
A Structural Model of Market Dynamics, and Why it Matters
by J. Khedair, and R. Kühn,
(pdf),
Comptes Rendus de l'Academie des Sciences (Physiqye) 20, 336348 (2019)
In this paper we explore an approach to understanding price fluctuations within a market via
considerations of functional dependencies between asset prices. Interestingly, this approach suggests
a class of models of a type used earlier to describe the dynamics of real and artificial neural networks.
Statistical physics approaches turn out to be suitable for an analysis of their collective properties.
In this paper, we first motivate the basic phenomenology and modelling arguments before moving
on to discussing some major issues with inference and empirical verification. In particular, we focus
on the natural creation of market states through the inclusion of interactions and how these then
interfere with inference. This is primarily addressed in a synthetic setting. Finally we investigate
real data to test the ability of our approach to capture some key features of the behaviour of financial
markets.

A Structural Model for Fluctuations in Financial Markets
by K. Anand, J. Khedair, and R. Kühn,
Phys. Rev. E 97 052312 (2018), preprint arXiv:1709.10277 (2017)
In this paper we provide a comprehensive analysis of a structural model for
the dynamics of prices of assets traded in a market which takes the form of an
interacting generalization of the geometric Brownian motion model. It is formally
equivalent to a model describing the stochastic dynamics of a system of analogue
neurons, which is expected to exhibit glassy properties and thus many metastable
states in a large portion of its parameter space. We perform a generating functional
analysis, introducing a slow driving of the dynamics to mimic the effect of slowly
varying macroeconomic conditions. Distributions of asset returns over various
time separations are evaluated analytically and are found to be fattailed in a
manner broadly in line with empirical observations. Our model also allows to
identify collective, interaction mediated properties of pricing distributions
and it predicts pricing distributions which are significantly broader than their
noninteracting counterparts, if interactions between prices in the model contain
a ferromagnetic bias. Using simulations, we are able to substantiate one of the
main hypotheses underlying the original modelling, viz. that the phenomenon of
volatility clustering can be rationalised in terms of an interplay between the
dynamics within metastable states and the dynamics of occasional transitions
between them.

Optimal Trading Strategies â€” A Time Series Approach
by P. A. Bebbington and R. Kühn,
JSTAT_05
P053209 (2016) (pdffile),
also at arXiv:1509.07953 (2015)
Motivated by recent advances in the spectral theory of autocovariance matrices,
we are
led to revisit a reformulation of Markowitzâ€™ meanvariance portfolio
optimization approach
in the time domain. In its simplest incarnation it applies to a single traded
asset and allows
an optimal trading strategy to be found which  for a given return  is
minimally exposed to market
price fluctuations. The model is initially investigated for a range of synthetic
price processes,
taken to be either second order stationary, or to exhibit second order
stationary increments.
Attention is paid to consequences of estimating autocovariance matrices from
small finite samples,
and autocovariance matrix cleaning strategies to mitigate against these are
investigated. Finally we apply
our framework to real world data.
 Contagion in an Interacting Economy
by P. Paga and R. Kühn,
JSTAT P03008 (2015)
preprint arXiv:1409.2625 (2014)
We investigate the credit risk model previously introduced by Hatchett and
Kühn under more
general assumptions. We show that the model is exactly solvable in the
$N\to\infty$limit and that the
exact solution is described by a messagepassing approach outlined by Karrer and
Newman,
generalized to include heterogeneous agents and couplings. We provide
comparisons with simulations
in the case of a scalefree graph.
 Derivatives and Credit Contagion in
Interconnected
Networks
by S. Heise and R. Kühn,
Eur. Phys. J. B 85, 115 (2012)
(New expanded version, 31 Jan 2012) DOI
10.1140/epjb/e2012207400
also available at
http://arxiv.org/abs/1202.3025
The importance of adequately modeling credit risk has once again been
highlighted in the recent financial crisis. Defaults tend to cluster around
times of economic stress due to poor macroeconomic conditions, but also
by directly triggering each other through contagion. Although credit default
swaps have radically altered the dynamics of contagion for more than a decade,
models quantifying their impact on systemic risk are still missing. Here, we
examine contagion through credit default swaps in a stylized economic network of
corporates and financial institutions. We analyse such a system using a
stochastic setting, which allows us to exploit limit theorems to exactly solve
the contagion dynamics for the entire system. Our analysis shows that, by
creating additional contagion channels, CDS can actually lead to greater
instability of the entire network in times of economic stress. This is
particularly pronounced when CDS are used by banks to expand their loan books
(arguing that CDS would offload
the additional risks from their balance sheets). Thus, even with complete
hedging through CDS, a significant loan book expansion can lead to considerably
enhanced probabilities for the occurrence of very large losses and very high
default rates in the system. Our approach adds a new dimension to research on
credit contagion, and could feed into a rational underpinning of an improved
regulatory framework for credit derivatives.
 Credit Contagion and Credit Risk
by J.P.L. Hatchett and R. Kühn,
Quantitative Finance 9, 373382
(2009)
(pdf)
We study a simple, solvable model that allows us to investigate
effects of credit contagion on the default probability of individual
firms, in both portfolios of firms and on an economy wide scale.
While the effect of interactions may be small in typical (most
probable) scenarios they are magnified, due to feedback, by
situations of economic stress, which in turn leads to fatter tails
in loss distributions of large loan portfolios.
 Intermittency in an Interacting Generalization of the
Geometric Brownian Motion Model
by R. Kühn and P. Neu,
J. Phys A 41, 324015 (2008) (pdf)
We propose a minimal interacting generalisation of the geometric Brownian
motion model, which turns out to be formally equivalent to a model describing
the dynamics of networks of analogue neurons. For suficiently strong
interactions, such systems may have many metastable states. Transitions
between metastable states are associated with macroscopic reorganisations
of the system, which can be triggered by random external forcing. Such a
system will exhibit intermittent dynamics within a large part of its parameter
space. We propose market dynamics as a possible application of this model, in
which case random external forcing would correspond to arrival of important
information. The emergence of a model of interacting prices of the type
considered here can be argued to follow naturally from a general argument
based on integrating out all nonprice degrees of freedom from the dynamics
of a hypothetical complete description of economic dependencies.
 A Solvable Model for Distribution
Networks on Random Graphs
by D. Nasiev, J. van Mourik and R. Kühn,
Phys. Rev. E 76, 041120 (2007)
We propose a simple model that captures the salient properties of distribution
networks, and study the possible occurrence of blackouts, i.e. sudden failings
of large portions of such networks. The model is defined on a random graph of
finite connectivity. The nodes of the graph represent hubs of the network,
while the edges of the graph represent the links of the distribution network.
Both, the nodes and the edges carry dynamical two state variables representing
the functioning or dysfunctional state of the node or link in question. We
describe a dynamical process in which the breakdown of a link or node is
triggered when the level of maintenance it receives falls below a given
threshold. This form of dynamics can lead to situations of catastrophic
breakdown, if levels of maintenance are themselves dependent on the functioning
of the net, once maintenance levels locally fall below a critical threshold due
to fluctuations. We formulate conditions under which such systems can be
analysed in terms of thermodynamic equilibrium techniques, and under these
conditions derive a phase diagram characterising the collective behaviour of the
system, given its model parameters. The phase diagram is confirmed
qualitatively and quantitatively by simulations on explicit realisations of the
graph, thus confirming the validity of our approach.
 Phase Transitions in Operational
Risk
by K. Anand and R Kühn
Phys. Rev. E 75, 016111 (2007) (pdf)
In this paper we explore the functional correlation approach to
operational risk. We consider networks with heterogeneous apriori
conditional and unconditional failure probability. In the limit of
sparse connectivity, selfconsistent expressions for the dynamical
evolution of order parameters are obtained. Under equilibrium
conditions, expressions for the stationary states are also obtained.
The consequences of the analytical theory developed are analyzed using
phase diagrams. We find coexistence of operational and nonoperational
phases, much as in liquidgas systems. Such systems are susceptible to
discontinuous phase transitions from the operational to nonoperational
phase via catastrophic breakdown. We find this feature to be robust
against variation of the microscopic modelling assumptions.
 Effects of Economic Interactions
on Credit Risk
by J.P.L. Hatchett and R. Kühn, J.
Phys. A 39 22312251
(2006)
(pdf)
We study a credit risk model which captures effects of economic
interactions on a firm's default probability. Economic interactions are
represented as a functionally defined graph, and the existence of both
cooperative, and competitive, business relations is taken into account.
We provide an analytic solution of the model in a limit where the
number of business relations of each company is large, but the overall
fraction of the economy with which a given company interacts may be
small. While the effects of economic interactions are relatively weak
in typical (most probable) scenarios, they are pronounced in situations
of economic stress, and thus lead to a substantial fattening of the
tails of loss distributions in large loan portfolios. This manifests
itself in a pronounced enhancement of the Value at Risk computed for
interacting economies in comparison with their noninteracting
counterparts.
 Adequate Capital and Stress Testing for Operational Risks
by R. Kühn and P. Neu,
reprint (pdf),
in: Operational Risk Modelling and Analysis: Theory and Practice,
M. Cruz (Editor), (Risk Waters Group, 2004), pp 273  289
We describe how the notion of sequential correlations naturally
leads to the quantification of operational risk. Our main point
is that functional dependencies between mutually supportive
processes give rise to nontrivial temporal correlations, which
can lead to the occurrence of collective risk events in the form
of bursts and avalanches of process failures, and crashes of
process networks. We show how the adequate capital for
operational risk can be calculated via a stochastic dynamics
defined on a topological network of interacting processes. One
of the main virtues of the present model is the suitability for
capital allocation and stress testing of operational risks.
 Credit Risk Enhancement in a Network of Interdependent Firms
by P. Neu and R. Kühn,
Physica A 342, 639655 (2004) (pdf)
We propose a dynamical model to study the impact of sequential
defaults in a network of economically interdependent firms on the loss
distribution for bank loan portfolios. Exploring the analogy to a lattice
gas model from physics, correlations between sequential defaults are modeled
as due to functionally defined, heterogeneous couplings between mutually
dependent business partners. In our model functional dependencies result
in an enhancement of credit risk and a reduced granularity of the loan
portfolio. We find that corporate dependencies may result in additional
extreme losses in the loan portfolio. In particular  depending on the
relationship between the firms  collective phenomena such as bursts and
avalanches of defaults can be observed in the model. In this context,
traditional credit risk models are inadequate because they underestimate
the required equity capital. Furthermore, our model setting is particularly
applicable for doing stress analyses of credit risk in loan portfolios.
 Functional correlation approach to operational risk in
banking organizations
by R. Kühn and P. Neu, Physica
A 322 650666 (2003) (pdf).
A ValueatRiskbased model is proposed to compute the adequate
equity capital necessary to cover potential losses due to operational
risks, such as human and system process failures, in banking
organizations. Exploring the analogy to a lattice gas model from
physics, correlations between sequential failures are modeled by as
functionally defined, heterogeneous couplings
between mutually supportive processes. In contrast to traditional risk
models
for market and credit risk, where correlations are described as
equaltimecorrelations by a covariance matrix, the dynamics of the
model shows collective phenomena such as bursts and avalanches of
process failures.