Large portfolio losses: A dynamic contagion model
Large portfolio losses: A dynamic contagion model
The Annals of Applied Probability
2009, Vol. 19, No. 1, 347–394
DOI: 10.1214/08-AAP544
c© Institute of Mathematical Statistics, 2009
LARGE PORTFOLIO LOSSES: A DYNAMIC CONTAGION MODEL
By Paolo Dai Pra, Wolfgang J. Runggaldier,
Elena Sartori and Marco Tolotti
University of Padova, University of Padova, University of Padova, and
Bocconi University and Scuola Normale Superiore
Using particle system methodologies we study the propagation of
financial distress in a network of firms facing credit risk. We investi-
gate the phenomenon of a credit crisis and quantify the losses that
a bank may suffer in a large credit portfolio. Applying a large devi-
ation principle we compute the limiting distributions of the system
and determine the time evolution of the credit quality indicators of
the firms, deriving moreover the dynamics of a global financial health
indicator. We finally describe a suitable version of the “Central Limit
Theorem” useful to study large portfolio losses. Simulation results are
provided as well as applications to portfolio loss distribution analysis.
1. Introduction.
1.1. General aspects. The main purpose of this paper is to describe prop-
agation of financial distress in a network of firms linked by business rela-
tionships. Once the model for financial contagion has been described, we
quantify the impact of contagion on the losses suffered by a financial insti-
tution holding a large portfolio with positions issued by the firms.
A firm experiencing financial distress may affect the credit quality of
business partners (via direct contagion) as well as of firms in the same sector
(due to an information effect).
We refer to direct contagion when the actors on the market are linked by
some direct partner relationship (e.g., firms in a borrowing-lending network).
Reduced-form models for direct contagion can be found—among others—in
Jarrow and Yu [27] for counterparty risk, Davis and Lo [13] for infectious
Received March 2007; revised April 2008.
AMS 2000 subject classifications. 60K35, 91B70.
Key words and phrases. Credit contagion, credit crisis, interacting particle systems,
large deviations, large portfolio losses, mean field interaction, nonreversible Markov pro-
cesses, phase transition.
This is an electronic reprint of the original article published by the
Institute of Mathematical Statistics in The Annals of Applied Probability,
2009, Vol. 19, No. 1, 347–394. This reprint differs from the original in pagination
and typographic detail.
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2 DAI PRA, RUNGGALDIER, SARTORI AND TOLOTTI
default, Kiyotaki and Moore [28], where a model of credit chain obligations
leading to default cascade is considered and Giesecke and Weber [23] for a
particle system approach. Concerning the banking sector, a microeconomic
liquidity equilibrium is analyzed by Allen and Gale [1].
Information effects are considered in information-driven default models;
here the idea is that the probability of default of each obligor is influenced
by a “not perfectly” observable macroeconomic variable, sometimes also
referred to as frailty. This dependence increases the correlation between the
default events. For further discussions on this point see Schönbucher [33] as
well as Duffie et al. [16] and Collin-Dufresne et al. [7].
1.2. Purpose and modeling aspects. We propose in this paper a direct
contagion model which is constructed in a general modeling framework
where information effects could also be included. In addition to modeling
contagion, with the approach that we shall develop we intend also to find
a way to explain what is usually referred to as the clustering of defaults
(or credit crises), meaning that there is evidence—looking at real data—of
periods in which many firms end up in financial distress in a short time.
A standard methodology to reproduce this real-world effect is to rely on
macroeconomic factors as indicators of business cycles. These factor models
seem to explain a large part of the variability of the default rates. What
these models do not explain is above all clustering: as Jarrow and Yu in
[27] argue, “A default intensity that depends linearly on a set of smoothly
varying macroeconomic variables is unlikely to account for the clustering of
defaults around an economic recession.”
A second issue that we would like to capture is—in some sense—more
“fundamental” and refers to the nature of a credit crisis. We shall propose a
model where the general “health” of the system is described by endogenous
financial indicators, endogenous in the sense that its dynamics depends on
the evolution of the variables of the system. Our aim is to show how a credit
crisis can be described as a “microeconomic” phenomenon, driven by the
propagation of the financial distress through the obligors.
Our model is to be considered within the class of reduced-form models
and is based on interacting intensities. The probability of having a default
somewhere in the network depends also on the state of the other obligors.
The first papers on interacting intensities appear to be those by Jarrow and
Yu [27], and Davis and Lo [13] on infectious default.
In our perspective the idea of a network where agents interact leads natu-
rally to the literature of particle systems used in statistical mechanics. This
point of view is quite new in the world of financial mathematics especially
when dealing with credit risk management. Among some very recent papers
we would like to mention the works by Giesecke and Weber [23], and [24]
for an interacting particle approach, the papers by Frey and Backhaus [19]
LARGE PORTFOLIO LOSSES 3
on credit derivatives pricing and Horst [26] on cascade processes. More de-
veloped is the use of particle and dynamical systems in the literature on
financial market modeling. It has been shown that some of these models
have “thermodynamic limits” that exhibit similar features compared to the
limiting distributions (in particular when looking at the tails) of market
returns time series. For a discussion on financial market modeling see the
survey by Cont [9] and the paper by Föllmer [18] that contains an inspiring
discussion on interacting agents.
Another reason to focus on particle systems is that they allow to study
a credit crisis as a microeconomic phenomenon and so provide the means
to explain phenomena such as default clustering that are difficult to explain
by other means. In fact, interacting particle systems may exhibit what is
called phase transition in the sense that in the limit, when the number N of
particles goes to infinity, the dynamics may have multiple stable equilibria.
The effects of phase transition for the system with finite N can be seen
on different time-scales. On a long time-scale we expect to observe what
is usually meant by metastability in statistical mechanics: the system may
spend a very long time in a small region of the state space around a stable
equilibrium of the limiting dynamics and then switch relatively quickly to
another region around a different stable equilibrium. This switch, of which
the rigorous analysis will be postponed to future work, occurs on a time-scale
proportional to ekN for a suitable k > 0, that could be unrealistic for financial
applications. The model we propose exhibits, however, a different feature
that can be interpreted as a credit crisis. For certain values of the initial
condition the system is driven toward a symmetric equilibrium, in which half
of the firms are in good financial health. After a certain time that depends
on the initial state, the system is “captured” by an unstable direction of this
symmetric equilibrium, and moves toward a stable asymmetric equilibrium;
during the transition to the asymmetric equilibrium, the volatility of the
system increases sharply, before decaying to a stationary value. All this
occurs at a time-scale of order O(1) (i.e., the time-scale does not depend
on N ).
1.3. Financial application. As already mentioned in Section 1.1, the ap-
plied financial aim of this paper is to quantify the impact of contagion on
the losses suffered by a financial institution holding a large portfolio with
positions issued by the firms. In particular, we aim at obtaining a dynamic
description of a risky portfolio in the context of our contagion model. The
standard literature on risk management usually focuses on static models
allowing to compute the distribution of a risky portfolio over a given fixed
time-horizon T . For a recent paper that introduces a discussion relating to
static and dynamic models see Dembo, Deuschel and Duffie [14].
4 DAI PRA, RUNGGALDIER, SARTORI AND TOLOTTI
We shall consider large homogeneous portfolios. Attention to large ho-
mogeneous portfolios becomes crucial when looking at portfolios with many
small entries. Suppose a bank is holding a credit portfolio with N = 10,000
open positions with small firms; it is quite costly to simulate the dynamics
of each single firm, taking into account all business ties. If the firms are
supposed to be exchangeable, in the sense that the losses that they may
cause to the bank in case of financial distress depend on the single firm only
via its financial state indicator, it is worth evaluating a homogeneous model
where N goes to infinity and then to look for “large-N” approximations.
This apparently restrictive assumption may be easily relaxed by considering
many homogeneous groups within the network (in this context see also [19]).
We shall provide formulas to compute quantiles of the probability of excess
losses in the context of our contagion model; we shall in fact determine the
entire portfolio loss distribution. Other credit risk related quantities can also
be computed, as we shall briefly mention at the end of Section 4.
We conclude this section by noticing that in recent years the challenging
issue of describing the time evolution of the loss process connected with port-
folios of many obligors has received more and more attention. Applications
can be found, for example, in the literature dealing with pricing and hedging
of risky derivatives such as CDOs, namely Collateralized Debt Obligations
(see, e.g., the papers by Frey and Backhaus [20], Giesecke and Goldberg [22]
and Schönbucher [34]).
We believe that our paper may be considered as an original contribution
to the modeling of portfolio loss dynamics: to our knowledge, this is the first
attempt to apply large deviations on path spaces (i.e., in a dynamic fashion)
for finance or credit management purposes. For a survey on existing large
deviations methods applied to finance and credit risk see Pham [31].
1.4. Methodology. Our interacting particle system, which describes the
firms in the network, will be Markovian, but nonreversible. Usually, when
the dynamics admit a reversible distribution, this distribution can be found
explicitly by the detailed balance condition [see (6) below]. In the model we
propose in this paper, and that will be introduced in Section 2, no reversible
distribution exists. This makes it difficult to find an explicit formula for the
stationary distribution. For this reason we have not pursued the “static”
approach consisting in studying the N → +∞ asymptotics of the stationary
distribution. We shall rather proceed in a way that in addition allows to
obtain nonequilibrium properties of the system dynamics. First we study the
N →∞ limiting distributions on the path space. To this effect we shall derive
an appropriate law of large numbers based on a large deviations principle.
We then study the possible equilibria of the limiting dynamics. This study
leads to considering different domains of attraction corresponding to each
of the stable equilibria. Finally, we study the finite volume approximations
LARGE PORTFOLIO LOSSES 5
(for finite but large N ) of the limiting distribution via a suitable version of
the Central Limit Theorem that allows to analyze the fluctuations around
this limit. As a consequence of the different domains of attraction of the
limiting dynamics one obtains for finite N and on ordinary time-scales an
interesting behavior of the system that has an equally interesting financial
interpretation, which was already alluded to at the end of Section 1.2. This
behavior will also be documented by simulation results.
Our interaction model is characterized by two parameters indicating the
strength of the interactions. Phase transition occurs in an open subset of the
parameter space, whose boundary is a smooth curve (critical curve) that we
determine explicitly. We shall derive the Central Limit Theorem in a fixed
time-interval [0, T ] for every value of the parameters. We do not consider in
this paper the Central Limit Theorem in the case when the time-horizon T
depends on N itself; it will be dealt with elsewhere. When T grows with N
we expect the behavior to depend more strongly on the parameters. In the
case when the parameters belong to the uniqueness region (the complement
of the closure of the region where phase transition occurs) we believe that
the Central Limit Theorem should be uniform in time, while in the phase
transition region the Central Limit Theorem should extend to any time-scale
strictly smaller than the metastability scale (which grows exponentially in
N ). On the critical curve one expects a critical time-scale (of order
N ) at
which large and non-Gaussian fluctuations are observed.
For real applications the interaction parameters have to be calibrated
to market data. In this paper we do not consider the issue of calibration
but rather present some simulation results of the loss behavior for different
values of the parameters.
The outline of the paper is as follows. The more detailed description of
the model will be given in Section 2. Section 3 is devoted to stating the
main limit theorems on the stochastic dynamics, in particular a law of large
numbers and a central limit theorem. The financial application, in particular
to large portfolio losses with specific examples, will be described in Section
4. Section 5 contains the proofs of the results stated in Sections 3 and 4. A
Conclusions section completes the paper.
2. The model.
2.1. A mean-field model. In this section we describe a mean-field interac-
tion model. What characterizes a mean-field model—within the large class
of particle systems—is the absence of a “geometry” in the configuration
space, meaning that each particle interacts with all the others in the same
way. This “homogeneity” assumption is clearly rather restrictive; neverthe-
less this kind of framework has been proposed by authors in different fields.
Among the others we quote Frey and Backhaus [19] for a credit risk model
6 DAI PRA, RUNGGALDIER, SARTORI AND TOLOTTI
and Brock and Durlauf [4] for their contribution to the Social Interaction
models. These models are used to capture the interaction of agents when
facing any kind of decision problems. As pointed out in [19], if we are con-
sidering a large group of firms belonging to the same sector (e.g., the energy
sector), then the ability of generating cash flows and the capacity of rais-
ing capital from financial institutions may be considered as “homogeneous”
characteristics within the group (and this assumption is quite common in
practice); we moreover recall that the final aim of this work is to study ag-
gregate quantities for a large economy such as the expected global health
of the system and large portfolio losses as well as related quantities. These
considerations allow us to avoid the (costly) operation of modeling a fully
heterogeneous set of firms.
Other approaches, different from the mean-field one, have also been pro-
posed in the literature: Giesecke and Weber have chosen a local-interaction
model (the Voter model1) assuming that each particle interacts with a fixed
number d of neighbors; it may be argued that the hypothesis that each firm
has the same (constant) number of partners is rather unrealistic. Cont and
Bouchaud (see [10]) suggest a random graph approach, meaning that the
connections are randomly generated with some distribution functions.
The philosophy behind our model can be summarized as follows:
• We introduce only a small number of variables that, however, have a
simple economic interpretation.
• We define dynamic rules that describe interaction between the variables.
• We keep the model as simple as possible; in particular, as we shall see,
we define it in such a way that it has some symmetry properties. On one
hand this may make the model less adherent to reality; on the other it
leads to exact computations and still allows to show what basic features
of the model produce phenomena such as clustering of defaults, phase
transition, etc. More generally, it allows to show how, contrary to most
models relying on macroeconomic factors, the “health” of the system can
here be described by endogenous financial indicators so that a credit crisis
can be viewed as a microeconomic phenomenon.
Consider a network of N firms. The state of each firm is identified by two
variables, that will be denoted by σ and ω [(σi, ωi) is the state of the ith
firm]. The variable σ may be interpreted as the rating class indicator : a low
value reflects a bad rating class, that is, a higher probability of not being
able to pay back obligations. The variable ω represents a more fundamental
indicator of the financial health of the firm and is typically not directly
1The Voter model assumes—roughly speaking—that the variable σi ∈ {−1,1} is more
likely to take a positive value if the majority of the nearest neighbors of i are in a positive
state and vice versa.
LARGE PORTFOLIO LOSSES 7
observable. It could, for example, be a liquidity indicator as in Giesecke
and Weber [23] or the sign of the cash balances as in Çetin et al. [5]. The
important fact is that, while there is usually a strong interaction between
σi and ωi, the nonobservability of ω makes it reasonable to assume that ωi
cannot directly influence the rating indicators σj for j 6= i.
In this paper we assume that the two indicators σi, ωi can only take two
values, that we label by 1 (“good” financial state) and −1 (financial distress).
In the case of portfolios consisting of defaultable bonds, we may then refer to
the rating class corresponding to σ = −1 also as “speculative grade” and that
corresponding to σ = +1 as “investment grade.” Although the restriction to
only two possible values may appear to be unrealistic, we believe that many
aspects of the qualitative behavior of the system do not really depend on
this choice. On the other hand, modulo having more complex formulae, the
results below can be easily extended to the case when these variables take
an arbitrary finite number of values.
In our binary variable model we are naturally led to an interacting in-
tensity model, where we have to specify the intensities or rates (inverse of
the average waiting times) at which the transitions σi 7→ −σi and ωi 7→ −ωi
take place. If we neglect direct interactions between the ωi’s, and we make
the mean-field assumption that the interaction between different firms only
depends on the value of the global financial health indicator
we are led to consider intensities of the form
σi 7→ −σi with intensity a(σi, ωi,mσN ),
ωi 7→ −ωi with intensity b(σi, ωi,mσN ),
where a(·, ·, ·) and b(·, ·, ·) are given functions. Since both financial health
and distress tend to propagate, we assume that a(−1, ωi,mσN ) is increasing
in both ωi and m
N , and a(1, ωi,m
N ) is decreasing. Similarly, b(σi,−1,m
and b(σi,1,m
N ) should be respectively increasing and decreasing in their
variables.
The next simplifying assumption is that the intensity a(σi, ωi,m
N) is
actually independent of m
N , that is, of the form a(σi, ωi). Although this
assumption amounts to a rather mild computational simplification, it allows
to show that aggregate behavior (phase transition, etc.) may occur even in
absence of a direct interaction between rating indicators.
Although a model of this generality could be fully analyzed, we make the
following choice of the intensities, inspired by spin-glass systems, to make
8 DAI PRA, RUNGGALDIER, SARTORI AND TOLOTTI
the model depend on only a few parameters:
σi 7→ −σi with intensity e−βσiωi ,
ωi 7→ −ωi with intensity e−γωim
Here β and γ are positive parameters which indicate the strength of the
corresponding interaction. Put differently, we are considering a continuous-
time Markov chain on {−1,1}2N with the following infinitesimal generator:
Lf(σ,ω) =
e−βσiωi∇σi f(σ,ω) +
e−γωjm
N∇ωj f(σ,ω),(3)
where ∇σi f(σ,ω) = f(σi, ω) − f(σ,ω) (analogously for ∇ωi ), and where the
jth component of σi is
σij =
σj , for j 6= i,
−σi, for j = i.
The rest of the paper is devoted to a detailed analysis of the above model.
We conclude this subsection with some general remarks on the model we
have just defined.
Remark 2.1.
• We have viewed the variable σ as a rating class indicator. Contrary to
the standard models for rating class transitions, our rating indicator σ
is not Markov by itself, but it is Markov only if paired with ω. This
property is in line with empirical data and with recent research in the
field of credit migration models. It is in fact well documented that real
data of credit migration between rating classes exhibit a “non-Markovian”
behavior. For a discussion on this topic see, for example, Christensen et
al. [6]. In that paper the authors propose a hidden Markov process to
model credit migration. The basic criticism to Markovianity is the fact
that the probability of being downgraded is higher for firms that have
been just downgraded. In order to capture this issue, the authors consider
an “excited” rating state (e.g., B∗ from which there is a higher probability
to be downgraded compared to the standard state B). This point of view is
not far from ours, even though the mechanism of the transition is different.
The downgrade to σ = −1 is higher when (σ = 1, ω = −1) compared to
(σ = 1, ω = 1).
• In our model, unlike other rating class models, we do not introduce a de-
fault state for firms; it could be identified as a value for the pair (σ,ω) for
which the corresponding intensities are identically zero, that is, a(σ,ω,m
N ) =
b(σ,ω,m
N ) = 0 for all values of m
N . This would have the effect of intro-
ducing a “trap state” for the system, changing drastically the long-time
LARGE PORTFOLIO LOSSES 9
behavior. Even in case of defaultable firms, however, our model could be
meaningful up to a time-scale in which the fraction of defaulted firms is
small.
• With a choice of the intensities as in (2) we introduce a form of symmetry
in our model, whereby the values σ = −1 and σ = +1 for the rating indica-
tor turn out to be equally likely. One could, however, modify the model in
order to make the value σ = −1 less (more) likely than the value σ = +1
and this could, for example, be achieved by letting the intensity for ωi
be of the form eωiφ(m
), where φ is an increasing, nonlinear and noneven
function. A possible “prototype” choice would be φ(x) = γ(x−K)+ + δ
with γ, δ > 0 and K ∈ (0,1). Note that with this latter choice we have
φ≥ 0 so that the value ωi = +1 (and hence also σi = +1) becomes more
likely. Such an asymmetric setup might be more realistic in financial appli-
cations but, besides leading to more complicated derivations, it depends
also on the specific application at hand. Since, as already mentioned, we
want to study a model that is as simple as possible and yet capable of
producing the basic features of interest, in this paper we concentrate on
the “symmetric choice” in (2). The large deviation approach to the Law
of Large Numbers developed in Sections 3.1 and 3.2 can be adapted to
the asymmetric setup (see Remark 3.5) with no essential difference. On
the other hand, our proof of the Central Limit Theorem in Section 3.3
may require more regularity on the function φ above. We leave this point
for further investigation.
2.2. Invariant measures and nonreversibility. Mean-field models as the
one we propose in this paper have already appeared, mostly in the statistical
mechanics literature (see in particular [12] and [8], from which we borrow
many of the mathematical tools). However, unlike what happens for the
models in the cited references, we now show that our model is nonreversible.
This implies that an explicit formula for the stationary distribution and its
N →∞ asymptotics is not available. It is thus appropriate to follow a more
specifically dynamic approach to understand the long-time behavior of the
system. As already mentioned, we shall thus first study the N →∞ limit of
the dynamics of the system, obtaining limit evolution equations. Then we
study the equilibria of these equations. This is not necessarily equivalent to
studying the N →∞ properties of the stationary distribution µN . However,
as we shall show later in this paper, this provides rather sharp information
on how the system behaves for t and N large.
The operator L given in (3) defines an irreducible, finite-state Markov
chain. It follows that the process admits a unique stationary distribution
µN , that is, a distribution such that, for each function f on the configuration
space of (σ,ω),
µN (σ,ω)Lf(σ,ω) = 0.(4)
10 DAI PRA, RUNGGALDIER, SARTORI AND TOLOTTI
This distribution reflects the long-time behavior of the system, in the
sense that, for each f and any initial distribution,
E[f(σ(t), ω(t))] =
µN (σ,ω)f(σ,ω).
The stationarity condition (4) is equivalent to
[µN (σ
i, ω)eβσiωi − µN (σ,ω)e−βσiωi ]
[µN (σ,ω
i)eγωim
N − µN (σ,ω)e−γωim
N ] = 0
for every σ,ω ∈ {−1,1}N .
Simpler sufficient conditions for stationarity are the so-called detailed bal-
ance conditions. We say that a probability ν on {−1,1}2N satisfies the de-
tailed balance condition for the generator L if
ν(σi, ω)eβσiωi = ν(σ,ω)e−βσiωi and
ν(σ,ωi)eγωim
N = ν(σ,ω)e−γωim
for every σ,ω. When the detailed balance conditions (6) hold, we say the sys-
tem is reversible: the stationary Markov chain with generator L and marginal
law ν has a distribution which is left invariant by time-reversal. In the case
(6) admits a solution, they usually allow to derive the stationary distribution
explicitly. This is not the case in our model. We have in fact:
Proposition 2.2. The detailed balance equations (6) admit no solution,
except at most for one specific value of N .
Proof. By way of contradiction, assume a solution ν of (6) exists. Then
one easily obtains
∇σi log ν(σ,ω) = −2βσiωi,
∇ωi log ν(σ,ω) = −2γωim
which implies
∇ωi ∇σi log ν(σ,ω) = 4βσiωi,
∇σi ∇ωi log ν(σ,ω) = 4N−1γωiσi.
This is not possible since ∇ωi ∇σi log ν(σ,ω)≡∇σi∇ωi log ν(σ,ω). �
LARGE PORTFOLIO LOSSES 11
3. Main results: law of large numbers and Central Limit Theorem. In
this section we state the results concerning the dynamics of the system
(σi[0, T ], ωi[0, T ])
i=1 in the limit as N → ∞. Note that for each value of
N we are considering a Markov process with generator (3). Thus, it would
be more accurate to denote by (σ
i [0, T ], ω
i [0, T ]) the trajectories of the
variables related to the ith firm in the system with N firms. For convenience,
we consider a fixed probability space (Ω,F , P ) where all D([0, T ])-valued
processes σ
i [0, T ], ω
i [0, T ] are defined, and the following conditions are
satisfied:
• for each N ≥ 1 the processes (σ(N)i [0, T ], ω
i [0, T ])
i=1 are Markov pro-
cesses with infinitesimal generator (3);
• for each N ≥ 1 the {−1,1}2-valued random variables (σ(N)i (0), ω
i (0))
are independent and identically distributed with an assigned law λ.
This last assumption on the initial distribution is stronger than what we
actually need to prove the results below; however, it allows to avoid some
technical aspects in the proof, that we consider not essential for the purposes
of the paper. The other point, concerning the fact of realizing all processes
in the same probability space, is not a restriction; we are not making any
assumption on the dependence of processes with different values of N , so
this joint realization is always possible. Its main purpose is to allow to state
a strong law of large numbers.
Our approach proceeds according to the following three steps, to which
correspond the three subsections below, namely:
(i) look for the limit dynamics of the system (N →∞);
(ii) study the equilibria of the limiting dynamics;
(iii) describe the “finite volume approximations” (for large but finite N )
via a central limit-type result.
3.1. Deterministic limit: large deviations and law of large numbers. In
what follows D([0, T ]) denotes the space of right-continuous, piecewise con-
stant functions [0, T ] → {−1,1}, endowed with the Skorohod topology (see
[17]). Let (σi[0, T ], ωi[0, T ])
i=1 ∈D([0, T ])2N denote a path of the process in
the time-interval [0, T ] for a generic T > 0. If f(σi[0, T ], ωi[0, T ]) is a function
of the trajectory of the variables related to a single firm, one is interested in
the asymptotic behavior of empirical averages of the form
f(σi[0, T ], ωi[0, T ]) =:
f dρN ,
where ρN is the sequence of empirical measures
δ(σi[0,T ],ωi[0,T ]).
12 DAI PRA, RUNGGALDIER, SARTORI AND TOLOTTI
We may think of ρN as a (random) element of M1(D([0, T ]) × D([0, T ])),
the space of probability measures on D([0, T ])×D([0, T ]) endowed with the
weak convergence topology.
Our first aim is to determine the limit of
f dρN as N →∞, for f con-
tinuous and bounded; in other words we look for the weak limit limN ρN
in M1(D([0, T ]) × D([0, T ])). This corresponds to a law of large numbers
with the limit being a deterministic measure. This limit, being an element
of M1(D([0, T ])×D([0, T ])), can be viewed as a stochastic process, and rep-
resents the dynamics of the system in the limit N →∞. The fluctuations of
ρN around this deterministic limit will be studied in Section 3.3 below, and
this turns out to be particularly relevant in the risk analysis of a portfolio
(Section 4).
The result we actually prove is a large deviation principle, which is much
stronger than a law of large numbers. We start with some preliminary no-
tions letting, in what follows, W ∈M1(D([0, T ])×D([0, T ])) denote the law
of the {−1,1}2-valued process (σ(t), ω(t)) such that (σ(0), ω(0)) has distri-
bution λ, and both σ(·) and ω(·) change sign with constant intensity 1. For
Q ∈M1(D([0, T ])×D([0, T ])) let
H(Q|W ) :=
dQ log
, if Q≪W and log dQ
∈L1(Q),
+∞, otherwise,
denote the relative entropy between Q and W . Moreover, ΠtQ denotes the
marginal law of Q at time t, and
t := γ
σΠtQ(dσ, dτ).
For a given path (σ[0, T ], ω[0, T ]) ∈D([0, T ]) ×D([0, T ]), let Nσt (resp. Nωt )
be the process counting the jumps of σ(·) [resp. ω(·)]. Define
F (Q) =
(1− e−βσ(t)ω(t))dt+
(1− e−ω(t)γ
t )dt
σ(t)ω(t−)dNσt +
ω(t)γ
whenever
(NσT +N
T )dQ<+∞,
and F (Q) = 0 otherwise. Finally let
I(Q) :=H(Q|W )− F (Q).
We remark that, if
(NσT +N
T )dQ = +∞, then H(Q|W ) = +∞ (this will
be shown in Section 5, Lemma 5.4) and thus also I(Q) = +∞.
LARGE PORTFOLIO LOSSES 13
Proposition 3.1. For each Q ∈ M1(D([0, T ]) × D([0, T ])), I(Q) ≥ 0,
and I(·) is a lower-semicontinuous function with compact level-sets [i.e., for
each k > 0 one has that {Q : I(Q) ≤ k} is compact in the weak topology].
Moreover, for A,C ⊆M1(D([0, T ])×D([0, T ])) respectively open and closed
for the weak topology, we have
lim inf
logP (ρN ∈A) ≥− inf
I(Q),(8)
lim sup
logP (ρN ∈C) ≤− inf
I(Q).(9)
This means that the distributions of ρN obey a large deviation principle
(LDP) with rate function I(·) (see, e.g., [15] for the definition and funda-
mental facts on LDP).
The proof of Proposition 3.1 is given in Section 5 and follows from ar-
guments similar to those in [12]. Various technical difficulties are due to
unboundedness and noncontinuity of F , which are related to the nonre-
versibility of the model.
The key step to derive a law of large numbers from Proposition 3.1 is
given in the following result, whose proof is also given in Section 5. In what
follows, for q ∈M1({−1,1}2) a probability on {−1,1}2, we define
mσq :=
σ,ω=±1
σq(σ,ω),
that can be interpreted as the expected rating under q.
Proposition 3.2. The equation I(Q) = 0 has a unique solution Q∗.
Moreover, if qt ∈M1({−1,1}2) denotes the marginal distribution of Q∗ at
time t, then qt is the unique solution of the nonlinear (McKean–Vlasov)
equation
= Lqt, t∈ [0, T ],
q0 = λ,
where
Lq(σ,ω) = ∇σ[e−βσωq(σ,ω)] +∇ω[e−γωmσq q(σ,ω)](11)
with (σ,ω) ∈ {−1,1}2.
From Propositions 3.1 and 3.2, it is easy to derive the following strong
law of large numbers.
14 DAI PRA, RUNGGALDIER, SARTORI AND TOLOTTI
Theorem 3.3. Let Q∗ ∈ M1(D([0, T ]) × D([0, T ])) be the probability
given in Proposition 3.2. Then
ρN → Q∗ almost surely
in the weak topology.
Proof. Let Q∗ be the unique zero of the rate function I(·) as given by
Proposition 3.2. Let BQ∗ be an arbitrary open neighborhood of Q
∗ in the
weak topology. By the upper bound in Proposition 3.1, we have
limsup
logP (ρN /∈BQ∗)≤− inf
Q/∈BQ∗
I(Q)< 0,
where the last inequality comes from lower semicontinuity of I(·), com-
pactness of its level sets and the fact that I(Q) > 0 for every Q 6=Q∗. In-
deed, if infQ/∈BQ∗ I(Q) = 0, then there exists a sequence Qn /∈BQ∗ such that
I(Qn) → 0. By the compactness of the level sets there exists then a sub-
sequence Qnk → Q̄ /∈BQ∗ . By lower semicontinuity it then follows I(Q̄) ≤
lim inf I(Qnk) = 0 which contradicts I(Q)> 0 for q 6=Q∗. By the above in-
equality we thus have that P (ρN /∈BQ∗) decays to 0 exponentially fast. By
a standard application of the Borel–Cantelli lemma, we obtain that ρn →Q∗
almost surely. �
3.2. Equilibria of the limiting dynamics: phase transition. Equation (10)
describes the dynamics of the system with generator (3) in the limit as N →
+∞. In this section we determine the equilibrium points, or stationary (in t)
solutions of (10), that is, solutions of Lqt = 0 and, more generally, the large
time behavior of its solutions. First of all, it is convenient to reparametrize
the unknown qt in (10).
Let q be a probability on {−1,1}2. Note that each f :{−1,1}2 → R can
be written in the form f(σ,ω) = aσ + bω + cσω + d. It follows that q is
completely identified by the expectations
mσµ :=
σ,ω=±1
σq(σ,ω),
mωµ :=
σ,ω=±1
ωq(σ,ω),(12)
mσωµ :=
σ,ω=±1
σωq(σ,ω).
In particular, if q = qt, the marginal of Q
∗ appearing in Proposition 3.2, then
we write mσt for m
, and similarly for mωt ,m
t . In order to rewrite (10) in
terms of the new variables mσt ,m
t , observe that
ṁσ =
σ,ω=±1
σq̇t(σ,ω) =
σ,ω=±1
σLqt.
LARGE PORTFOLIO LOSSES 15
On the other hand, a straightforward computation shows that, for every
probability q,
σ,ω=±1
σLq = 2sinh(β)mωq − 2cosh(β)mσq ,
giving
ṁσt = 2sinh(β)m
t − 2cosh(β)mσt .
By making similar computations for mωt ,m
t , it is shown that (10) can be
rewritten in the following form:
ṁσt = 2sinh(β)m
t − 2cosh(β)mσt ,
ṁωt = 2sinh(γm
t )− 2cosh(γmσt )mωt ,(13)
ṁσωt = 2sinh(β) + 2sinh(γm
t − 2(cosh(β) + cosh(γmσt ))mσωt ,
with initial condition mσ0 =m
λ , m
λ . Note that m
t does
not appear in the first and in the second equation in (13); this means that
the differential system (13) is essentially two-dimensional: first one solves
the two-dimensional system (on [−1,1]2)
(ṁσt , ṁ
t ) = V (m
t ),(14)
with V (x, y) = (2 sinh(β)y − 2cosh(β)x,2 sinh(γx)− 2y cosh(γx)), and then
one solves the third equation in (13), which is linear in mσωt . Note also
that to any (mσ∗ ,m
∗ ) satisfying V (m
∗ ) = 0, there corresponds a unique
mσω∗ :=
sinh(β)+mσ∗ sinh(γm
cosh(β)+cosh(γmσ∗ )
such that (mσ∗ ,m
∗ ) is an equilibrium (stable
solution) of (13). Moreover, ifmσt →mσ∗ as t→ +∞, then mσωt →mσω∗ . Thus,
to discuss the equilibria of (13) and their stability, it is enough to analyze
(14) and for this we have the following proposition, where by “linearly stable
equilibrium” we mean a pair (x̄, ȳ) such that V (x̄, ȳ) = 0, and the linearized
system (ẋ, ẏ) =DV (x̄, ȳ)(x− x̄, y − ȳ) is stable, that is, the eigenvalues of
the Jacobian matrix DV (x̄, ȳ) have all negative real parts.
Theorem 3.4. (i) Suppose γ ≤ 1
tanh(β)
. Then (14) has (0,0) as a unique
equilibrium solution, which is globally asymptotically stable, that is, for every
initial condition (mσ0 ,m
0 ), we have
(mσt ,m
t ) = (0,0).
(ii) For γ < 1
tanh(β)
the equilibrium (0,0) is linearly stable. For γ = 1
tanh(β)
the linearized system has a neutral direction, that is, DV (0,0) has one zero
eigenvalue.
16 DAI PRA, RUNGGALDIER, SARTORI AND TOLOTTI
(iii) For γ > 1
tanh(β)
the point (0,0) is still an equilibrium for (14), but
it is a saddle point for the linearized system, that is, the matrix DV (0,0)
has two nonzero real eigenvalues of opposite sign. Moreover (14) has two
linearly stable solutions (mσ∗ ,m
∗ ), (−mσ∗ ,−mω∗ ), where mσ∗ is the unique
strictly positive solution of the equation
x= tanh(β) tanh(γx),(15)
mω∗ =
tanh(β)
mσ∗ .(16)
(iv) For γ > 1
tanh(β)
, the phase space [−1,1]2 is bipartitioned by a smooth
curve Γ containing (0,0) such that [−1,1]2 \ Γ is the union of two disjoint
sets Γ+,Γ− that are open in the induced topology of [−1,1]2. Moreover
(mσt ,m
t ) =
(mσ∗ ,m
∗ ), if (m
0 ) ∈ Γ+,
(−mσ∗ ,−mω∗ ), if (mσ0 ,mω0 ) ∈ Γ−,
(0,0), if (mσ0 ,m
0 ) ∈ Γ.
Proof. See Section 5. �
Remark 3.5. The results in this section are specific to our model with
the symmetry properties as induced by the specification of the intensities
in (2). With an asymmetric setup such as described in Remark 2.1, (15)
becomes
x= tanh(β) tanh(φ(x))
thus allowing more flexibility in the position of the equilibria. In particular,
by letting φ(x) = γ(x−K)+ + δ, while still having three equilibria, we may
choose their relative position by suitably choosing the values for γ,K, δ.
Notice that in this way we also increase the number of parameters in our
model.
3.3. Analysis of fluctuations: Central Limit Theorem. Having established
a law of large numbers ρN →Q∗, it is natural to analyze fluctuations around
the limit, that is, the rate at which ρN converges to Q
∗ and the asymptotic
distribution of ρN −Q∗.
To study the asymptotic distribution of ρN −Q∗ there are at least the
following two possible approaches:
(i) An approach based on a functional central limit theorem using a
result in [2] that relates large deviations with the Central Limit Theorem
(see [35], Chapter 3, for some results in this direction).
LARGE PORTFOLIO LOSSES 17
(ii) A weak convergence-type approach based on uniform convergence of
the generators (see [17]).
In this paper we shall follow an approach of the second type; more pre-
cisely we shall provide a dynamical interpretation of the law of large numbers
discussed in Theorem 3.3. Let ψ :{−1,1}2 → R, and define ρN (t) by
ψdρN (t) :=
ψ(σi(t), ωi(t)).
In other words, ρN (t) is the marginal of ρN at time t and we also have
N (t) =m
ρN (t)
. Note that, for each fixed t, ρN (t) is a probability on {−1,1}2,
and so, by the considerations leading to (12), it can be viewed as a three-
dimensional object. Thus (ρN (t))t∈[0,T ] is a three-dimensional flow. A simple
consequence of Theorem 3.3 is the following convergence of flows:
(ρN (t))t∈[0,T ] → (qt)t∈[0,T ] a.s.,(17)
where the convergence of flows is meant in the uniform topology. Since the
flow of marginals contains less information than the full measure of paths,
the law of large numbers in (17) is weaker than the one in Theorem 3.3.
However, the corresponding fluctuation flow
N(ρN (t)− qt))t∈[0,T ]
is also a finite-dimensional flow, and it allows for a very explicit characteriza-
tion of the limiting distribution. The following theorem gives the asymptotic
behavior of this fluctuation flow; its proof is given in Section 5.
Theorem 3.6. Consider the following three-dimensional fluctuation pro-
cess:
xN (t) :=
N(mσρN (t) −m
yN (t) :=
N(mωρN (t) −m
zN (t) :=
N(mσωρN (t) −m
Then (xN (t), yN (t), zN (t)) converges as N →∞, in the sense of weak con-
vergence of stochastic processes, to a limiting three-dimensional Gaussian
process (x(t), y(t), z(t)) which is the unique solution of the following linear
stochastic differential equation:
dx(t)
dy(t)
dz(t)
 =A(t)
 dt+D(t)
dB1(t)
dB2(t)
dB3(t)
(18)
18 DAI PRA, RUNGGALDIER, SARTORI AND TOLOTTI
where B1,B2,B3 are independent, standard Brownian motions,
A(t) = 2
− cosh(β)
−γmωt sinh(γmσt ) + γ cosh(γmσt )
sinh(γmσt ) + γm
t cosh(γm
t )− γmσωt sinh(γmσt )
sinh(β) 0
− cosh(γmσt ) 0
0 −(cosh(β) + cosh(γmσt ))
D(t)D∗(t)
−mσωt sinh(β) + cosh(β) 0
0 −mωt sinh(γmσt ) + cosh(γmσt )
−mσt sinh(β) +mωt cosh(β) mσt cosh(γmσt )−mσωt sinh(γmσt )
−mσt sinh(β) +mωt cosh(β)
mσt cosh(γm
t )−mσωt sinh(γmσt )
−mσωt sinh(β) + cosh(β)−mω sinh(γmσt ) + cosh(γmσt )
and (x(0), y(0), z(0)) have a centered Gaussian distribution with covariance
matrix
1− (mσλ)2 mσωλ −mσλmωλ mωλ −mσλmσωλ
mσωλ −mσλmωλ 1− (mωλ)2 mσλ −mσωλ mωλ
mωλ −mσλmσωλ mσλ −mσωλ mωλ 1− (mσωλ )2
.(19)
Theorem 3.6 guarantees that, for each t > 0, the distribution of (xN (t),
yN (t), zN (t)) is asymptotically Gaussian, and provides a method to compute
the limiting covariance matrix. Indeed, denote by Σt the covariance matrix
of (x(t), y(t), z(t)). A simple application of Itô’s rule to (18) shows that Σt
solves the Lyapunov equation
=A(t)Σt + ΣtA(t)
∗ +D(t)D∗(t).(20)
In order to solve (20), it is convenient to interpret Σ as a vector in R3×3 =
3⊗R3. To avoid ambiguities, for a 3×3 matrix C we write vec(C) whenever
we interpret it as a vector. It is easy to check that (20) can be rewritten as
follows
d(vec(Σt))
= (A(t)⊗ I + I ⊗A(t)) vec(Σt) + vec(D(t)D∗(t)),(21)
where “⊗” denotes the tensor product of matrices. Equation (21) is linear,
so its solution can be given an explicit expression and can be computed after
LARGE PORTFOLIO LOSSES 19
having solved (13). More importantly, the behavior of Σt for large t can be
obtained explicitly as follows.
A. Case γ < 1
tanh(β)
. In this case we have shown in Theorem 3.4 that the
solution (mσt ,m
t ) of (13) converges to (0,0, tanh(β)) as t→ +∞.
In particular, one immediately obtains the limits
A := lim
A(t), DD∗ := lim
D(t)D∗(t).(22)
A direct inspection (see the Appendix) shows that A has three real strictly
negative eigenvalues. Moreover, the eigenvalues of the matrix A × I +
I × A are all of the form λi + λj where λi and λj are eigenvalues of
A, and therefore they are all strictly negative. It follows from (21) that
limt→+∞ Σt = Σ where
vec(Σ) = −(A⊗ I + I ⊗A)−1 vec(DD∗).(23)
B. Case γ > 1
tanh(β)
. Also in this case, by Theorem 3.4, the limit
(mσt ,m
exists. Disregarding the exceptional case in which the initial condition of
(13) belongs to the stable manifold Γ introduced in Theorem 3.4(iv), the
limit above equals either (mσ∗ ,m
∗ ), or (−mσ∗ ,−mω∗ ,mσω∗ ), depending
on the initial condition, where (mσ∗ ,m
∗ ) are obtained by Theorem
3.4(iii). In both cases one obtains as in (22) the limits A and DD∗, and
we show in the Appendix that also in this case the eigenvalues of A are
real and strictly negative, so that limt→+∞ Σt = Σ is obtained as in (23).
C. Case γ = 1
tanh(β)
. In this case, as shown in the Appendix, the limiting
matrix A is singular; it follows that the limit limt→+∞ Σt does not exist,
as one eigenvalue of Σt grows polynomially in t. This means that, for
critical values of the parameters, the size of normal fluctuations around
the deterministic limit grows in time. Similarly to what is done in [8]
for reversible models, it is possible to determine the critical long-time
behavior of the fluctuation by a suitable space–time scaling in the model,
giving rise to nonnormal fluctuations. More precisely, one can show the
following convergence in distribution:
N1/4(m·ρN (
Nt)−m·(
N→∞−→ Z
where Z is non-Gaussian. This result is contained in [32].
We now state an immediate corollary of Theorem 3.6 concerning the fluc-
tuations of the global health indicator; this will be used in the next section
on large portfolio losses.
20 DAI PRA, RUNGGALDIER, SARTORI AND TOLOTTI
Corollary 3.7. As N →∞ we have that
N [mσρN (t) −m
converges in law to a centered Gaussian random variable Z with variance
V (t) = Σ11(t),(24)
where Σ(t) solves (20) and mσt solves (13).
We conclude this section with the following:
Remark 3.8. The evolution equation (20) for the covariance matrix Σt
is coupled with the McKean–Vlasov equation (13), and their joint behavior
exhibits interesting aspects even before the system gets close to the stable
fixed point. In particular, in the case γ > 1
tanh(β)
, if the initial condition is
sufficiently close to the stable manifold Γ, the system (13) spends some time
close to the symmetric equilibrium (0,0) before drifting to one of the stable
equilibria. A closer look at (20) shows that when the system is close to
the neutral equilibrium, the covariance matrix Σ grows exponentially fast in
time, causing sharp peaks in the variances. This is related to the credit crisis
mentioned in the Introduction. A more detailed discussion on this point is
given in the next section, in relation with applications to portfolio losses.
4. Portfolio losses. We address now the problem of computing losses in
a portfolio of positions issued by the N firms. A rather general modeling
framework is to consider the total loss that a bank may suffer due to a
risky portfolio at time t as a random variable defined by LN (t) =
iLi(t).
Different specifications for the single (marginal) losses Li(t) can be chosen
accounting for heterogeneity, time dependence, interaction, macroeconomic
factors and so on. A punctual treatment of this general modeling framework
can be found in the book by McNeil, Frey and Embrechts [29]. For a com-
parison with the most widely used industry examples of credit risk models
see Frey and McNeil [21], Crouhy, Galai and Mark [11] or Gordy [25]. The
same modeling insights are also developed in the most recent literature on
risk management and large portfolio losses analysis; see [14, 19, 23, 26] for
different specifications.
In this paper we adopt the point of view of Giesecke and Weber [23]. The
idea is to compute the aggregate losses as a sum of marginal losses Li(t),
of which the distribution is supposed to depend on the realization of the
variable σi, that is, on the rating class. In particular, conditioned on the
realization of σ, the marginal losses will be assumed to be independent and
identically distributed (the independence condition can be weakened; see
LARGE PORTFOLIO LOSSES 21
Example 4.4 below). More precisely, we assume given a suitable conditional
distribution function Gx, x ∈ {−1,1}, namely
Gx(u) := P (Li(t) ≤ u|σi(t) = x)(25)
where the first and second moments are well defined, namely
l1 :=E(Li(t)|σi(t) = 1)<E(Li(t)|σi(t) = −1) =: l−1(26)
v1 := Var(Li(t)|σi(t) = 1), v−1 := Var (Li(t)|σi(t) = −1).(27)
The inequality in (26) specifies that we expect to lose more when in financial
distress.
The aggregate loss of a portfolio of volume N at time t is then defined as
LN (t) =
Li(t).
We recall the definition of the global health indicatorsm
N (t) :=
i=1 σi(t),
and mσt :=
σ dqt where qt solves the McKean–Vlasov equation [see (10)].
We also introduce a deterministic time function, which will be seen to
represent an “asymptotic” loss when the number of firms goes to infinity,
namely
L(t) =
(l1 − l−1)
mσt +
(l1 + l−1)
.(28)
We state now the main result of this section.
Theorem 4.1. Assume Li(t) has a distribution of the form (25). Then
for t ∈ [0, T ] with generic T > 0 and for any value of the parameters β > 0
and γ > 0, we have
LN (t)
−L(t)
→ Y ∼N(0, V̂ (t))
in distribution, where L(t) has been defined in (28) and
V̂ (t) =
(l1 − l−1)2V (t)
(1 +mσt )v1
(1−mσt )v−1
,(29)
with V (t) as defined in (24).
Proof. See Section 5. �
22 DAI PRA, RUNGGALDIER, SARTORI AND TOLOTTI
Remark 4.2. The Gaussian approximation in Theorem 4.1 leads in
particular to
P (LN (t) ≥ α) ≈N
NL(t)−α
V̂ (t)
.(30)
By the symmetry of the model, the above Gaussian approximation for the
losses is appropriate for a wide (depending on N ) range of values of α. If we
modify the model to become asymmetric as discussed in Remark 2.1 and,
more precisely, we modify it so that σ = −1 becomes much less likely than
σ = +1, then for a “realistic” value of N , the number of firms with σi = −1
could be too small for the Gaussian approximation to be sufficiently precise.
One could then rather consider a Poisson-type approximation instead.
We shall now provide examples illustrating possible specifications for the
marginal loss distributions where, without loss of generality, we assume a
unitary loss (e.g., loss due to a corporate bond) when a firm is in the bad
state.
We start with a very basic example where we assume that the marginal
losses (when conditioned on the value of σ) are deterministic. This means
that the riskiness of the loss portfolio is related only to the number of firms
in financial distress and so we can use directly the results of Section 3, in
particular of Corollary 3.7.
Example 4.3. Suppose that marginal losses are described as follows:
Li(t) =
1, if σi(t) = −1,
0, if σi(t) = 1.
On the other hand
LN (t) =
1− σi(t)
Recalling that m
N (t) =
i σi(t), by Corollary 3.7 [see also (30)], we can
compute various risk measures related to the portfolio losses such as the
following Var-type measure:
P (LN (t) ≥ α) = P
N −NmσN (t)
N (t) ≤
N − 2α
(−2α+ (1−mσt )N√
V (t)
(−2α+ 2L∞(t)N√
V (t)
where L∞(t) := limN→∞
LN (t)
= limN→∞
1−σi(t)
1−mσt
LARGE PORTFOLIO LOSSES 23
Fig. 1. Excess loss in a large portfolio (N = 10,000) for different values of the parameters
γ and β compared with the independence case.
Looking at a portfolio of N = 10,000 small firms, we compute the excess
loss probability for different values of the parameters β,γ comparing them
with the benchmark case where there is no interaction at all, that is, where
β = γ = 0 (“independence case”). In Figure 1 we show the cumulative prob-
ability of having excess losses for the same portfolios. In this figure we see
that, when the dependence increases, variance and risk measures increase as
well.
More general specifications are already suggested in the existing literature.
For example, one could consider the losses to depend also on a random
exogenous factor Ψ; more precisely, the marginal losses Li(t) are independent
and identically distributed conditionally to the realizations of the σi(t)’s and
of Ψ. The conditional distributions
Gx(u) := P (Li(t) ≤ u|σi(t) = x,Ψ)
are random variables, as well as the corresponding moments l1, l−1, v1, v−1.
In particular in the following example we apply our approach to a very
tractable class of models, the “Bernoulli mixture models.” This kind of mod-
eling has been used in the context of cyclical correlations, that is in models
where exogenous factors are supposed to characterize the evolution of the in-
dicator of defaults (the classical factor models). In the context of contagion-
based models this class was first introduced by Giesecke and Weber in [23].
24 DAI PRA, RUNGGALDIER, SARTORI AND TOLOTTI
Example 4.4 (Bernoulli mixture models). Assume that the marginal
losses Li(t) are Bernoulli mixtures, that is,
Li(t) =
1, with probability P (σi(t),Ψ),
0, with probability 1−P (σi(t),Ψ),
where the mixing derives not only from the rating class indicator σi(t) of firm
i, but also from an exogenous factor Ψ ∈ Rp that represents macroeconomic
variables that reflect the business cycle and thus allow for both contagion
and cyclical effects on the rating probabilities.
Notice that, with the above specification, the quantities defined in (26)
and (27) now depend on the random factor Ψ, that is,
l1 = P (1,Ψ), v1 = P (1,Ψ)(1−P (1,Ψ)) and analogously for l−1, v−1.
Consequently, the asymptotic loss function L(t) as well as the variance of the
Gaussian approximation V̂ (t) defined in (28) and (29) are also functions of
Ψ. With a slight abuse of notation we shall write Lψ(t) [respectively V̂ψ(t)]
for the asymptotic loss (variance) at time t given that Ψ = ψ.
Next we give a possible expression for the mixing distribution for P (σ,Ψ)
that is in line with existing models on contagion. Let a and bi, i= 1,2, be
nonnegative weight factors. Let us assume for simplicity that Ψ ∈ R is a
Gamma distributed random variable. Define then
P (σ,Ψ) = 1− exp
−aΨ− b1
This specification follows the CreditRisk+ modeling structure, even though
in the standard industry examples direct contagion is not taken into account.
Notice that the factor 1−σ
increases the probability of default for the firms
in the bad rating class (σ = −1). Using (30) we have that
P (LN (t) ≥ α) ≈
NLψ(t)−α
NV̂ψ(t)
dfΨ(ψ),
where fΨ is the density function of the Gamma random variable Ψ.
In Figure 2 we plot the excess loss probability in the case where a= 0.1,
b1 = 1, b2 = 0.5 and β = 1.5 is supposed to be fixed. We compare different
specifications for Ψ and γ. In particular we consider the following cases:
Ψ = 4.5, γ = 0.6; Ψ = 4.5, γ = 1.1;
Ψ ∼ Γ(2.25,2), γ = 1.1.
The shape of the excess losses suggests that the loss may be sensibly higher
in the case of high uncertainty about the value of the macroeconomic factor
[Ψ ∼ Γ(2.25; 2)] and in the case of high level of contagion (γ = 1.1). Notice
that in all three situations we are in the subcritical case, since the critical
value for γ is γc = 1/ tanh(β) ≃ 1.105. This also implies that the equilibrium
value is the same in the three situations and depends only on Ψ.
LARGE PORTFOLIO LOSSES 25
Fig. 2. Loss amount in a large portfolio (N = 10,000) in the case of marginal losses
which (depending on the rating class) are distributed as Bernoulli random variables for
which the parameter depends on Ψ.
Remark 4.5. Notice that the asymptotic loss distribution in the above
Bernoulli mixture model does not only depend on a mixing parameter as
in standard Bernoulli mixture models but, via L(t), it depends also on the
value mσt of the asymptotic average global health indicator. Moreover, com-
pared to Giesecke and Weber [23], we are able to quantify the time-varying
fluctuations of the global indicator mσ
ρN (t)
. We shall see that this may sen-
sibly influence the distribution of losses in particular when looking at two
different time horizons T1 and T2 before and after a credit crisis.
Remark 4.6. Further examples may be considered, in particular when
the distribution of the marginal losses Li(t) depends on the entire past
trajectory of the rating indicator σi, taking, for example, into account how
long the firm has been in the bad state. Instead of depending simply on
σi(t), the distribution of Li(t) could then be made dependent on Si(t) :=
((1−σi(s))/2)ds≥δt}
with δ ∈ (0,1), which is equal to 1 if firm i has spent
a fraction δ of time in the bad state. Corresponding to (32) we would then
Li(t) =
1, with probability P (Si(t),Ψ),
0, with probability 1− P (Si(t),Ψ).
This model is not a straightforward extension of Example 4.4. In fact the
theory developed above, in particular the Central Limit Theorem result in
26 DAI PRA, RUNGGALDIER, SARTORI AND TOLOTTI
Section 3.3, does not appear to be strong enough to handle it. For this
purpose an approach based on a functional central limit theorem that was
alluded to at the beginning of Section 3.3 would be more appropriate. This,
however, goes beyond the scope of the present paper.
Let us point out that in the examples above we have considered only the
problem of computing large portfolio losses which led to examples where we
computed (approximately) the quantiles P (LN (t) ≥ α) where α is a (large)
integer. From here, one could then compute the probability that the loss
ratio
LN (t)
belongs to a given interval and this would then allow to compute
(approximately) for our contagion model also other quantities in a risk-
sensitive environment. In any case notice that Theorem 4.1 provides the
entire asymptotic distribution for the portfolio losses.
In the previous examples we have described large portfolio losses at a
predetermined time horizon T for different specifications of the conditional
loss distribution. In what follows, we shall describe in more detail how the
phenomenon of a credit crisis may be explained in our setting and how this
issue may influence the quantification of losses. This dynamic point of view
on risk management that accounts for the possibility of a credit crisis in the
market, is one of the main contributions of this work.
As one could expect, the possibility of having a credit crisis is related
to the existence of particular conditions on the market, more precisely to
certain levels of interaction between the obligors (i.e., the parameters β and
γ) and certain values of the state variables describing the rating classes and
the fundamentals (i.e., σ and ω).
4.1. Simulation results. To illustrate the situation we shall now present
some simulation results. We shall proceed along two steps: the first one
relates more specifically to the particle system, the second to the portfolio
losses.
Step 1 (Domains of attraction). In Section 3.2 we have characterized all
the equilibria of the system depending on the values of the parameters. In
particular we have shown that for supercritical values, by which we mean
γ > 1
tanh(β)
, there are two asymmetric equilibrium configurations in the space
(mσ,mω) that, for our symmetric model, are symmetric to one another and
are defined as (mσ∗ ,m
∗ ) and (−mσ∗ ,−mω∗ ).
In particular, Theorem 3.4 allows to characterize their domains of at-
traction, that is, the sets of initial conditions that lead the trajectory to
one of the equilibria, and we shall denote them by Γ+ and Γ−. Numerical
simulations provide diagrams as in Figure 3.
LARGE PORTFOLIO LOSSES 27
Fig. 3. Domains of attraction Γ+ for (mσ∗ ,m
∗ ) and Γ
− for (−mσ∗ ,−m
∗ ) and
their boundary Γ for β = 1 and varying γ. Here the critical value for γ is
γc := 1/ tanh(β) ≃ 1.313.
Step 2 (Credit crises). We show results from numerical simulations that
detect the crises when the values of the parameters are supercritical and the
initial conditions are “near” the boundary of the domains of attraction, that
is, near Γ. Given the symmetry of our model, the behavior of the system will
be perfectly symmetric when starting in either Γ+ or Γ−, but the typical
credit crisis corresponds to what happens in Γ−, so that below we shall
illustrate this latter case. The analysis in an asymmetric model would be
analogous.
In Figure 4 we have plotted a trajectory starting in (mσ0 ,m
0 ) ∈ Γ− but
near the boundary. It can be seen that the path moves toward (mσ,mω) =
(0,0) and then leaves it decaying to the stable equilibrium.
Concerning the time evolution, we see in Figure 5 that, for an initial
condition in Γ− and near the boundary, the variable mσt (the same would
happen also with mωt that for clarity is not plotted) is first attracted to the
unstable value zero, around which it spends a long time before moving to
the stable equilibrium value mσ∗ . This can be explained, in financial terms,
as follows:
Suppose that at the initial time the market conditions are such that
(mσ,mω) are in Γ− but close to the curve Γ. Then for a while the system
moves close to the stable manifold Γ toward (0,0), until it gets “captured”
28 DAI PRA, RUNGGALDIER, SARTORI AND TOLOTTI
Fig. 4. Domains of attraction Γ+ for (mσ∗ ,m
∗ ) and Γ
− for (−mσ∗ ,−m
∗ ) and phase dia-
gram of (mσt ,m
t ) with initial conditions (m
0 ) = (0.6,−0.85) when β = 1 and γ = 2.3
[here γc = 1/ tanh(β) ≃ 1.313].
by the unstable direction of the equilibrium point (0,0). Since the system
configuration belongs to Γ−, the new stable equilibrium that the system is
attracted to is given by (−mσ∗ ,−mω∗ ).
This situation represents (in a stylized manner) what we intend as a
credit crisis: the state (0,0) may be considered as a “credit bubble,” the
decay toward the stable equilibrium mimics a credit crisis (i.e., a crash in
the credit market).
As soon as the system moves away from (0,0), the uncertainty (volatility)
increases quickly and the credit quality indicators move to the stable con-
figuration changing completely the picture of the market (the speed of the
convergence depends on the level of interaction).
This situation is also well illustrated by the loss probability computed
before and after the crisis (i.e., in certain time instants T1 and T2). In Figure
6 we see the excess probability of suffering a loss larger than x for the
case of Example 4.4 with an exogenous parameter Ψ ∼ Γ(2.25; 2). One can
see that before the crisis both the expected loss and the variance may be
underestimated as well as the corresponding risk measures. Put differently,
a model that does not distinguish between stable and unstable equilibria
LARGE PORTFOLIO LOSSES 29
Fig. 5. Trajectory of mσt and V (t) with initial conditions m
0 =−0.5, m
0 = 0.395 when
β = 1.5 and γ = 2.1 [here γc = 1/ tanh(β) ≃ 1.105]. We have marked by (∗) the time hori-
zons T1 = 2 and T2 = 10 before and after the crisis where in Figure 6 we shall compute the
excess loss probabilities.
(does not take credit crises into account) may underestimate the excess loss
probability, since it does not recognize in the given situation the possibility
of a sudden crash.
Finally we mention the fact that for different levels of interaction we can
distinguish between a smoothly varying business cycle and a crisis. When
β and γ, the parameters describing the level of interaction, are sufficiently
small, the business cycle (described in our simple model by the proportion of
firms in the rating classes) evolves smoothly and the induced variance (level
of uncertainty about the number of bad rated firms) is lower compared to
the crisis case. In Figure 7 we show this fact for two levels of β and γ, both
supercritical.
5. Proofs.
5.1. Proofs of Propositions 3.1 and 3.2. One of the main tools in this
proof is the Girsanov formula for Markov chains. Since a Markov chain is a
functional of the multivariate point process that counts the jumps between
all pairs of states, this formula can be derived from the corresponding Gir-
sanov formula for point processes (see, e.g., [3], Section 4.2). We state it here
for completeness.
30 DAI PRA, RUNGGALDIER, SARTORI AND TOLOTTI
Fig. 6. Excess probability of losses in a portfolio of N = 10,000 obligors, β = 1.5 and
γ = 2.1 computed in T1 = 2 and T2 = 10, namely before and after the crisis in the case of
Example 4.4 with Ψ ∼ Γ(2.25; 2) [here γc = 1/ tanh(β) ≃ 1.105].
Fig. 7. Trajectories of mσt and V (t) for different levels of interaction, that is, letting β
and γ vary. In the case of higher values we really see a crisis and a corresponding peak in
the uncertainty in the market. In the case of smaller values the number of bad rated firms
decreases smoothly to a new equilibrium, that is, toward a bad business cycle. The critical
values for γ are, respectively, 1/ tanh(1.5) ≃ 1.105 and 1/ tanh(0.9) ≃ 1.396.
LARGE PORTFOLIO LOSSES 31
Proposition 5.1. Let S be a finite set, and (X(t))t∈[0,T ], (Y (t))t∈[0,T ]
two S-valued Markov chains with infinitesimal generators, respectively,
Lf(x) =
y 6=x
Lx,y[f(y)− f(x)],
Mf(x) =
y 6=x
Mx,y[f(y)− f(x)].
Assume X(0) and Y (0) have the same distribution, and denote by PX and
PY the law of the two processes on the appropriate set of trajectories in
the time-interval [0, T ]. Assume that whenever Mx,y = 0 also Lx,y = 0. Then
PX ≪ PY , and
(x([0, T ]))
= exp
y 6=x(t)
(Mx(t),y −Lx(t),y)dt+
Lx(t−),x(t)
Mx(t−),x(t)
where x(t−) := lims↑t x(s), log
= 1 and Nt is the counting process that
counts the jumps of the trajectory x([0, T ]).
In what follows we denote by PN the law on the path space of (σ[0, T ],
ω[0, T ]) ∈ (D([0, T ]))2N under the interacting dynamics, with initial condi-
tions such that (σ
i (0), ω
i (0))
i=1 are independent and identically dis-
tributed with an assigned law λ (see beginning of Section 3). As in Section
3.1 we let W ∈ M1(D([0, T ]) × D([0, T ])) denote the law of the {−1,1}2-
valued process (σ(t), ω(t)) such that (σ(0), ω(0)) has distribution λ, and
both σ(·) and ω(·) change sign with constant rate 1. By W⊗N we mean the
product of N copies of W . We begin with some preliminary lemmas.
Lemma 5.2.
dW ⊗N
(σ[0, T ], ω[0, T ]) = exp[NF (ρN (σ[0, T ], ω[0, T ]))],(33)
where F is the function defined in (7).
Proof. Let (N
t (i))
i=1 be the multivariate counting process which counts
the jumps of σi for i= 1, . . . ,N , and (N
t (i))
i=1 be the multivariate counting
process which counts the jumps of ωi for i= 1, . . . ,N . Since each jump of the
trajectory (σ[0, T ], ω[0, T ]) is counted by exactly one of the above counting
processes, Proposition 5.1 applied to this case yields
dW ⊗N
(σ[0, T ], ω[0, T ])
32 DAI PRA, RUNGGALDIER, SARTORI AND TOLOTTI
= exp
(1− e−βσi(t)ωi(t))dt+
log e−βσi(t
−)ωi(t
−) dN
t (i)
(1− e−γωi(t)m
ρN (t))dt
log e
−γωi(t−)mσ
ρN (t
−) dN
t (i)
Since, with probability 1 with respect to W ⊗N , there are no simultaneous
jumps, we have
log e−βσi(t
−)ωi(t
−) dN
t (i)=
−β (−σi(t))ωi(t)dNσt (i)
log e
−γωi(t−)m
ρN (t
−) dN
t (i)=
−γ (−ωi(t))mσρN (t) dN
t (i),
from which (33) follows easily after having observed that, W⊗N almost
surely,
(NσT +N
T )dρN <+∞,
and that simultaneous jumps of σ and ω do not occur under dW ⊗N . �
The main problem in the proof of Proposition 3.1 is related to the fact
that the function F in (7) is neither continuous nor bounded. The following
technical lemmas have the purpose of circumventing this problem. In what
follows, we let
Q ∈M1(D[0, T ]2) :
(NσT +N
T )dQ<+∞
.(34)
We first define, for r > 0 and Q∈ I ,
Fr(Q) =
(r− e−βσ(t)ω(t))dt+
(r− e−ω(t)γ
t )dt
(βσ(t)ω(t−)− log r)dNσt(35)
(ω(t)γ
− log r)dNωt
LARGE PORTFOLIO LOSSES 33
Note that F = F1. Moreover, Lemma 5.2 can be easily extended to show
dW ⊗Nr
(σ[0, T ], ω[0, T ]) = exp[NFr(ρN (σ[0, T ], ω[0, T ]))],(36)
where Wr is the law of the {−1,1}2-valued process σ(t), ω(t) such that (σ(0),
ω(0)) has distribution λ, and both σ(·) and ω(·) change sign with constant
rate r.
Lemma 5.3. For 0< r ≤ min(e−β , e−γ), Fr is lower semicontinuous on
I . For r ≥max(eβ, eγ), Fr is upper semicontinuous.
Proof. By definition of weak topology the fact that the map
(r− e−βσ(t)ω(t))dt+
(r− e−ω(t)γ
t )dt
is continuous is rather straightforward (since Q-expectations of bounded
continuous functions in D([0, T ]) are continuous in Q). Thus we only have
to deal with the term
(βσ(t)ω(t−)− log r)dNσt
(ω(t)γ
− log r)dNωt
We show that for 0< r≤ min(e−β , e−γ) the expression in (37) is lower semi-
continuous in Q ∈ I . This shows that Fr is lower semicontinuous. The case
r ≥max(eβ, eγ) is treated similarly.
For ε > 0 consider the function ϕε :D[0, T ]→ R defined by
ϕε(η) :=
, if η(t) jumps for some t ∈ (0, ε],
0, otherwise.
Given η ∈ D([0, T ]) we define η(s) for s > T by letting η(s) ≡ η(T ). Then,
letting θt denote the shift operator, we have that, for t ∈ [0, T ], θtη is the
element of D([0, T ]) given by θtη(s) := η(t+ s). Consider now two functions
f, g :{−1,1}2 → R, and define fε, gε :D[0, T ]2 → R by
fε(σ[0,T ], ω[0,T ]) := inf{f(σ(t), ω(t)) : t ∈ (0, ε)},
and similarly for gε. Then define
Φε(σ[0,T ], ω[0,T ]) :=
fε(θtσ, θtω)ϕε(θtσ)dt+
gε(θtσ, θtω)ϕε(θtω)dt.
The key to the continuation of the proof below are the following two proper-
ties of Φε. These properties are essentially straightforward, and their proofs
are omitted:
34 DAI PRA, RUNGGALDIER, SARTORI AND TOLOTTI
• Φε is continuous and bounded on {(σ[0,T ], ω[0,T ]) :NσT +NωT <+∞}.
• Suppose f, g ≥ 0. Then, assuming σ[0,T ], ω[0,T ] have a finite number of
jumps, Φε(σ[0,T ], ω[0,T ]) increases when ε ↓ 0 to
f(σt− , ωt−)dN
g(σt− , ωt−)dN
Therefore by monotone convergence
f(σt− , ωt−)dN
g(σt− , ωt−)dN
= sup
Φε(σ[0,T ], ω[0,T ])dQ.
In particular, the map
f(σt− , ωt−)dN
g(σt− , ωt−)dN
is lower semicontinuous on I .
Now, for r≤ min(e−β , e−γ), the function f(σ,ω) =−βσω− log r is nonnega-
tive. As for the function g, that should be −ω(t)γQt − log r, we notice that it
is not a function of (σ,ω), but rather a function of (σ,ΠtQ), thus depending
explicitly on t and Q. However, due to its boundedness and the fact that γ
is continuous in Q uniformly in t, σ, the argument above applies with minor
modifications thus leading to the conclusion of the proof. �
Lemma 5.4. Let Q∈M1(D([0, T ])2) be such that H(Q|W )<+∞. Then
Q ∈ I . The same result applies if Wr replaces W .
Proof. By the entropy inequality (see (6.2.14) in [15])
NσT dQ≤ log
T dW +H(Q|W ).
But NσT has Poisson distribution under W , so
T dW <+∞. By applying
the same argument to NωT , the proof is completed. This proof extends with
no modifications to the case r 6= 1. �
Lemma 5.5. The function
I(Q) :=H(Q|W )−F (Q)
is lower semicontinuous on M1(D[0, T ]2).
LARGE PORTFOLIO LOSSES 35
Proof. It is well known (see [15], Lemma 6.2.13) that the entropy
H(Q|W ) is lower semicontinuous in Q in all of M1(D([0, T ])2). Moreover,
by definition, F (Q) < +∞ for every Q, and so we have H(Q|W ) = I(Q)
whenever H(Q|W ) = +∞. Since, by Lemma 5.4, H(Q|W ) = +∞ for Q /∈ I ,
we are left to prove the following two statements:
(i) I(Q) is lower semicontinuous in I .
(ii) If H(Q|W ) = +∞ and Qn →Q weakly, then I(Qn)→ +∞.
The following key identity, which holds for r > 0, is a simple consequence
of the definition of relative entropy and of the Girsanov formula for Markov
chains.
H(Q|Wr) =H(Q|W ) +
=H(Q|W ) + 2T (r− 1) + log r
(NσT +N
T )dQ.
In particular, by Lemma 5.4, we have that H(Q|W )<+∞ ⇐⇒ H(Q|Wr)<
+∞. A simple consequence of (38) is then the following:
I(Q) =H(Q|Wr)− Fr(Q),(39)
where the difference in (39) is meant to be +∞ whenever H(Q|Wr) = +∞
[which is equivalent to H(Q|W ) = +∞].
We are now ready to prove (i) and (ii). To prove (i) it is enough to choose
r ≥max(eβ , eγ) and use Lemma 5.3. Moreover, for the same choice of r, the
stochastic integrals in (35) are nonpositive, so Fr(Q) ≤ 2Tr. Therefore, if
H(Q|W ) = +∞ and Qn→Q,
lim inf I(Qn) ≥ lim infH(Qn|Wr)− 2Tr = +∞,
where the last equality follows from lower semicontinuity of H(·|Wr) and
H(Q|Wr) = +∞. Thus (ii) is proved. �
Lemma 5.6. The function I(Q) has compact level sets, that is, for every
k > 0 the set {Q : I(Q) ≤ k} is compact.
Proof. Choosing, as above, r ≥max(eβ , eγ), we have that Fr(Q) ≤ 2Tr
for every Q. Thus, by (39),
{Q : I(Q) ≤ k} ⊆ {Q :H(Q|Wr)≤ k+ 2Tr}.
Since (see [15], Lemma 6.2.13) the relative entropy has compact level sets,
{Q : I(Q) ≤ k} is contained in a compact set. Moreover, by lower semiconti-
nuity of I , {Q : I(Q)≤ k} is closed, and this completes the proof. �
36 DAI PRA, RUNGGALDIER, SARTORI AND TOLOTTI
Lemma 5.7. For every r > 0 there exists δ > 1 such that
lim sup
exp[δNFr(ρN )]dW
r <+∞.
Proof. We give the proof for r = 1; the modifications for the general
case are obvious. The proof consists of rather simple manipulations. The idea
can be summarized as follows. If δ = 1, then by Lemma 5.2, exp[δNF (ρN )]
is the Radon–Nikodym derivative of PN with respect to W
⊗N , and therefore
has expectation 1. For δ > 1, we write δF (ρN ) = F1(ρN ) +F2(ρN ) in such a
way that F2 is bounded and exp[NF1(ρN )] is a Radon–Nikodym derivative
of a probability with respect to W⊗N . More specifically, observe that, using
δNF (ρN ) =
(δ − δe−βσi(t)ωi(t))dt+
δβσi(t)ωi(t
t (i)
(δ − δe−γωi(t)m
ρN (t))dt
δγωi(t)m
ρN (t
−) dN
t (i)
(1− e−δβσi(t)ωi(t))dt+
δβσi(t)ωi(t)dN
t (i)
(1− e−δγωi(t)m
ρN (t))dt+
δγωi(t)m
ρN (t)
t (i)
(δ − δe−βσi(t)ωi(t) − (1− e−δβσi(t)ωi(t)))dt
(δ − δe−γωi(t)m
ρN (t) − (1− e−δγωi(t)m
ρN (t)))dt
=NF1(ρN ) +NF2(ρN ),
where
NF1(ρN ) :=
(1− e−δβσi(t)ωi(t))dt+
δβσi(t)ωi(t)dN
t (i)
(1− e−δγωi(t)m
ρN (t))dt+
δγωi(t)m
ρN (t)
t (i)
LARGE PORTFOLIO LOSSES 37
NF2(ρN ) :=
(δ − δe−βσi(t)ωi(t) − (1− e−δβσi(t)ωi(t)))dt
(δ − δe−γωi(t)m
ρN (t) − (1− e−δγωi(t)m
ρN (t)))dt.
Note that exp[NF1(ρN )] has the same form of exp[NF (ρN )] after having
replaced β by δβ. In particular,
exp[NF1(ρN )]dW
⊗N = 1. Moreover, it is
easy to see that
F2(ρN )≤ T (2δ − δ(e−β + e−γ)− 2 + eδβ + eδγ).
Putting all together, we obtain
exp[δNF (ρN )]dW
≤ exp[NT (2δ − δ(e−β + e−γ)− 2 + eδβ + eδγ)]
exp[NF1(ρN )]dW
= exp[NT (2δ − δ(e−β + e−γ)− 2 + eδβ + eδγ)],
from which the conclusion follows easily. �
Completing the proof of Proposition 3.1. It remains to show
the upper and the lower bounds (9) and (8). We prove them separately; our
main tool is the Varadhan Lemma in the version in [15], Lemmas 4.3.4 and
4.3.6.
We deal first with the upper bound (9). Take r≥ max(eβ , eγ), so that the
function Fr in (35) is upper semicontinuous. Denote by PN the distribution
of ρN under PN , and by WN its distribution under W⊗Nr . By (36)
(Q) = exp[NFr(Q)].(40)
By Sanov’s theorem (Theorem 6.2.10 in [15]), the sequence of probabilities
WN satisfies a large deviation principle with rate function H(Q|Wr). Since
Fr is upper semicontinuous and satisfies the superexponential estimate in
Lemma 5.7, we can apply Lemma 4.3.6 in [15], together with identity (39),
to obtain the upper bound (9). The lower bound (8) is proved similarly,
by taking 0< r≤ min(e−β , e−γ), so that Fr becomes lower semicontinuous,
using (40) again and Lemma 4.3.4 in [15]. �
The remaining part of this section is devoted to the proof of Proposition
3.2. It mainly consists in giving an alternative representation of the rate
function I(Q).
38 DAI PRA, RUNGGALDIER, SARTORI AND TOLOTTI
Let now Q ∈M1(D([0, T ]) ×D([0, T ])). We associate with Q the law of
a time-inhomogeneous Markov process on {−1,1}2 which evolves according
to the following rules:
with intensity e−βσω,
with intensity exp
σ,τ∈{−1,1}
σΠtQ(σ, τ)
−γωmσ
ΠtQ = e−γ
and with initial distribution λ. We denote by PQ the law of this process. In
other words, PQ is the law of the Markov process on {−1,1}2 with initial
distribution λ and time-dependent generator
LQt f(σ,ω) = e−βσω∇σf(σ,ω) + e
−γωmσ
ΠtQ∇ωf(σ,ω).
Lemma 5.8. For every Q ∈M1(D([0, T ])×D([0, T ])) such that I(Q)<
+∞, we have
I(Q) =H(Q|PQ).
Proof. We begin by observing that, since by assumption I(Q) <∞,
we have H(Q|W )<+∞ and so by Lemma 5.4 it follows that Q ∈ I , which
implies that the integrals below are well defined. Using again Girsanov’s
formula for Markov chains in Proposition 5.1, we obtain
(σ[0, T ], ω[0, T ])dQ
(1− e−βσ(t)ω(t))dt+
(1− e−γω(t)
σΠtQ(dσ, dτ))dt
(−βσ(t−)ω(t−))dNσt
−γω(t−)
σΠt−Q(dσ, dτ)
(1− e−βσ(t)ω(t))dt+
(1− e−γω(t)
σΠtQ(dσ,dτ))dt
σ(t)ω(t)dNσt + γ
σΠtQ(dσ, dτ)
(1− e−βσ(t)ω(t))dt+
(1− e−ω(t)γ
t )dt
LARGE PORTFOLIO LOSSES 39
σ(t)ω(t)dNσt +
ω(t)γ
= F (Q).
Finally, just observe that
I(Q) =
dQ log
dQ log
dQ log
=H(Q|PQ). �
Completing the proof of Proposition 3.2. By properness of the
relative entropy [H(µ|ν) = 0 ⇒ µ = ν], from Lemma 5.8 we have that the
equation I(Q) = 0 is equivalent to Q= PQ. Suppose Q∗ is a solution of this
last equation. Then, in particular, qt := ΠtQ
∗ = ΠtP
Q∗ . The marginals of a
Markov process are solutions of the corresponding forward equation that,
in this case, leads to the fact that qt is a solution of (10). This differential
equation, being an equation in finite dimension with locally Lipschitz coeffi-
cients, has at most one solution in [0, T ]. Since PQ
is totally determined by
the flow qt, it follows that equation Q= P
Q has at most one solution. The
existence of a solution follows from the fact that I(Q) is the rate function
of a LDP, and therefore must have at least one zero, indeed, by (8) with
A= M1(D[0, T ] ×D[0, T ]), we get infQ I(Q) = 0. Since I is lower semicon-
tinuous, this inf is actually a minimum. �
5.2. Proof of Theorem 3.4. We first observe that the square [−1,1]2
is stable for the flow of (14), since the vector field V (x, y) points inward
at the boundary of [−1,1]2. It is also immediately seen that the equation
V (x, y) = 0 holds if and only if x= tanh(β) tanh(γx) and y = 1
tanh(β)
x. More-
over a simple convexity argument shows that x= tanh(β) tanh(γx) has x= 0
as unique solution for γ ≤ 1
tanh(β)
, while for γ > 1
tanh(β)
a strictly positive so-
lution, and its opposite, bifurcate from the null solution. We have therefore
found all equilibria of (14).
We now remark that (14) has no cycles (periodic solutions). Indeed, sup-
pose (xt, yt) is a cycle of period T . Then by the Divergence Theorem
[V1(xt, yt)ẋt + V2(xt, yt)ẏt]dt=
divV (x, y)dxdy,(41)
where V1, V2 are the components of V and C is the open set enclosed by the
cycle. But a simple direct computation shows that divV (x, y)< 0 in all of
[−1,1]2, so that (41) cannot hold.
40 DAI PRA, RUNGGALDIER, SARTORI AND TOLOTTI
It follows by the Poincaré–Bendixon theorem that every solution must
converge to an equilibrium as t→ +∞. This completes the proof of (i). The
matrix of the linearized system is
DV (0,0) =
−2cosh(β) 2 sinh(β)
2γ −2
from which also (ii) and (iii) are readily shown. It remains to show (iv).
For γ > 1
tanh(β)
, we let vs be an eigenvector of the negative eigenvalue of
DV (0,0). By the Stable Manifold Theorem (see Section 2.7 in [30]), the
set of initial conditions that are asymptotically driven to (0,0) form a one-
dimensional manifold Γ that is tangent to vs at (0,0). Since any solution
converges to an equilibrium point, and solutions starting in Γc cannot cross
Γ (otherwise uniqueness would be violated), the remaining part of statement
(iv) follows.
5.3. Proof of Theorem 3.6.
Proof. One key remark is the fact that the stochastic process (mσ
ρN (t)
ρN (t)
ρN (t)
) is a sufficient statistic for our model; in this context this
means that its evolution is Markovian. This can be proved by checking that
if we apply the generator L in (3) to a function of the form ϕ(mσ
ρN (t)
ρN (t)
ρN (t)
), then we obtain again a function of (mσ
ρN (t)
ρN (t)
ρN (t)
). A long
but straightforward computation actually gives
Lϕ(mσρN (t),m
ρN (t)
,mσωρN (t)) = [KNϕ](m
ρN (t)
,mωρN (t),m
ρN (t)
where
KNϕ(ξ, η, θ)
(j,k)∈{−1,1}2
[jξ + kη + jkθ + 1]
e−βjk
ξ − 2
j,η, θ − 2
− ϕ(ξ, η, θ)
+ e−γξk
ξ, η− 2
k,θ− 2
− ϕ(ξ, η, θ)
This implies that KN is the infinitesimal generator of the three-dimensional
Markov process (mσ
ρN (t)
ρN (t)
ρN (t)
). Note now that (xN (t), yN (t), zN (t))
is obtained from (mσ
ρN (t)
ρN (t)
ρN (t)
) through a time dependent, linear
invertible transformation. We call Tt this transformation, that is,
Tt(ξ, η, θ) = (
N(ξ −mσt ),
N(η −mωt ),
N(θ −mσωt ))
LARGE PORTFOLIO LOSSES 41
(the dependence onN of Tt is omitted in the notation). Therefore (xN (t), yN (t),
zN (t)) is itself a (time-inhomogeneous) Markov process, whose infinitesimal
generator HN,t can be obtained from (42) as follows:
HN,tf(x, y, z) =KN [f ◦ Tt](T−1t (x, y, z)) +
[f ◦ Tt](T−1t (x, y, z)).
A simple computation gives then
HN,tf(x, y, z)
(j,k)∈{−1,1}2
+ jmσt + km
t + jkm
t + 1
e−βj k
x− 2√
j, y, z − 2√
− f(x, y, z)
+ e−γ(x/
x, y− 2√
k,z − 2√
−f(x, y, z)
Nṁσt fx(x, y, z)−
Nṁωt fy(x, y, z)−
Nṁσωt fz(x, y, z),
where fx stands for
, and similarly for the other derivatives. At this
point we compute the asymptotics of HN,tf(x, y, z) as N → +∞, assum-
ing f :R3 → R a C3 function with compact support. First of all we make a
Taylor expansion of terms like
x− 2√
j , y, z − 2√
− f(x, y, z)
= − 2√
fx(x, y, z)−
fz(x, y, z)(44)
fxx(x, y, z) +
fzz(x, y, z) +
fxz(x, y, z) + o
e−γ(x/
N) = 1− γ
.(45)
Note that, since all derivatives of f are bounded, the remainder in (44) is
) uniformly in (x, y, z) ∈ R3. Moreover, the remainder in (45) is o( 1√
uniformly for x in a compact set. Therefore, since f has compact support,
when we use (44) and (45) to replace the corresponding terms in (43), we
obtain remainders whose bounds are uniform in R3. When (44) and (45) are
plugged into (43), all terms of order
N coming from the sum over (j, k) ∈
42 DAI PRA, RUNGGALDIER, SARTORI AND TOLOTTI
{−1,1}2 are canceled by the terms
Nṁσt fx(x, y, z) −
Nṁωt fy(x, y, z) −√
Nṁσωt fz(x, y, z). It follows then by a straightforward computation that
t∈[0,T ]
x,y,z∈R3
|HN,tf(x, y, z)−Htf(x, y, z)|= 0,
where
Htf(x, y, z) = 2{fx[−x cosh(β) + y sinh(β)]
+ fy[−γxmωt sinh(γmσt ) + γx cosh(γmσt )− y cosh(γmσt )]
+ fz[x sinh(γm
t ) + γxm
t cosh(γm
− γxmσωt sinh(γmσt )− z cosh(β)− z cosh(γmσt )]
+ fxx[−mσωt sinh(β) + cosh(β)]
+ fyy[−mωt sinh(γmσt ) + cosh(γmσt )]
+ fzz[−mσωt sinh(β) + cosh(β)
−mωt sinh(γmσt ) + cosh(γmσt )]
+ 2fxz[−mσt sinh(β) +mωt cosh(β)]
+ 2fyz[m
t cosh(γm
t )−mσωt sinh(γmσt )]}
is the infinitesimal generator of the linear diffusion process (18). Using Theo-
rem 1.6.1 in [17], the proof is completed if we show that (xN (0), yN (0), zN (0))
converges as N → +∞, in distribution to (x(0), y(0), z(0)). This last state-
ment follows by the standard Central Limit Theorem for i.i.d. random vari-
ables; indeed, by assumption, (σi(0), ωi(0)) are independent with law λ,
and (19) is just the covariance matrix under λ of (σ(0), ω(0), σ(0)ω(0)). It
should be pointed out that Theorem 1.6.1 in [17] does not deal explicitly
with time-dependent generators, as is the case here. To fix this point it
is enough to introduce an additional variable, τ(t) := t, and consider the
process α(t) := (x(t), y(t), z(t), τ(t)), whose generator is time-homogeneous.
This argument, together with the fact that the convergence of HN,tf(x, y, z)
to Htf(x, y, z) is uniform in both (x, y, z) and t, completes the proof. �
5.4. Proof of Theorem 4.1. We start with a technical lemma.
Lemma 5.9. For t∈ [0, T ] we have the convergence in distribution
j lσj(t)
−L(t)
→X ∼N
(l1 − l−1)2V (t)
where L(t) is defined in (28) and V (t) in (24).
LARGE PORTFOLIO LOSSES 43
Proof. Define, for x ∈ {−1,1}, the quantity ANx (t) as the number of σi
that, at a given time t, are equal to x. We may then write
AN1 (t)
−1(t)
. Recall moreover that for N →∞, mσN (t) →mσt . We
then have
j lσj(t)
−L(t)
1 (t) + l−1A
−1(t)
−L(t)
N (t)
+ l−1
1−mσN (t)
−L(t)
(l1 + l−1)
(l1 − l−1)
N (t)−
(l1 − l−1)
mσt −
(l1 + l−1)
(l1 − l−1)
N (t)−m
→X ∼N
(l1 − l−1)2V (t)
where the last convergence follows from Corollary 3.7 noticing that m
N (t) =
ρN (t)
Proof of Theorem 4.1. We have to check that
LN (t)
−L(t)
→ Y ∼N(0, V̂ (t)),
where V̂ (t) is defined in (29).
Separating the firms according to whether their σj(t) is +1 or −1,
j Lj(t)
−L(t)
j:σj(t)=1
Lj(t) +
j:σj(t)=−1Lj(t)
−L(t)
We then add and subtract
j lσj(t) to obtain
j:σj(t)=1
(Lj(t)− l1)
j:σj(t)=−1(Lj(t)− l−1)
j lσj(t)
−L(t)
Since we have only independence conditionally on σ(t), we need to check
whether the CLT still applies. Let us show the convergence of the corre-
sponding characteristic functions:
LN (t)−NL(t)√
44 DAI PRA, RUNGGALDIER, SARTORI AND TOLOTTI
j:σj(t)=1
(Lj(t)− l1)√
j:σj(t)=−1(Lj(t)− l−1)√
j lσj(t) −NL(t)√
∣σ(t)
The last of the three terms is measurable with respect to the sigma algebra
generated by σ(t) so that we can take it out from the inner expectation. Be-
cause of the conditional independence we can separate the remaining terms
in the product of conditional expectations:
j:σj(t)=1
(Lj(t)− l1)√
∣σ(t)
j:σj(t)=−1(Lj(t)− l−1)√
∣σ(t)
By conditional independence,
j:σj(t)=1
(Lj(t)− l1)√
∣σ(t)
AN1 (t)
Lj(t)− l1√
∣σ(t)
1− v1
)]AN1 (t)
where the last equality follows because l1 and v1 are the first two conditional
moments of Lj(t).
Recalling that
AN1 (t)
converges almost surely to
we have
1− v1
)]AN1 (t)
= lim
1− v1
AN1 (t)
AN1 (t)
)]AN1 (t)
= exp
1 +mσt
The same argument holds for the terms where σj(t) = −1. Since
−1(t)
1−mσt
, we have
1− v−1
AN−1(t)
AN−1(t)
−1(t)
= exp
1−mσt
LARGE PORTFOLIO LOSSES 45
Finally, recall from Lemma 5.9 that
lσj (t)
−NL(t)
converges to X ∼N(0,
(l1−l−1)2V (t)
), so that
j lσj(t) −NL(t)√
= exp
(l1 − l−1)2V (t)
Thus, denoting by E[· · · |σ(t)] the inner conditional expectation in (48), we
have shown that
E[· · · |σ(t)] = exp
(l1 − l−1)2V (t)
1 +mσt
× exp
1−mσt
= exp
V̂ (t)
By the Dominated Convergence Theorem, taking the limit as N → +∞ in
(48), we can interchange the limit with the outer expectation, and the proof
is completed. �
6. Conclusions and possible extensions. In this paper we have described
propagation of financial distress in a network of firms linked by business
relationships.
We have proposed a model for credit contagion, based on interacting par-
ticle systems, and we have quantified the impact of contagion on the losses
suffered by a financial institution holding a large portfolio with positions
issued by the firms.
Compared to the existing literature on credit contagion, we have proposed
a dynamic model where it is possible to describe the evolution of the indica-
tors of financial distress. In this way we are able to compute the distribution
of the losses in a large portfolio for any time horizon T , via a suitable version
of the central limit theorem.
The peculiarity of our model is the fact that the changes in rating class
(the σ variables) are related to the degree of health of the system (the
global indicator mσ). There is a further characteristic of the firms that is
summarized by a second variable ω (a liquidity indicator) and that describes
the ability of the firm to act as a buffer against adverse news coming from
the market. The evolution of the pair (σ,ω) depends on two parameters β
and γ, which indicate the strength of the interaction.
The fact that our model leads to endogenous financial indicators that de-
scribe the general health of the systems has allowed us to view a credit crisis
as a microeconomic phenomenon. This has also been exemplified through
simulation results.
46 DAI PRA, RUNGGALDIER, SARTORI AND TOLOTTI
The model we have proposed in this paper exhibits some phenomena
having interesting financial interpretation. There are many extensions that
could make the model more flexible and realistic, allowing also calibration to
real data. One of them, concerning the symmetry of the model, has already
been mentioned in Remark 2.1. Other more substantial extensions are the
following:
• In real applications, the variable σ denoting the rating class is not binary;
one could extend the model by taking σ to be valued in a finite, totally
ordered set.
• One could assume the fundamental values ωi to be R+-valued, and evolv-
ing according to the stochastic differential equation
dωi(t) = ωi(t)[f(m
N (t))dt+ g(m
N (t))dBi(t)] + dJi(t),
where f and g are given functions, the Bi(·) are independent Brownian
motions, and Ji(·) is a pure jump process whose intensity is a function of
ωi(t) and m
N (t).
• An interesting extension of the above model consists in letting the func-
tions a(·, ·, ·) and b(·, ·, ·) in (1) be random rather than deterministic;
in particular they may depend on (possibly time-dependent) exogenous
macroeconomic variables.
• The mean-field assumption may be weakened by assuming that the rate
at which ωi changes depends on an i-dependent weighted global health of
the form
N,i :=
where J : [0,1]2 → R is a function describing the interaction between pairs
of firms. In other words, the ith firm “feels” the information given by the
rating of the other firms in a nonuniform way.
Other generalizations could be useful, in particular to introduce inhomogene-
ity in the model. In principle, the extensions listed above could be treated
by the same techniques used in this paper.
APPENDIX: THE EIGENVALUES OF THE MATRIX A IN
THEOREM 3.6
We begin by writing down explicitly the limit matrix A:
− cosh(β)
−γ sinh(γm
cosh(γmσ∗ )
sinh(γmσ∗ ) + γ cosh(γm
sinh(γmσ∗ ) + γm
∗ cosh(γm
∗ ) + γ
sinh(β) +mσ∗ sinh(γm
cosh(β) + cosh(γmσ∗ )
sinh(γmσ∗ )
LARGE PORTFOLIO LOSSES 47
sinh(β) 0
− cosh(γmσ∗ ) 0
0 −(cosh(β) + cosh(γmσ∗ ))
where for the first term in the second row we have used (16). By direct
computation, one shows that the eigenvalues of A are given by the following
expressions:
λ1 = −2(cosh(β) + cosh(γmσ∗ )),
λ2 = −
cosh(β) + cosh(γmσ∗ )
(cosh(β)− cosh(γmσ∗ ))
sinh(β)
cosh(γmσ∗ )
,(49)
λ3 = −
cosh(β) + cosh(γmσ∗ )
(cosh(β)− cosh(γmσ∗ ))
sinh(β)
cosh(γmσ∗ )
Note that these eigenvalues are all real, and that clearly λ1, λ2 < 0. Moreover,
λ3 < 0 if and only if
< cosh2(γmσ∗ )(50)
where γc =
tanh(β)
(a) If γ < γc, then by part (i) in Theorem 3.4 we have m
∗ = 0. In this
case (50) holds, because
< 1 = cosh2(γ · 0).
In this case the matrix A has three different real eigenvalues, all strictly
negative.
(b) If γ = γc, we still have m
∗ = 0, but it is immediately seen that λ3 = 0.
(c) Finally, if γ > γc, set y = γm
∗ ; by (15) we have
mσ∗ =
tanh(γmσ∗ ) ⇔ y =
tanh(y).(51)
Then (50) is equivalent to showing that
< cosh2(y)(52)
48 DAI PRA, RUNGGALDIER, SARTORI AND TOLOTTI
and from (51) we obtain
tanh(y)
sinh(y)
cosh(y)< cosh(y)< cosh2(y)
because y/ sinh(y)< 1 and cosh(y)< cosh2(y), since y = γmσ∗ > 0 if γ > γc.
Then, in this case too, the matrix A has three different real eigenvalues, all
strictly negative.
Acknowledgment. The authors would like to acknowledge the extremely
careful reading of the paper and the useful suggestions made by an anony-
mous referee.
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50 DAI PRA, RUNGGALDIER, SARTORI AND TOLOTTI
P. Dai Pra
W. J. Runggaldier
E. Sartori
Dipartimento di Matematica Pura ed Applicata
University of Padova
63, Via Trieste
I-35121-Padova
Italy
E-mail: daipra@math.unipd.it
runggal@math.unipd.it
esartori@math.unipd.it
M. Tolotti
Istituto di Metodi Quantitativi
Bocconi University
25, Via Sarfatti
I-20136 Milano
Italy
Scuola Normale Superiore
Italy
E-mail: tolotti@unibocconi.it
Department of Applied Mathematics
University of Venice
Venice
Italy
E-mail: tolotti@unive.it
mailto:daipra@math.unipd.it
mailto:runggal@math.unipd.it
mailto:esartori@math.unipd.it
mailto:tolotti@unibocconi.it
mailto:tolotti@unive.it
	Introduction
	General aspects
	Purpose and modeling aspects
	Financial application
	Methodology
	The model
	A mean-field model
	Invariant measures and nonreversibility
	Main results: law of large numbers and Central Limit Theorem
	Deterministic limit: large deviations and law of large numbers
	Equilibria of the limiting dynamics: phase transition
	Analysis of fluctuations: Central Limit Theorem
	Portfolio losses
	Simulation results
	Proofs
	Proofs of Propositions 3.1 and 3.2
	Proof of Theorem 3.4
	Proof of Theorem 3.6
	Proof of Theorem 4.1
	Conclusions and possible extensions
	Appendix: The eigenvalues of the matrix A in Theorem 3.6
	Acknowledgment
	References
	Author's addresses
