excess in the short term money market. At

the same time, banks whose reserves are depleted seek to eliminate the deficit,

and the implied liquidity risk, by acquiring overnight loans. This creates

an interesting dynamics in the interbank network, whereby flows of nonloan

transactions create imbalances of the banks’ reserves, which in turn engender

112

6. Conclusion

flows of overnight loans to remove the imbalances. The study reported in

Chapter 2 confirms this dynamics and provides empirical constraints on the

extent of the connection between loan and nonloan flows.

Comparing the interbank networks on different days during the week re-

veals that about 80% of nonloan flows persist for two days or more, although

the amount of persistent flows can change significantly. For the overnight loan

flows, only 50% of the flows persist. Furthermore, persistent loan flows carry

about 65% of the total value of the loan flows, whereas persistent nonloan

flows account for as much as 96% of the total value of nonloan flows. There

is no significant correlation between individual loan and nonloan flows, i.e.

overnight loans and other payments are linked via imbalances only.

As for the net flows, the number of transactions per net flow is approx-

imated well by a power law with the exponent α = −1 for nonloans and

α = −1.4 for overnight loans. Out of about 470 net nonloan flows per day

there are 110 flows that consist of a single transaction (usually these occur be-

tween small banks); more than 1000 transactions can be present in a net flow

between two large banks. Similarly, out of about 60 loan flows per day there

are as many as 40 flows that consist of a single transaction (between small

banks); loan flows between large banks can consist of 30 individual loans or

more. Given the small number of commercial banks in Australia compared

to other countries, where the interbank networks have been studied, it is not

surprising that the shape of the degree distribution of the Australian inter-

bank network is difficult to infer. It is inconsistent with a power law, which

has been observed in many countries, and is close to an exponential distribu-

tion, although this could not be rigorously confirmed. The loan and nonloan

networks are disassortative with an assortativity coefficient of about −0.4 for

nonloans and −0.1 for loans on average. The topology of the net flows is

found to be highly variable, with many circular and transitive flow structures

present.

The study of the interbank network based on five days provides a valuable

insight into the dynamics of the interbank network. However, to understand

statistical properties of the network and its variability requires follow-up stud-

ies based on longer sequences of data. A longer sequence of data will also allow

to constrain the contribution from interbank loans with two-day maturity or

longer. In addition, the intraday timing of transactions will allow to inves-

tigate the dynamics at more finely grained time scales, which is significant

for understanding the interbank network’s reaction to external events such

113

6. Conclusion

as changes in target interest rates or other relevant economic news. These

future studies are essential for uncovering the dynamical laws that govern the

dynamics of the interbank flows composed of high-value transactions, particu-

larly given that the internal dynamics of monetary flows in interbank networks

has been neglected so far. The relationship between loan and nonloan flows via

the daily imbalances of the banks that was reported in Chapter 2 is a natural

one but it has not been addressed quantitatively in the studies of the banking

networks in other countries. Therefore, there is a need for similar studies in

order to confirm this relationship in other countries where the properties and

the institutional design of the banking system may be different.

Another area of future studies that could stem from research reported in

Chapter 2 concerns multi-agent numerical simulations of transactional flows

in banking systems. These simulations are of particular interest as they al-

low one to probe various mechanism that determine bank’s behaviour in the

overnight loan market in response to changes in their reserve accounts. An

appropriate model for nonloan transactions also needs to be investigated. A

naive approach where nonloan payments are assumed to be randomly taken

from a suitable distribution may be incorrect. Indeed, a significant reduction

in the reserve account of a bank indicates the preponderance of outgoing pay-

ments (by value) over the incoming payments. If this occurs, it may reduce

the likelihood of further outgoing payments, since the funds of the bank’s

customers have been reduced. Similarly, a significant increase in the reserve

account of a bank may increase the likelihood of outgoing payments. Since

the overnight loans have to be paid back (with interest) on the next day (or

the day after next in case of two-day loans), the largely random dynamics

of nonloan payments has an effect on the following days and therefore the

dynamics on a given day cannot be considered independent of previous days.

This may have serious implications for stability of the interbank network and

the continuous operation of the real time gross settlement system.

6.2 Wealth distributions

The studies reported in Chapter 3 seek to refine our understanding of asset

exchange systems, and the giver scheme in particular, by proposing an efficient

technique for numerically computing the shape of the wealth distribution in

the steady state. The technique also provides an interesting example of em-

ploying a numerical inverse Laplace transform to solve the master equation

114

6. Conclusion

that describes the detailed balance of the giver scheme by matching influx and

outflow of agents at every value of wealth. In the steady state, the Laplace

transform of the master equation yields a functional equation that is amenable

to analysis, e.g. by Taylor series, which gives a recursive expression for the

moments of the wealth distribution. The functional equation is solved numer-

ically by iterations in the complex plane; numerical experiments reveal that

the convergence does not depend on the shape of the initial approximation or

details of the grid. The solution of the functional equation thus obtained is

then fed into the numerical inverse Laplace transform, which yields the wealth

distribution for the specified value of the transfer parameter.

The procedure for computing the wealth distribution in the giver scheme

compares favourably with the direct approach of estimating the distribution by

running a multi-agent simulation of the asset exchanges. It is computationally

faster and provides much better precision than estimating the distribution

by computing the histogram of the agents’ wealth. In particular, it gives a

handle on the asymptotic behaviour of the distribution function at extreme

values of wealth where the number of agents is small. It is confirmed that the

asymptotic behaviour is not exponential, even though a closed-form expression

for the tail has not been found. The dependence of the wealth distribution on

the value of the transfer parameter and the corresponding changes in wealth

inequality are investigated in Chapter 3. The Gini coefficient, a common

measure of income inequality, is computed for a range of different values of

the transfer parameter.

The wealth distributions obtained for small values of the transfer parame-

ter are found to be qualitatively different from those for values close to unity,

which corresponds to the case when the givers concede most of their wealth in

a single exchange (the gambling scenario). Namely, the former distributions

(small values of the transfer parameter) are peaked around the mean of the

distribution and are characterised by low level of inequality (Gini coefficient

close to zero), whereas the latter (values close to unity) are approximately

power-law in shape with overlaid oscillations and show high level of inequality

(Gini coefficient close to unity). This unexpected oscillatory pattern in the

wealth distribution is found to be approximately periodic on logarithmic scale

with the period inversely proportional to the fraction of wealth retained by

the givers.

The giver scheme is a closed system with constant amount of total wealth

and no friction. It can be expected to exhibit the usual traits of Boltzmann

115

6. Conclusion

entropy, which measures the level of disorder in the system, i.e. the entropy

is expected to rise steadily as the system evolves towards the steady state.

However, multi-agent simulations reported in Chapter 3 demonstrate that the

entropy of the system evolves in a non-monotonic fashion, in stark conflict

with the expectations based on its behaviour in physical systems such as an

ideal gas. The Boltzmann H-theorem, according to which entropy cannot

decrease, is not applicable to the giver scheme since its microscopic rules of

exchange are not symmetric with respect to time reversal. Indeed, the exact

rules that reconstruct past behaviour of system in the reverse time order can

be worked out easily but they are different from the rules describing exchange

when time order is normal. Therefore, Boltzmann entropy is not a faithful

measure of disorder in a multiplicative asset transfer system.

The study reported in Chapter 3 raises a number of questions, which could

inform future research efforts in this area. Firstly, the technique of using the

master equation and the Laplace transform is applicable to asset transfer sys-

tems with rules different from those employed in the giver scheme. Secondly,

since Boltzmann entropy fails to behave properly in the giver scheme, other

measures on entropy, e.g. Tsallis or Renyi entropy, need to be investigated

in the context of asset exchange systems that lack time symmetry. Thirdly,

an intriguing area of future research concerns exchange systems where the

total amount of money in the systems is not conserved and, moreover, can be

produced endogenously by the system’s participants.

The giver scheme and most other exchange models investigated by econo-

physicists view money as a commodity that cannot be created by the agents,

which is consistent with the perspective commonly accorded to consumers

and companies in mainstream economics. According to the mainstream view,

base money (currency) is created by the central bank and is multiplied by

commercial banks through fractional reserves, such that the banks can loan

the deposited money as long as they retain a certain fraction of all deposits

(the reserve). However, this picture conflicts with how money is actually cre-

ated by commercial banks, which create new money when they make loans

(McLeay and Radia, 2014). The actual process of money creation is endoge-

nous and is the opposite of what is implied by the money multiplier theory.

In practise, loans created by the bank engender new deposits, which in turn

raise the amount of base money as the central banks attempts to accommo-

date the demand for cash in the economy. Modeling the banking system and

its impact on the economy in this light presents a timely opportunity to make

116

6. Conclusion

a contribution which is both theoretically interesting and rich with significant

social implications in terms of the institutional design of the banking system.

6.3 Critique of fast money flow theory

The objective of Chapter 4 is to analyse the lattice gauge model of fast money

flow dynamics. The assumptions of the theory are reviewed; a careful deriva-

tion of the Euler-Lagrange equations that determine the model dynamics are

given. It is found that the dynamics of the model reported by Ilinski (2001)

is inconsistent with the Lagrangian derived there. For instance, Ilinski (2001)

observes that the oscillations in exchange rate are slowly decaying with time

for a certain combination of initial values of the model variables. However, it

is shown in Chapter 4 that the oscillations persist indefinitely with no decay

in this case. The inconsistency is traced to an algebraic error in the deriva-

tion of the Euler-Lagrange equations given in Ilinski (2001). Furthermore, it

is shown that the constraint on the initial values of model variables used by

Ilinski (2001) is unrealistic. If the constraint is relaxed, the dynamics become

unstable with the exchange rate either growing exponentially or decaying ex-

ponentially to zero. In light of these results, the implications for technical

analysis are re-evaluated and it is found that the model provides no support

for technical trading.

Furthermore, it is observed in Chapter 4 that the continuous form of the

action has not been sufficiently motivated. The part of the action that de-

scribes the dynamics of the exchange rate, which is identified with a field, is

obtained by taking the limit ∆t → 0, whereas the part that represents the

effect of the exchange rate on the number of agents in each currency uses a

finite value of ∆t as one of the input parameters of the model. The transition

from discrete evolution of the state vector, which represents the number of

agents in each currency at each time step, to continuous evolution described

by a Hamiltonian is unjustified, since the transition matrix can be degenerate

and therefore its action cannot be identified with the evolution operator.

6.4 On memory in sandpiles

In chapter 5, the Abelian sandpile is introduced as a toy model that captures

some of the features of financial markets in order to explore the connection

between observed changes in the market, e.g. price changes in the foreign ex-

117

6. Conclusion

change (FX) market, and underlying structural features of the market, which

are not observed. The emphasis of the study is on analysing Abelian sandpiles

rather than establishing a plausible model of the FX market. To that end,

chapter 5 investigates memory effects in the sandpile. It examines the hidden

structural changes resulting from grain drops and avalanches and relates those

changes to the observed avalanches by analysing the sandpile evolution with

a hidden Markov model to capture patterns in the average intensity of the

avalanches.

The investigation reported in chapter 5 concerns two-dimensional Abelian

sandpiles on a square grid. The quantitative measure of memory loss employed

in the study relies on computing site occupancy fractions, which are equal to

the number of sites occupied by a specific charge (ranging from 0 to 3 in

a two-dimensional sandpile) normalised by the total number of sites. Site

occupancy fractions are closely related to the probabilities that a given site

has a specific charge; the two concepts coincide in the limit of the infinite

sandpile in a recurrent configuration. Each of the four occupancy fractions

varies as the sandpile evolves. The distribution of each occupancy fraction

computed from a large number of configurations is found to be approximated

well by a Gaussian.

The memory loss is determined by comparing sandpile configurations sep-

arated by a certain time delay measured in grain drops. Occupancy fractions

are computed for the absolute difference of these configurations and the frac-

tion’s variation with time delay is investigated. As time delay increases, the

fractions approach constant values characteristic of the absolute difference

maps of unrelated configurations. The analysis described in chapter 5 shows

that the most common value in the absolute difference map is 1, followed by

0 and then 2, with 3 being the least common. It is interesting to note that

as far as the sandpile configuration is concerned the most common charge is

3, followed by 2, 1, and 0, in order of frequency. The dynamics of memory

loss is revealed most conveniently by comparing the fractions for the absolute

difference maps of time-delayed configurations with their expected values for

the absolute difference maps of unrelated configurations, for which time delay

is effectively infinite.

The simulations show that the memory of a given configuration is lost

in two stages, each characterised by approximately exponential decay as mea-

sured by the rate of approach of the fractions to the expected values. The first

stage is characterised by a faster decay; its time-scale is proportional to the

118

6. Conclusion

linear size of the sandpile. The second stage scales with the number of sites

in the sandpile (linear size squared) and consequently lasts much longer than

the first stage. It is not clear at present what is responsible for the two-stage

pattern of memory loss. It is left to future research to uncover the detailed

mechanisms responsible for this behaviour and develop an analytical model of

the fractions as a function of the time delay from the initial configuration.

Chapter 5 also attempts to answer the following question: can memory

loss be detected in the sequence of observed quantities that characterise the

sandpile’s activity, e.g. avalanche size (which in the financial context may cor-

respond to large exchange rate movements in the foreign exchange market, for

example). To eliminate “shot noise” due to random grain drops, the average

avalanche size over a specific time window is fed as a sequence into a hidden

Markov model. The hidden Markov model looks for patterns in the time series

of average avalanche activity; the analysis is repeated for a range of averaging

periods. The hidden Markov output demonstrates that the sandpile retains

memory on time scales that are less than the long time scale. On longer time

scales, the output depends randomly on the time scale, which is qualitatively

similar to the output obtained from the same time series shuffled randomly.

In other words, no pattern is detected by the hidden Markov model on time

scales exceeding the long time scale. The hidden Markov analysis demon-

strates that all memory is lost on long time scales, in accord with the previous

study based on the occupation fractions.

The work presented in chapter 5 contributes to the line of research devoted

to analysing temporal correlations in the Abelian sandpiles by introducing the

innovation of hidden Markov analysis. It can be extended easily to other sand-

pile models like the Manna model or the Oslo model, and conceivably to other

dynamical systems that exhibit self-organised criticality. Even in the context

of the standard Abelian sandpile model, it is desirable to extend the study’s

results by considering larger sandpiles, longer sequences of data, and higher-

dimensional sandpiles. An important direction for future research is to identify

the internal elements of the sandpile that are responsible for the dynamics of

memory loss exhibited by ensemble-averaged fractions. Due to averaging over

many realisations this dynamics is independent of random driving by grain

drops and therefore genuinely reflects the internal structure of the sandpile.

The new idea of analysing sandpile dynamics with hidden Markov models mer-

its future development. Such models can be used in a number of ways, one of

which is to feed to the model other sandpile characteristics besides avalanche

119

6. Conclusion

size, or even combinations of two or more quantities.

Unlike the study based on the occupation fractions, the hidden Markov

analysis is agnostic with respect to the internal structure of the dynamical

system it is applied to, yet it is sensitive to the memory effects, which depend

on the internal structure of the system, as the above results demonstrate.

Therefore, hidden Markov models can in principle be used as a diagnostic tool

of the internal structure of financial systems, where the internal structure is

hidden as in the case of the FX markets.

120

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133

Minerva Access is the Institutional Repository of The University of Melbourne

Author/s:

SOKOLOV, ANDREY

Title:

Application of non-equilibrium statistical mechanics to the analysis of problems in financial

markets and economy

Date:

2014

Persistent Link:

http://hdl.handle.net/11343/44218

File Description:

Application of non-equilibrium statistical mechanics to the analysis of problems in financial

markets and economy