Journal of Statistical Physics (2019) 175:495–511https://doi.org/10.1007/s10955-019-02264-w

AMarkov Modulated Dynamic Contagion Processwith Application to Credit Risk

Puneet Pasricha1 · Dharmaraja Selvamuthu1

Received: 14 September 2018 / Accepted: 8 March 2019 / Published online: 13 March 2019© Springer Science+Business Media, LLC, part of Springer Nature 2019

AbstractSelf-exciting point processes are applied to various fields such as seismology, finance, neu-rophysiology, criminology, biology etc to model the clustering/contagion phenomenon andextreme risk events. This article proposes an analytically tractable point process, a gener-alization of the classical Hawkes process, with the intensity process following a Markovmodulated mean-reverting affine jump-diffusion process with contagion effects. The pro-posed process has both self-exciting and externally-exciting jumps that represent the effectsof endogenous and exogenous events. This article derives the closed-form expressions, fordistributional properties such as the first ordermoment, probability generating function (PGF)of the point process and Laplace transform of the intensity process, which makes it computa-tionally efficient. The application of the proposed process to price the synthetic collateralizeddebt obligations (CDOs) in a top-down framework is also presented.

Keywords Contagion process · Affine jump diffusion process · Credit risk ·Regime-switching

1 Introduction

Point processes have been a classical modeling tool in various fields such as seismology,credit risk, insurance, biology etc. In many of the applications, one is most interested tomodel the extreme events such as clustering of defaults in credit risk modeling, accidents innuclear power generation etc. The point processes featuring contagion or positive feedbackphenomenonmay provide an appropriate tool that generates the extreme events. For instance,so called cluster processes and self-exciting point processes have widely been applied tomodel default contagion [8], high frequency financial markets [15], earthquake aftershocks

Communicated by Irene Giardina.

B Dharmaraja Selvamuthudharmar@maths.iitd.ac.in

Puneet Pasrichamaz148097@maths.iitd.ac.in

1 Department of Mathematics, Indian Institute of Technology Delhi, Hauz Khas, New Delhi 110016, India

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[29], insurance claims following a catastrophe [6,7], to describe the after-pulse phenomenonin photomultiplier tubes (PMTs) [30] etc. Further, one important class of stochastic processesrelated to point processes known as branching processes [19] were applied to model thenuclear chain reactions in the project of Manhattan. Such models are also being used in thestudy and modeling of extremes such as the accidents in nuclear power generation, “cyberrisk” events etc.

The extreme events that occur in more complex and non-linear systems are generated bymechanisms such as bifurcations, tipping points, positive feedback and regime changes. Ithas been observed that extreme events often occur around regime changes. Further, in orderto get understanding of these dynamics, a powerful approach is to decompose the fluctuationsand activities into endogeneous and exogeneous parts [35].

Motivated by these model requirements, in this article, a Markov modulated dynamiccontagion process is proposed in order to capture the impact of changes in the environmentalconditions and various exogenous factors on the intensity of the point process. The proposedprocess is a natural extension of the the models by Dong et al. [10] and Dassios and Zhao [9]in a Markov modulated framework motivated by [5,33] and thus gives a minor but significantcontribution to the literature. The proposed point process can be applied to various problemsin earthquake modeling, credit risk, insurance or economics where one needs to model thecontagious arrival of different events such as aftershocks, catastrophes, claim arrivals ordefaults. Further, the proposed process incorporates both endogenous and exogenous factorsaffecting the intensity. Closed form expressions for distributional properties of the proposedprocess such as first order moment, Laplace transform of the intensity process and probabilitygenerating function of the proposed point process have been derived, naturally follows fromDassios and Zhao [9], which makes it computationally efficient.

In order to demonstrate the application of the proposed process, we consider a top-downapproach in order to model the portfolio default risk and perform the pricing of syntheticCDOs (Errais et al. [17], Giesecke et al. [16]). In contrast to the bottom up approach thatmodels the loss process of individual constituents of the portfolio and then couple them, thetop-down approach models the portfolio loss without referring to the individual constituents.The occurrence of defaults in the portfolio are assumed to be driven by the proposed pointprocess and a closed form expression for the expected loss is obtained. Finally, the sensitivityof prices with respect to various parameters of the model is also demonstrated.

The remainder of this paper is organized as follows. Section 2 gives a brief literaturereview in the field of credit risk modeling. Section 3 presents a mathematical definition of theMarkovmodulated dynamic contagion processwith diffusion. Section 4 gives the distributionproperties of the proposed model such as the moments, the Laplace transform of the intensityprocess, and the probability generating function of the point process. Section 5 applies theseresults to the valuation of a portfolio credit derivative in a top-down framework and alsopresents the sensitivity analysis with respect to various parameters of the proposed model.Section 6 concludes the paper with future work.

2 Literature Survey

In credit risk modeling, one is most interested to model the default event since it effectsthe present value of default-able securities. In order to model the default, two categories ofmodels are proposed in the literature namely structural models and intensity based models.Point processes have widely been applied to both forms of the credit risk models (Jarrow et

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al. [21], Zhou [37]) since they are an appropriate modeling tool to address the discontinuitycaused by an unexpected default or a change in the credit quality of the reference entity.

Poisson process is the most commonly used point process in credit risk due to its compu-tational tractability. For instance, Jarrow and Turnbull [22] modeled the default time as thefirst jump time of a Poisson process. However, many empirical studies on default data haveshown that Poisson process is often unrealistic since the hazard function is not necessarilya constant. Hence, many generalizations of the Poisson process are proposed in order toaccount for the non-suitability of the Poisson process to default data. For instance, Duffieand Singleton [13] considered that the hazard rate is not a constant but a function of time,i.e., ht at any time t . In other words, the conditional probability of default in a small timeintervalΔt , given no default till time t is htΔt . Later, Lando [24] proposed a doubly stochas-tic Poisson process (also known as Cox process) assuming that the default intensity is nota deterministic function of time but it is driven by a stochastic process. They assumed thatthe default intensity is affected by a set of variables like market variables or macroeconomicfactors.

However, recent financial crisis which led to the collapse of some major financial insti-tutions such as Lehman Brothers, has emphasized the significance of contagion risk in thefinancial market. A number of empirical studies on default data and bond prices have alsofound evidences of dependence among large jumps in credit spreads of a number of issuers.Hence, it is important to analyze and quantify the contagion feature of event arrivals. Also,Aït-Sahalia et al. [2] argued that “what makes financial crisis take place is typically not theinitial jump, but the amplification that occurs subsequently over hours or days, and the factthat other markets become affected as well”. Although doubly stochastic Poisson processesprovides computationally tractable modeling approach but it fails to incorporate clusteringof jumps due to the independent increments property.

These empirical observations motivated the development of models that are capable ofcapturing the feedback phenomena and contagion phenomenon. In this direction, the pioneerwork is by Jarrow and Yu [23] who observed that in order to explain the clustering ofdefaults, models with default intensity depending linearly only on macroeconomic factors isnot adequate. They introduced a model with contagion effects where default of the referencefirm increases the default intensity of the counterparty. Errais et al. [16] observed using realdata on defaults that the default clustering could be addressed more consistently using self-exciting Hawkes processes originally proposed by Hawkes (1971a,b). They extended theidea by Jarrow and Yu [23] to model the default contagion by including various parametersin order to capture some specific features of arrival of endogenous or exogenous events suchas the magnitude of the impact, frequency and time decay. Dassios and Jang [8] proposeda dynamic contagion process, which is a generalization of the classical Hawkes processand doubly stochastic Poisson process with shot noise intensity. Their model is capable toincorporate jumps due to both self-excitation and external-excitation. Using the martingaleapproach, they obtained distribution properties of the proposed process. Further, Dassios andZhao [9] extended the dynamic contagion process by Dassios and Zhao [8] by includingthe independent diffusion component in the intensity governing the proposed process whichensures that the intensity process is not deterministic between the two jumps in the intensityprocess.

Recently, there is an increasing interest in the application of regime-switching models tostudy various problems in financial mathematics. These models are motivated by the fact thatthe prices of financial securities are affected significantly by the state of the macro-economywhich itself is subjected to various unpredictable regimes. The state of the economy changesto different states due to various financial factors such as management decisions, business

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conditions or someothermacroeconomic factors. Thesemodelswere introduced to thefield offinance by Hamilton [18]. In a regime-switching modeling framework, the market is assumedto be in various states depending on the state of the economy. Regime-switching models arenow been widely applied in modeling of credit risk. For instance, Xu and Wang [36] studiesthe stability of the Markov modulated skew-CIR process. Liang and Wang [25] proposeda common Poisson shock model in a Markov modulated framework to price kth-to-defaultbasket default swaps. Yang and Siu [32] studied the problem of pricing bond options underHull-White model in a regime-switching framework. Bo et al. [4] considered the problem offinding the default probability under the assumption that the asset dynamics are driven bya regime-switching reflected stochastic processes. Recently, Wang et al. [34] considered aMarkov modulated Hawkes process with step-wise decay with an application to earthquakemodeling. Cohen and Elliot [5] obtained optimal filter and smoother for hidden chain basedon the jumps of aMarkovmodulated self exciting counting process. Dong et al. [10] proposedaMarkovian regime-switching shot noise process and studied its applications in bond pricingand insurance.

The self-exciting point processes are now widely applied in various fields (in seismology(Adamopoulos [1]; Ogata [28,29]), in credit risk (Errais et al. [16], Dassios and Zhao [8])).For instance, Meyer et al. [27] introduced a self-exciting spatio-temporal point process topredict the incidence of invasive meningococcal disease which can be transmitted betweeninfected humans and sometimes forms epidemics. For a review on self-exciting point processand applications in different field, one can refer Reinhart [31], Hawkes [20].

3 The ProposedModel

In this section, firstly we present a regime-switching modeling framework followed by amathematical definition of the proposed Markov modulated dynamic contagion process withdiffusion. The proposed process is considered in regime-switching framework where dynam-ics of the regimes ismodeled by aMarkov chain. Therefore, inwhat follows, thewordMarkovmodulated or regime-switchingwill be used interchangeably. In the next subsection,wederiveits key distributional properties, such as the moments, the Laplace transform of the intensityprocess, and the PGF of the proposed point process.

3.1 TheModel Description

In this subsection, we discuss the Markov modulated framework followed by the proposedmodel.

3.1.1 Regime Switching Framework

Consider a finite time horizon T > 0 and a continuous-time modeling framework. Assumethat the filtered probability space (Ω,F, P,Ft∈[0,T ]) models the uncertainty in the econ-omy. The filtration F := {Ft , t ∈ [0, T ]} is assumed to satisfy the usual conditions (i.e.,complete and right continuous) and P is a risk-neutral probability measure. Let E denotesthe expectation with respect to the risk neutral measure P .

Consider a time-homogeneous continuous time, irreducible and observable Markov chainX := {X(t), t ∈ [0, T ]} on the probability space (Ω,F, P,Ft ) that describes the dynam-ics of economy over time. Assume that the Markov chain has a finite state space and its

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A Markov Modulated Dynamic Contagion Process... 499

generator matrix is given by Q = (qi j ), i, j = 1, 2, . . . , N . Without loss of generality,assume that the state space of X is the set of unit vectors E = {e1, e2, . . . , eN }, ei =(0, . . . , 0, 1, 0, . . . , 0)∗ ∈ R

N , where ∗ denotes the transpose of a vector or a matrix. Thestates of the chain X represent various states of an economy such as recession or economicexpansion etc. Let FX := {F X

t , t ∈ [0, T ]} denotes the natural filtration generated by thechain X. We have the following semi-martingale representation for the Markov chain asproposed by Elliott et al. (2008) [14]

X(t) = X(0) +∫ t

0Q∗X(s)ds + L(t), (1)

where {L(t), t ∈ [0, T ]} is anRN valued, (FX , P)-martingale. In the rest of the article, 〈·, ·〉denotes the Euclidean inner product in RN , i.e., for any x, y ∈ R

N , 〈x, y〉 = ∑Ni=1 xi yi .

3.1.2 Markov Modulated Dynamic Contagion Process with Diffusion

Definition 1 A Markov modulated dynamic contagion process with diffusion (MMDCPD)is a point process N (t) = {Tk}k=1,2,... with non-negative stochastic intensity (conditional){λ(t), t ∈ [0, T ]} given by

dλ(t) = δ(η(t) − λ(t))dt + σ(t)√

λ(s)dW (t) + d

⎛⎝N (t)∑

j=1

Z j

⎞⎠ + d

⎛⎝M(t)∑

j=1

Y j

⎞⎠ , (2)

where

– λ(0) denotes the initial intensity at time t = 0.– η(t) represents the mean-reverting level of the intensity process. We assume that η(t) is

modulated by the Markov chain X as follows:

η(t) := η(t, X(t)) := 〈η, X(t)〉with η := (η1, η2, . . . , ηN ) and ηi > 0 for all i = 1, 2, . . . , N .

– δ denotes themean-reverting rate of the intensity process andwe assume it to be a positiveconstant.

– W := {W (t), t ≥ 0} is a standard Brownian motion.– σ(t) represents the volatility of the diffusion part in the dynamics of the intensity process

{λ(t), t ≥ 0}. We assume that σ(t) is governed by the Markov chain X in the followingway:

σ(t) := σ(t, X(t)) := 〈σ, X(t)〉,where σ := (σ1, σ2, . . . , σN ) and σi > 0 for all i = 1, 2, . . . , N .

– {Zk}k=1,2,... is a sequence of independent and identically distributed (i.i.d.) positive ran-domvariableswith distribution functionG(z), z > 0 (satisfying the suitable integrabilityconditions) at the corresponding jump times {Tk}k=1,2,.... Here, {Zk}k=1,2,... representsthe magnitude of jumps in the intensity process {λ(t), t ≥ 0} due to self-excitation.

– The process {M(t), t ≥ 0} with arrival times {T j } j=1,2,... is a regime-switching Poissonprocess that reflects the external jumps effecting the intensity process. For instance, inthe context of credit risk, they represent the jumps emerging out of the macroeconomicevents affecting all the firms, such as financial crisis or economic downturn. We assumethat the intensity ρ(t) of the process {M(t), t ≥ 0} is driven by the Markov chain X asfollows:

ρ(t) := ρ(t, X(t)) := 〈ρ, X(t)〉

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where ρ := (ρ1, ρ2, . . . , ρN ) and ρi > 0 for all i = 1, 2, . . . , N .– {Yk}k=1,2,... is a sequence of i.i.d. positive random variables with distribution func-

tions Ht (y), y > 0 at the corresponding jump times {Tk}k=1,2,.... Here, {Yk}k=1,2,...

reflects the magnitudes of the jumps in the intensity process {λ(t), t ≥ 0} due tothe external excitation. We assume these distribution functions to be dependent on theMarkov chain X in a way that given X(t), the conditional distribution of the magni-tude of jump occurring at time t is Ht (·) = 〈H(·), X(t)〉 supported on (0,∞), whereH(·) = (H1(·), . . . , HN (·))∗ ∈ R

N . Further, we assume that the distribution functionssatisfy the suitable integrability conditions.

– We assume that {Z j } j=1,2,..., {Y j } j=1,2,..., {Tj } j=1,2,... and {W (t), t ≥ 0} are indepen-dent of each other and all are independent of {X(t), t ≥ 0}. Furthermore, we assumethat the jump processes {M(t), t ≥ 0}, {N (t), t ≥ 0} and {X(t), t ≥ 0} do not havecommon jumps.

We assume that the parameters satisfy the Feller condition σ 2i < 2δηi [3]. One can observe

that the intensity process {λ(t), t ≥ 0} decays exponentially after excitation due to self-excited or externally excited jumps. Further, the diffusion term included in the intensityprocess {λ(t), t ≥ 0} makes the intensity, a stochastic process in between two jumps of thepoint process {N (t), t ≥ 0}. We now introduce the information structure for the proposedmodel. Let FW := {FW (t), t ≥ 0}, be the natural filtration generated by the the process{W (t), t ≥ 0}. Similarly, letFN := {FN

t , t ≥ 0} be the P-complete right continuous naturalfiltration generated by the point process {N (t), t ≥ 0} with respect to which the intensityprocess {λ(t), t ≥ 0} is adapted. Since the Markov chain {X(t), t ≥ 0} is observable, theinformation available is given by the enlarged filtration G := {Gt , , t ≥ 0}, where, for eacht ∈ [0, T ], Gt := FN

t∨FW

t∨F X

t , the minimal σ -field.

3.1.3 Infinitesimal Generator

Because of the assumption of exponential decays, the process (λ(t), N (t), X(t)) is a Markovprocess. The intensity process λ(t) decreases with the rate δ(λ(t) − a(t)) and experiencesan additive positive jump on the arrival of an external event with rate ρ(t) and jump sizefollowing the distribution Ht (y). Similarly, λ(t) experiences an additive positive jump of sizethat have distribution function G(z) due to self-excitation with rate λ(t).

Write

f(λ, n, t) = ( f (λ, n, e1, t), f (λ, n, e2, t), . . . , f (λ, n, eN , t))∗.

Hence,

f (λ, n, x, t) = 〈f(λ, n, t), x〉.The proposed process belongs to the class of affine processes [11,12]. The infinitesimal gen-erator A of the process (λ(t), N (t), X(t), t), which acts on a suitable function f (λ, n, x, t)within its domain Ω(A), is given by

A f (λ, n, x, t) = ∂ f

∂t− δ(λ − η(t))

∂ f

∂λ+ 1

2σ 2(t)λ

∂2 f

∂λ2+ 〈f(λ, n, t), Q∗x〉

+∫ ∞

0ρ(t)( f (λ + y, n, x, t) − f (λ, n, x, t))dHt (y)

+∫ ∞

0λ( f (λ + z, n + 1, x, t) − f (λ, n, x, t))dG(y) (3)

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where the domain of f is such that f (λ, n, x, t) is differentiable with respect to each ofthe argument λ, t and x , for all possible values of λ, n, x and t , and satisfies the followingconditions ∣∣∣∣

∫ ∞

0( f (λ + y, n, x, t) − f (λ, n, x, t)) dHt (y)

∣∣∣∣ < ∞,

∣∣∣∣∫ ∞

0( f (λ + z, n + 1, x, t) − f (λ, n, x, t)) dG(z)

∣∣∣∣ < ∞.

The proposed model is an amalgamation of the models by Dong et al. [10] and Dassiosand Zhao [9] in a way that the intensity process is incorporated with self-exciting jumps anda diffusion component in the regime-switching framework.

4 Distribution Properties

In this section, we derive the distribution properties of the proposed process. Following thenotations in Dassios and Zhao [9], denote the first order moment and Laplace transforms ofthe jump sizes given by Y j and Z j in λ(t) of Equation (2) respectively by

μ1G =∫ ∞

0zdG(z), g(u) =

∫ ∞

0e−uzdG(z)

μ1H t =∫ ∞

0ydHt (y), ht (u) =

∫ ∞

0e−uzdHt (z).

Define κ = δ − μ1G . Here, we assume that the first order moment and Laplace transformsdefined above exists and are finite.

4.1 Joint Laplace transform–PGF of (�(T),N(T))

In the following theorem, the conditional joint Laplace transform-PGF of the processes{λ(t), t ≥ 0} and {N (t), t ≥ 0} is derived for a fixed time T . Further, the Laplace transformof λ(T ) and the PGF of N (T ) are derived as a corollary to this theorem in further subsections.

Theorem 1 For time t ∈ [0, T ] and the constants v ≥ 0, 0 ≤ θ ≤ 1, the conditional jointLaplace transform-PGF of (λ(T ), N (T )) is given by

E(θ(N (T )−N (t))e−vλ(T ) | Ft ) = e−B(t,X(t))λ(t)〈Ψ (t)1, X(t)〉, (4)

where B(t, X(t)) = 〈B(t), X(t)〉 and B(t) is a vector (B1(t), B2(t), . . . , BN (t)) whereBi (t), i = 1, 2, . . . , N is governed by the following non-linear ordinary differential equation(ODE)

B ′i (t) = δBi (t) + 1

2σ 2i B

2i (t) + θ g(Bi (t)) − 1,

with boundary condition Bi (T ) = v. Ψ (t) is a fundamental matrix corresponding to thefollowing matrix-valued system of linear ODEs:

dΨ (t)

dt+ Δ(t)Ψ (t) = 0, (5)

with boundary condition Ψ (T ) = 1. Here, Δ(t) = Q + diag(S(t)) and 1 = (1, 1, . . . , 1)∗.The quantity diag(S(t)) represents a N × N diagonal matrix with i th diagonal entry givenby

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502 P. Pasricha, D. Selvamuthu

Si (t) = δηi B(t) + ρi (hi (B(t)) − 1), i = 1, 2, . . . , N . (6)

Proof Consider the following function f (λ, n, x, t) with an exponential affine form

f (λ, n, x, t) = An(t)e−B(t,x)λD(t, x), (7)

with boundary conditions B(T , x) = 〈1, v〉 and D(T , x) = 〈1, x〉. Write

B(t) = (B(t, e1), B(t, e2), . . . , B(t, eN ))∗,D(t) = (D(t, e1), D(t, e2), . . . , D(t, eN ))∗.

Then, B(t, x) = 〈B(t), x〉, D(t, x) = 〈D(t), x〉. Let Bi (t) = B(t, ei ), i = 1, 2, . . . , N .Substituting f (λ, n, x, t) from Equation (7) to Equation (3), we have(

− B ′(t, x)λ + nA′(t)A(t)

+ D′(t, x)D(t, x)

)f + B(t, x)δ(λ − η(t)) f + 1

2σ 2(t)B2(t, x)λ f

+ f

D(t, x)〈QD(t), x〉 + f

∫ ∞

0ρ(t)(eB(t,x)y − 1)dHt (y)

+ λ f∫ ∞

0(A(t)eB(t,x)z − 1)dG(z) = 0,

0 = d〈D(t), x〉dt

+ 〈D(t), x〉[n

(A′(t)A(t)

)+ δ〈η, x〉B(t, x) +

∫ ∞

0

(eB(t,x)y − 1

)〈ρ, x〉d〈H(y), x〉

+ λ

(−B ′(t, x) + δB(t, x) + 1

2〈σ, x〉2B2(t, x) + A(t)g(B(t, x)) − 1

) ]〈QD(t), x〉.

Since the Equation (8) is true for all λ, n and x = ei , therefore, the coefficients of λ, n andx must vanish. Hence, it holds that

A′(t)A(t)

= 0, (8)

B ′i (t) − δBi (t) − 1

2σ 2i B

2i (t) − A(t)g(Bi (t)) + 1 = 0, Bi (T ) = v, i = 1, 2, . . . , N

(9)

0 = d〈D(t), x〉dt

+ 〈D(t), x〉[δ〈η, x〉〈B(t), x〉

+∫ ∞

0

(e〈B(t),x〉y − 1

)〈ρ, x〉d〈H(y), x〉] + 〈QD(t), x〉. (10)

Equation (10) can be written as

dD(t)

dt+ Δ(t)D(t) = 0, D(T ) = 1,

whereΔ(t) = Q+diag(S(t)) and diag(S(t)) is an N ×N diagonal matrix with i th diagonalentry given by

Si (t) = δηi Bi (t) + ρi

∫ ∞

0

(eBi (t)y − 1

)dHi (y).

It is easy to see that A(t) = θ . In order to solve the Equation (10), let Ψ (t) denotes thefundamental matrix to the following matrix-valued linear ODEs

dΨ (t)

dt+ Δ(t)Ψ (t) = 0, Ψ (T ) = 1. (11)

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A Markov Modulated Dynamic Contagion Process... 503

There exists a unique solution to the system of ODEs (11) on [0, T ], sinceΔ(t) is continuousover [0, T ]. Hence, we have

D(t) = Ψ (t)1.

Therefore, D(t, x) = 〈Ψ (t)1, x〉. Since the unknowns in f (λ, n, x, t) are obtained fromA f = 0, by property of the infinitesimal generator, the process {θN (t)e−B(t,x)λ(t)D(t, X(t)),0 ≤ t ≤ T } is a martingale. In other words, we have

E(θ(N (T )−N (t))e−vλ(T ) | Ft ) = e−B(t,X(t))λ(t)〈Ψ (t)1, X(t)〉,and hence the result (4) follows. �

4.2 Laplace Transform of �(T)

In this subsection, the conditional Laplace transform of λ(T ) given λ(0) is derived followingthe martingale approach as a corollary to Theorem 1.

Theorem 2 Assume that δ > μ1G. The Laplace transform λ(T ) conditional on λ(0) at timet = 0 with X(0) = ei , is given by

E(e−vλ(T ) | F0) = e−G−1

v,1,ei(T )λ(0)〈Ψ (0)1, ei 〉, (12)

where

Gv,1,ei (L) =∫ v

L

du

δu + 12σ

2i u

2 + g(u) − 1,

and Ψ (0) is a fundamental matrix solution to the linear, matrix-valued ODE given in (5).

Proof Setting θ = 1 and t = 0 in Equation (4), we have

E(e−vλ(T ) | F0) = e−Bi (0)λ(0)〈Ψ (0)1, X(0)〉,where Bi (0) is obtained by solving the following non-linear ordinary differential equations

− B ′i (t) + δBi (t) + 1

2σ 2i B

2i (t) + θ g(Bi (t)) − 1 = 0, (13)

with boundary conditions Bi (T ) = v. Ψ (0) is a fundamental matrix to the matrix valuedsystem of linear ODEs given in (5). The solution to Equation (13) is given in Theorem 3.1in Dassios and Zhao (2017) [9] and Bi (0) can be obtained as

Bi (0) = G−1v,1,ei

(T ), Gv,1,ei (L) =∫ v

L

du

δu + 12σ

2i u

2 + g(u) − 1.

�

4.3 Probability Generating Function of N(T)

In this subsection, the conditional PGF of N (T ) is derived following themartingale approach.

Theorem 3 Assuming that δ > μ1G, the probability generating function of N (T ) conditionalon λ(0), N (0) = 0 and X(0) = ei at time t = 0, is given by

E(θN (T ) | F0) = e−G−1

0,θ,ei(T )λ(0)〈Ψ (0)1, ei 〉, (14)

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504 P. Pasricha, D. Selvamuthu

where

G0,θ,ei (L) =∫ L

0

du

1 − δu − g(u) − 12σ

2i u

2, 0 ≤ θ < 1,

and Ψ (0) is a fundamental matrix solution to the linear, matrix-valued ODE given in (5).

Proof Setting v = 0 and t = 0 in Equation (4), we have

E(θN (T ) | F0) = e−Bi (0)λ(0)〈Ψ (0)1, X(0)〉, (15)

where Bi (0) is obtained by solving the following non-linear ordinary differential equations

− Bi (t) + δBi (t) + 1

2σ 2i B

2i (t) + θ g(Bi (t)) − 1 = 0, (16)

with boundary conditions Bi (T ) = 0. The solution to Equation (16) is given in Theorem 3.4in Dassios and Zhao (2017) [9] and Bi (0) is given by

Bi (0) = G−10,θ,ei

(T ), G0,θ,ei (L) =∫ L

0

du

1 − δu − g(u) − 12σ

2i u

2, 0 ≤ θ < 1.

�

4.4 First Order Moments of �(T) and N(T)

In this subsection, the expectation of λ(T ) and N (T ) are derived using the martingaleapproach.

Theorem 4 (Expectation of λ(T )) The expectation of λ(T ) conditional on information attime t is given by

E(λ(T ) | Ft ) ={e−(δ−μ1G )(T−t)λ(t) + D(t, X(t)) if κ �= 0

λ(t) + D(t, X(t)) if κ = 0, (17)

where D(t, X(t)) is obtained from the following matrix-valued ODE

d〈D(t), x〉dt

+ 〈QD(t), x〉 = S(t), D(T ) = diag(0),

and S(t) is a diagonal matrix with i th entry given by

Si (t) ={

(−δηi (t) − ρiμ1H i )e−(δ−μ1G )(T−t) if κ �= 0

−δηi (t) − ρiμ1H i if κ = 0.

Proof Consider the function f (λ, n, x, t) = A(t)λ+D(t, x) and substitute into the Equation(3), we have

λA′(t) + D′(t, x) + A(t)(δ(η(t) − λ))

+∫ ∞

0ρ(t)A(t)ydHt (y) + λ

∫ ∞

0zA(t)dG(z) + 〈D(t, x), Q∗x〉 = 0,

λ(A′(t) − δA(t) + A(t)μ1G

) + D′(t, x) + δA(t)η(t)

+∫ ∞

0ρ(t)A(t)ydHt (y) + λ

∫ ∞

0zA(t)dG(z) + 〈D(t, x), Q∗x〉 = 0. (18)

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A Markov Modulated Dynamic Contagion Process... 505

Since Equation (18) is true for all values of λ and x = ei , the following must hold

A′(t) − δA(t) + A(t)μ1G = 0, (19)d〈D(t), x〉

dt+ 〈QD(t), x〉 = S(t), D(T ) = diag(0), (20)

with A(T ) = 1 and D(T , x) = diag(0), where S(t) is a diagonal matrix with i th entry givenby

Si (t) = −δA(t)ηi (t) − ρi A(t)μ1H i .

Now, we obtain the expectation of λ(T ) for different values of κ .

Case 1When κ = δ − μ1G = 0, the solution of Equation (19) is given by A(t) = 1. Hence,we have the following

E(λ(T ) | Ft ) = λ(t) + D(t, X(t)),

where D(t, X(t)) can be obtained solving the system of equations (20) with the boundarycondition D(T , X(T )) = diag(0).

Case 2 When κ = δ − μ1G �= 0, the solution of Equation (19) is given by A(t) =e−(δ−μ1G )(T−t). Then, we have the following

E(λ(T ) | Ft ) = e−(δ−μ1G )(T−t)λ(t) + D(t, X(t)),

where D(t, X(t)) can be obtained solving the system of equations (20) with the boundarycondition D(T , X(T )) = diag(0). Thus, we obtain E(λ(T ) | Ft ) as given in Equation (17).

�Corollary 1 When there is only one regime, i.e., N = 1, the Equation (20) reduces to

dD(t)

dt= −δηA(t) − ρμ1H A(t), D(T ) = 0,

which can be solved to obtain

D(t) = −ρμ1H + δη

δ − μ1G

(e−(δ−μ1G )(T−t) − 1

).

Therefore, from Equation (17), we have

E(λ(T ) | Ft ) ={e−(δ−μ1G )(T−t)

(λ(t) − ρμ1H +δη

δ−μ1G

)+ ρμ1H +δη

δ−μ1Gif κ �= 0

λ(t) + (δη + ρμ1G )(T − t) if κ = 0.

which is same as that obtained in Dassios and Zhao [8].

Theorem 5 (Expectation of N (T )) The expectation of N (T ) conditional on λ(t), X(t) andN (t) is given by

E(N (T ) | Ft ) = N (t) + A(t)λt + D(t, X(t)), (21)

where A(t) is given by

A(t) ={ 1

δ−μ1G

(1 − e−(δ−μ1G )(T−t)

) + e−(δ−μ1G )(T−t) if κ �= 01 + T − t if κ = 0

,

and D(t, X(t)) is given by Equation (23).

123

506 P. Pasricha, D. Selvamuthu

Proof Consider the function f (λ, n, x, t) = n + A(t)λ + D(t, x) and substitute into theEquation (3), we have

A′(λ) + D′(t, x) − δ(λ − η(t))A(t) + 〈D(t), Q∗x〉 +∫ ∞

0ρ(t)A(t)ydHt (y)

+λ

∫ ∞

0(A(t)z + 1)dG(z) = 0.

Since the above equation is true for all values of λ, n and x = ei , therefore coefficients of λ

and x must vanish. Hence, it holds that

A′(t) − δA(t) + A(t)μ1G + 1 = 0 with A(T ) = 1, (22)dD(t)

dt+ 〈QD(t), x〉 = diag(S(t)) D(T ) = diag(0), (23)

where

Si (t) = (−δηi − ρiμ1H i )A(t).

Now, we obtain the expectation of N (T ) for different values of κ .

Case 1 When κ = δ − μ1G = 0, the solution of Equation (22) is given by

A(t) = 1 + T − t .

Case 2 When κ = δ − μ1G �= 0, the solution of Equation (22) is given by

A(t) = 1

δ − μ1G

(1 − e−(δ−μ1G )(T−t)

)+ e−(δ−μ1G )(T−t).

Hence, we can obtain the expectation of N (T ) conditional on λ(t), X(t) and N (t) as givenby Equation (21). �

5 Applications

The proposed model have potential to be used in numerous fields where we need to modelcontagion/clustering risk. Inmanynatural andphysical systems, the risk of extremes is amajorrisk and it is known that the extreme events are generated bymechanisms such as bifurcations,tipping points, positive feedback and regime changes. The proposed model is capable ofaddressing the factors such as positive feedback and regime changes and hence is a goodcandidate for modeling such scenarios. Although, the proposed model can be used in variousdirections such as modeling high frequency financial markets [15], earthquake aftershocks[29], insurance claims following a catastrophe [6,7], to describe the after-pulse phenomenonin photomultiplier tubes (PMTs) [30] etc. In this section, we present an application in thefield of credit risk modeling to obtain CDOs fair premium.

5.1 Valuation of CDOs: A Top-Down Approach

Portfolio credit derivatives such as CDOs play an important role as they allow investorsto buy or sell insurance on the underlying portfolio partially or in full. CDOs are bilateralcontracts in which one party called protection seller provides default protection against thedefaults in the reference portfolio. On the other hand, the other party called protection buyer

123

A Markov Modulated Dynamic Contagion Process... 507

pays a premium to the protection seller in return to the protection provided. The payoffof a CDO contract depends on the cumulative loss due to observed number of defaults inthe underlying portfolio of reference entities. There are two approaches namely bottom-up approach and top-down approach to obtain the premium. In bottom-up approach, thedefault process of each portfolio constituent is modeled individually and then portfolio lossis obtained considering the relationship among the portfolio constituents. On the other hand,in a top-down approach, portfolio loss is modeled in whole without specific reference tothe individual constituents. Modeling the loss of portfolio without referring to individualconstituents hasmany advantages as it resolvemodeling and estimation issue and the obtainedpricing formula is mathematically tractable.

In this section, we apply the results developed in Sect. 4 to study the valuation of CDOsin a top-down framework. The cumulative loss of the credit portfolio L(t) is modeled by theproposedMarkovmodulated contagion process N (t)with intensity process given in Equation(2), in line with Longstaff and Rajan [26]. The occurrence of a jump (i.e., default) inducesinstantaneous loss in the portfolio which is assumed to be a constant denoted by . Hence,the proposed framework can be summarized as

dL(t)

1 − L(t)= dN (t) (24)

which can be solved applying Ito’s Lemma. The solution of Equation (24) is given by

L(t) = 1 − exp

⎛⎝−

N (t)∑i=1

⎞⎠ = 1 − exp (− N (t)) . (25)

A protection seller agrees to pay to the protection buyer against the losses in the portfoliogiven a default, L(ti ) − L(ti−1) in return to the premium paid periodically by the protectionbuyer proportional to the current outstanding 1 − L(ti ) until the whole notional value ofthe portfolio is lost. The present value of the payments by the protection seller known asprotection leg is given by

E

{∫ T

0e− ∫ t

0 rududLt

}=

n∑i=1

e−r ti E(L(ti ) − L(ti−1)). (26)

Similarly, the expected value of premium paid by the protection buyer known as premiumleg is given by

n∑i=1

E

{exp

(−

∫ ti

0rudu

)(ti − ti−1)c(T )(1 − L(ti ))

}

= c(T )

n∑i=1

(ti − ti−1)e−r ti E(1 − L(ti )), (27)

where r is the short-rate and is assumed to be a constant. The time points t1 < t2 < . . . <

tn = T are the coupon payment dates and c is the T -year index coupon rate. The coupon ratec(T ) can be obtained by equating the the value of protection leg and premium leg, i.e.,

c(T ) =∑n

i=1 e−r ti E(L(ti ) − L(ti−1))∑n

i=1(ti − ti−1)e−r ti E(1 − L(ti )). (28)

In order to calculate the coupon rate, we need to find the expected loss E[L(t)] which canbe obtained from Theorem 3 by replacing θ by e− .

123

508 P. Pasricha, D. Selvamuthu

Table 1 Values of the parametersin the base case

Parameters Values Parameters Values

r 0.03 0.5

λ0 1 δ 2

βz 1.5 N 2

βiy 1.5 βy 1.5

a [0.5 ;0.9] σ [0.02; 0.1]

ρ [0.3;0.9] βy [1.25;1.75]

Q

( −0.5 0.51 −1

)X(0) e1

Time (in Years)

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pro

babi

lity

of D

efau

lt

N=2N=1

1 1.5 2 2.5 3 3.5 4 4.5 5 1 1.5 2 2.5 3 3.5 4 4.5 5

Time (in Years)

0

20

40

60

80

100

120

140

160

CD

O P

rem

ium

N=2N=1

Fig. 1 Comparison of probability of default and CDO premium obtained from the contagion process in regimeswitching framework and no regime framework

5.2 Numerical Illustrations

In order to develop some intuition for various parameters governing regime-switching behav-ior in the proposed model, we provide sensitivity analysis of the probability of default andthe spread of CDOs. Also, a comparative analysis between the proposed model and themodel without regime-switching is given to illustrate the impact of considering differentregimes. For performing the sensitivity analysis and comparison, value of only one parame-ter is changed at one time while the value of the other parameters are kept same as that in thebase case. For the numerical purpose, the jump sizes of self exciting component and externalexcitement components are assumed to follow exponential distribution with mean βy andβz respectively. The parameters values considered in the base case are listed in Table 1. Thepremium payment dates are considered to be per quarter and hence the units of the parame-ters are considered per quarter. In this chapter, since our focus is on proposing a theoreticalmodel of pricing, we just make sensitivity analysis by assuming some parameters withoutperforming any calibration.

Figure 1 gives the comparison of the probability of default and CDO premium for differentmaturity times for the regime-switching model and model with no regimes. From the leftfigure in Fig. 1, we observe that probability of default is an increasing function of time T forboth the models as expected. Right hand figure in Fig. 1 shows that CDO premium is also anincreasing function of the probability of default since more chances of default events, more

123

A Markov Modulated Dynamic Contagion Process... 509

Time (in Years)

-50

0

50

100

150

200

250

300C

DO

Pre

miu

m

λ0=0.5

λ0=1

λ0=1.5

λ0=2

1 1.5 2 2.5 3 3.5 4 4.5 5 1 1.5 2 2.5 3 3.5 4 4.5 5

Time (in Years)

-20

0

20

40

60

80

100

120

140

160

CD

O P

rem

ium

δ=0.5δ=1δ=1.5δ=2

Fig. 2 Variation of CDO premium against the parameters governing the clustering of jumps, i.e., λ0 and δ inregime switching framework and no regime framework

Fig. 3 CDO Premium againsttime T for different values of theloss on each default

1 1.5 2 2.5 3 3.5 4 4.5

Time (in Years)

0

50

100

150

200

250

300

350

CD

O P

rem

ium

loss=0.2loss=0.4loss=0.6loss=0.8

risk protection seller is taking and hence higher is the premium. Further, we can observe thatthere is a significant difference between the default probability for the models with regimesand no regime thus explaining the significance of considering regime-switching framework.Possible reason for this behavior is the dependence of various parameters governing defaulton the state of the economy. The proposed process is capable of capturing the effect ofeconomic conditions on the probability of default and hence address more aspects of the risk.Thus, it has potential to model the risks during the recession period which involve clustersof bad economic events.

Figure 2 present the variation of CDO premium against the time for different values ofinitial default intensity λ0 and mean-reverting rate δ. We observe that premium increaseswith the increase in the initial intensity λ0 but decreases with increase in value of δ. Sincejump clustering increases (decreases) with increase in the value of λ0 (δ), Figure 2 concludesthat jump clustering increases the premium. Figure 3 gives the variation of premium againstthe loss that occur on each default . More the value of , higher the loss on each default andhence higher the premium.

123

510 P. Pasricha, D. Selvamuthu

6 Conclusion

This article proposes a dynamic contagion point process whose intensity process is modeledby a Markov modulated jump-diffusion process with self exciting component. Some impor-tant distributional properties such as Laplace transform, probability generating function ofthe proposed model are discussed following the martingale method. One possible implemen-tation of the proposed model to the pricing of CDOs in a top-down framework is discussed.In the context of credit risk, the proposed model addresses the clustering of arrival events thatdepends on the state of the economy and hence is capable to capture the clustering of badevents during economic downturn. The potential future work could be to consider the processin multidimensional setup to account for the mutually exciting jumps and their applicationsin managing portfolio credit risk. Further, we wish to develop the calibration methods for theproposed model.

Acknowledgements Authors are thankful to the editor and two anonymous reviewers for their valuable sug-gestions and comments which helped improve the paper to great extent. The Puneet Pasricha is thankful toCouncil of Scientific and Industrial Research (CSIR) India for the financial grant.

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