Invariance properties in the dynamic gaussian copula model *

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Invariance properties in the dynamic gaussian copula model *

Stéphane Crépey, Shiqi Song

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Stéphane Crépey, Shiqi Song. Invariance properties in the dynamic gaussian copula model *. 2017.

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INVARIANCE PROPERTIES IN THE DYNAMIC GAUSSIAN COPULA MODEL^∗

Stéphane Crépey

and Shiqi Song

Abstract. We prove that the default times (or any of their minima) in the dynamic Gaussian copula model of Crépey, Jeanblanc, and Wu (2013) are invariance times in the sense of Crépey and Song (2017), with related invariance probability measures different from the pricing measure. This reflects a departure from the immersion property, whereby the default intensities of the surviving names and therefore the value of credit protection spike at default times. These properties are in line with the wrong-way risk feature of counterparty risk embedded in credit derivatives, i.e. the adverse dependence between the default risk of a counterparty and an underlying credit derivative exposure.

Keywords:counterparty credit risk, wrong-way risk, Gaussian copula, dynamic copula, immersion property, invariance time, CDS.

Mathematics Subject Classification: 91G40, 60G07.

1. Introduction

This paper deals with the mathematics of the dynamic Gaussian copula (DGC) model of Crépey, Jeanblanc, and Wu (2013) (see also Crépey, Bielecki, and Brigo (2014, Chapter 7) and Crépey and Nguyen (2016)). As developed in Crépey et al. (2014, Section 7.3.3), this model yields a dynamic meaning to the ad hoc bump sensitivities that were used by traders for hedging CDO tranches by CDS contracts before the subprime crisis. From a more topical perspective, it can be used for counterparty risk computations on CDS portfolios. Related models include the one-period Merton model of Fermanian and Vigneron (2015, Section 6) or other variants commonly used in credit and counterparty risk softwares.

The dynamic Gaussian copula model has been assessed from an engineering perspective in previous work, but a detailed mathematical study, including explicit computation of the main model primitives, has been deferred to the present paper.

1.1. Invariance Times and Probability Measures

We work on a filtered probability space(Ω,G,A,Q). Given aGstopping timeτand a subfiltrationFofG,FandG satisfying the usual conditions, letJandSdenote the survival indicator process ofτand its optional projection known as the Azéma supermartingale ofτ,i.e.

Jt=1{τ >t}, St=^oJt=Q(τ > t| Ft), t≥0.

The following conditions are studied in Crépey and Song (2015, 2017).

∗This research benefited from the support of the “Chair Markets in Transition”, Fédération Bancaire Française, and of the ANR project 11-LABX- 0019.

1Université d’Évry Val d’Essonne, Laboratoire de Mathématiques et Modélisation d’Évry and UMR CNRS 8071, 91037 Évry Cedex, France.

2Université d’Évry Val d’Essonne, Laboratoire de Mathématiques et Modélisation d’Évry and UMR CNRS 8071, 91037 Évry Cedex, France.

© EDP Sciences, SMAI 2017

Condition (B). AnyGpredictable processUadmits anFpredictable reduction, i.e. anFpredictable process, denoted by U⁰,that coincides withUonK0, τK.

For any left-limited processY,we denote byY^τ−=JY + (1−J)Yτ−the processY stopped beforeτ.

Condition (A). Given a constant time horizonT >0, there exists a probability measurePequivalent toQonFT such that(F,P)local martingales stopped beforeτare(G,Q)local martingales on[0, T].

If the conditions (B) and (A) are satisfied, then we say thatτis an invariance time andPis an invariance probability measure. If, in addition, ST > 0almost surely, then Fpredictable reductions are uniquely defined on(0, T]and any inequality between twoGpredictable processes on(0, τ]implies the same inequality between theirFpredictable reductions on(0, T](see Song (2014a, Lemma 6.1)); invariance probability measures are uniquely defined onFT,so that one can talk of the invariance probability measureP(as the specification of an invariance probability measure outsideFTis immaterial anyway).

2. Dynamic Gaussian Copula Model

In the paper we prove that, given a constant time horizonT >0,the default timesτi(or any of their minima) in the DGC model are invariance times, with related invariance probability measuresPuniquely defined and not equal toQon FT.This reflects a departure from the immersion property, whereby the default intensities of the surviving names and therefore the value of credit protection spike at default times, as observed in practice. This feature makes the DGC model appropriate for dealing with counterparty risk on credit derivatives (notably, portfolios of CDS contracts) traded between a bank and its counterparty, respectively labeled as−1and0,and referencing credit names 1 ton,for some positive integer n.Accordingly, we introduce

N={−1,0,1, . . . , n}andN^?={1, . . . , n}

and we focus onτ =τ−1∧τ0in the paper. However, analog properties hold for any minimum of theτiand, in particular, for theτithemselves.

2.1. The model

We consider a family of independent standard linear Brownian motionsZandZⁱ, i∈N. For%∈[0,1), we define Btⁱ=√

%Zt+p

1−%Ztⁱ. (2.1)

Letς be a continuous function onR+ withR

R+ς²(s)ds= 1andα²(t) = R+∞

t ς²(s)ds > 0for allt ∈ R+. For any i ∈N,lethi be a continuously differentiable strictly increasing function fromR^∗+toR,with derivative denoted byh˙i, such thatlims↓0hi(s) =−∞andlims↑+∞hi(s) = +∞. We define

τi=h⁻¹_i Z +∞

0

ς(u)dB_uⁱ

=h⁻¹_i √

% Z +∞

0

ς(u)dZu+p 1−%

Z +∞

0

ς(u)dZ_uⁱ

, (2.2)

fori∈N. The random times(τi)i∈N follow the standard one-factor Gaussian copula model of Li (2000) (a DGC model in abbreviation), with correlation parameter%and with marginal survival functionΦ◦hiofτi,where

Φ(t) = Z +∞

t

√1

2πe^−x22dx, t∈R is the standard normal survival function. Note that, if% <1, theτiavoid each other:

Q(τi=τj) = 0, for anyi6=jinN.

2.2. Density Property

By multivariate density default model, we mean a model with anFconditional density of the default times (see e.g. the condition (DH) in Pham (2010, page 1800)), given some reference subfiltrationFofG. This is the multivariate extension of the notion of a density time, first introduced in an initial enlargement setup in Jacod (1987) and revisited in a progressive enlargement setup in Jeanblanc and Le Cam (2009) (under the name of initial time) and El Karoui, Jeanblanc, and Jiao (2010, 2015a,b).

First we prove that the DGC model is a multivariate density model with respect to the natural filtrationB= (Bt)t≥0of the Brownian motionsZandZⁱ, i∈N. We introduce the following processes.

mⁱ_t= Z t

0

ς(u)dBⁱ_u and m¯ⁱ_t= Z ∞

t

ς(u)dB_uⁱ =h(τi)−mⁱ_t, i∈N.

The standard normal density function is denoted by φ(x) = 1

√2πe^−x22, x∈R.

Theorem 2.1. The dynamic Gaussian copula model is a multivariate density model of default times (with respect to the filtrationB), with conditional Lebesgue density

pt(ti, i∈N) =∂t−1. . . ∂tnQ(τi< ti, i∈N| B_t) of theτi, i∈N, given, for any nonnegativeti, i∈N, andt∈R+,by

pt(ti, i∈N) = Z

R

φ(y)Y

i∈N

φ

hi(ti)−mⁱt+α(t)√

%y α(t)√

1−%

h˙i(ti) α(t)√

1−%dy. (2.3)

Proof. The conditional density functionpgivenBtcan be computed thanks to the independence of increments of the processesZ, Zⁱ, i∈N. Actually, for anyt≥0, we can write

τi=h⁻¹_i (mⁱ_t+√

%ξ+p

1−%ξi), i∈N,

whereξis a real normal random variable with varianceα²_t, where(ξj)j∈N is a centered Gaussian vector independent ofξ with homogeneous marginal variancesα²_tand zero pairwise correlations, and where the familyξ, ξi, i∈N,is independent ofBt. See Crépey et al. (2014, page 172)¹.

2.3. Computation of the intensity processes

Note that theτi areB_∞ measurable, but they are notBstopping times. In the DGC model, the full model filtration G= (Gt)t≥0is taken as the progressive enlargement of the Brownian filtrationBby theτi, i ∈N, augmented so as to satisfy the usual conditions, i.e.

Gt=∩s>t(Bs∨ _

i∈N

σ(τi∧s)), t≥0. (2.4)

In this section we prove that theτiare totally inaccessibleGstopping times with intensities that we compute explicitly.

1Or Crépey et al. (2013, page 3) in the journal version.

ForI⊆Nandj∈N,we define:

ρ^I = %

|I|%+ 1, (σ^I)²= (1−%) |I|%+ 1

|I|%+ 1−%, λ^I = % (|I| −1)%+ 1, Z_t^j,I(u) =hj(u)−m^j_t

α(t) −λ^IX

i∈I

¯ mⁱ_t α(t). Fort≥0, let

It={i∈N :τi≤t}

(representing the set of obligors inN that are in default at timet) and let ρt=ρ^It, σt=σ^It, J_t=N\ I_t. Forσ >0, ρ∈[0,1]andJ ⊆N, we define the functions

ΦJ,ρ,σ(z_J) =Q(ξj > zj, j∈J), ψ^j_J,ρ,σ z_J

=−∂zjΦJ,ρ,σ

ΦJ,ρ,σ

z_J

, j∈J, (2.5)

wherezJ = (zj)j∈Jis a real vector and(ξj)j∈N is a centered Gaussian vector with homogeneous marginal variancesσ² and pairwise correlationsρ.Note the following:

Lemma 2.1. ForI=N\J, the family of random variables

ξj− ρ

(|I| −1)ρ+ 1 X

i∈I

ξi

!

j∈J

defines a centered Gaussian vector independent of σ(ξi, i ∈ I), with homogeneous marginal variances and pairwise correlations, respectively given as

σ²(1−ρ) |I|ρ+ 1

|I|ρ+ 1−ρ and ρ

|I|ρ+ 1. (2.6)

Lemma 2.2. Foru >0,

E[1{uj<τ_j,j∈J}|Bt∨σ(τI)] = ΦJ,ρ^I,σ^I(Zt^j,I(uj), j∈J). (2.7) Proof. Forj∈Janduj ∈R, the conditionuj< τjis equivalent to

Zt^j,I(uj) =hj(uj)−m^j_t

α(t) −λ^IX

i∈I

¯ mⁱ_t

α(t) < m¯^j_t

α(t)−λ^IX

i∈I

¯ mⁱ_t

α(t) (2.8)

Noting thatm^j_t∈ Bt,m¯ⁱ_t∈ Bt∨σ(τI), i∈I, the desired result follows by an application of Lemma 2.1.

Lemma 2.3. For everyt >0andI⊆Nwe have, writingJ =N\IandτI = (τi)i∈I :

{τi≤t < τj :i∈I, j∈J} ∩ Gt={τi≤t < τj:i∈I, j∈J} ∩(Bt∨σ(τ_I)). (2.9) Proof. Let theτ(i)be the increasing ordering of theτi,with alsoτ(0) = 0andτ(n+1) =∞.According to the optional splitting formula which holds in any multivariate density model of default times (see Song (2014b)), for anyGoptional processY, there exists aO(B)⊗ B([0,∞]ⁿ)-measurable functionsY⁽ⁱ⁾, i∈N,such that

Y =

n

X

i=0

Y⁽ⁱ⁾(τ−1-τ(i), . . . , τn-τ(i))1[τ_(i),τ_(i+1)), (2.10)

wherea - b denotesaif a ≤ b and∞if a > b, for a, b ∈ [0,∞]. Since G_t = σ(Yt)andY⁽ⁱ⁾(τ1 - τ(i), . . . , τn - τ(i))1[τ_(i),τ_(i+1))is a function ofB_tandτ_I on{τi ≤t < τj :i∈I, j∈J},this implies (2.9).

Theorem 2.2. For anyj∈N, τjadmits a(G,Q)intensity given by

γ_t^j=1{t<τj}

h˙j(t)

α(t)ψ_J^j_t,ρt,σt Z_t^j,It(t), j∈ Jt

, t∈R⁺. (2.11)

Proof. Letl∈N. For boundedBtmeasurable functionsF, for measurable bounded functionf, for0≤t≤s <∞, we look at

E[F f(τ_I)1{τi≤t<τj:i∈I,j∈J}1{t<τl≤s}].

We need only to considerl∈J. Then, using (2.7) to pass to the third line and conditioning in conjunction with the tower rule to pass to the fourth line:

E[F f(τI)1{τ_i≤t<τ_j,i∈I,j∈J}1{s<τ_l}]

= E[F f(τ_I)1{τi≤t,i∈I}E[1{t<τj,j∈J}1{s<τl}|Bt∨σ(τ_I)]]

= E[F f(τ_I)1{τi≤t,i∈I}ΦJ,ρ^I,σ^I(Zt^j,I(uj), j∈J)]

whereuj =texceptul=s,

= E[F f(τI)1{τi≤t<τj,i∈I,j∈J}

Φ_{J,ρI ,σI}(Z_t^j,I(u_j),j∈J) Φ_{J,ρI ,σI}(Z_t^j,I(t),j∈J) ]

= E[F f(τI)1{τi≤t<τj,i∈I,j∈J}

ΦJt ,ρt ,σt(Z_t^j,It(u_j),j∈J_t) ΦJt ,ρt ,σt(Z_t^j,It(t),j∈J_t)].

(2.12)

With the formula (2.9), we conclude

E[1{t<τ_l≤s}|Gt] =E[1{t<τ_l}|Gt]−E[1{s<τ_l}|Gt]

=1{t<τl}(1−ΦJt,ρt,σt(Zt^j,It(uj), j∈ Jt) ΦJt,ρt,σt(Z_t^j,It(t), j∈ J_t) ).

The stated result follows by an application of the Laplace formula of Dellacherie (1972, Chapter V, Theorem T54) (see also Dellacherie and Doléans-Dade (1971) or Knight (1991)).

2.4. Computation of the drift of the Brownian motion

Next we study the processesBⁱ, i∈N, in the filtrationG. Thanks to Theorem 2.1, the DGC model is a multivariate density model. According to Jacod (1987), this implies the following:

Lemma 2.4. The processesBⁱ, i∈N,areGsemimartingales.

By virtue of Jeanblanc and Song (2013, Theorem 6.4), another consequence of the multivariate density property is the martingale representation property.

Theorem 2.3. LetWⁱ, fori∈N, denote the martingale part inGofBⁱ. Let

dM_tⁱ=d1^τi≤t−γ_tⁱdt, t >0,

where the process γⁱ is defined in (2.11). Then, the martingale representation property holds in G with respect to (Wⁱ, Mⁱ, i∈N).

This section is devoted to the computation of the martingalesWⁱ. We begin with the following remark on the Gaussian processesBⁱ(cf. Lemma 2.1).

Lemma 2.5. ForJ⊆N andI=N\J, the family of processes

B^j− %

(|I| −1)%+ 1 X

i∈I

Bⁱ

!

j∈J

(2.13)

is a continuous Lévy process (multivariate Brownian motion with drift) independent ofσ(Bⁱ, i ∈ I), of homogeneous marginal variances and pairwise correlations, equal to, respectively,

(σ^I)²= (1−%) |I|%+ 1

|I|%+ 1−%and ρ^I = %

|I|%+ 1. (2.14)

Proof. This follows by computing the brackets of the continuous local martingales and applying the Lévy processes characterization.

Lemma 2.6. Fork∈Iand0≤t≤s≤s⁰,

E[B_s^k0−Bs^k|Bt∨σ(τI)] = (_α¹2 t

Rs⁰

sςudu) ¯m^kt. Proof. Fork∈I, for0≤t≤s≤s⁰ <∞,

Bs^k0−Bs^k−( 1 α²_t

Z s⁰ s

ςudu) ¯m^kt (2.15)

is a centered Gaussian random variable, independent ofm¯^kt, with variance (s⁰−s)−2 1

α²_t( Z s⁰

s

ςudu)²+ 1 α²_t(

Z s⁰ s

ςudu)²= (s⁰−s)− 1 α²_t(

Z s⁰ s

ςudu)².

Hence, fork∈I,

E[B_s^k0−B_s^k|Bt∨σ(τI)] =E[B_s^k0−B_s^k|Bt∨σ( ¯mⁱ_t, i∈I)]

= E[B_s^k0−B_s^k|Bt∨σ( ¯m^k_t)∨σ(ξi, i∈I\ {k})]

whereξiform a Gaussian family independent ofBt∨σ(B^k), constructed with (2.13),

= E[B_s^k0−B_s^k|σ( ¯m^k_t)] = (_α¹2 t

Rs⁰

sςudu) ¯m^k_t because of (2.15).

In the sequel we find it sometimes convenient to denote stochastic integration (or integration against measures) byand the Lebesgue measure on the half-line byλ.

Theorem 2.4. ForJ ⊆Nandk∈J, define the function

b^k_J(z, x) =E[1{zj<ξ_j,j∈J}(ξk+x)], x∈R,z= (zj, j∈J),

where(ξj, j∈J)is a Gaussian family of homogeneous marginal variances(σ^I)²and pairwise correlationsρ^I. For any k∈N, define the process

β_t^k= ς(t) α(t)



1{k∈It}

¯ m^k_t

α(t)+1{k /∈It}

b^k_J

t((Zt^j,It(t), j∈ Jt), λ^ItP

i∈I_t

¯ mⁱ_t α(t)) ΦJ_t,ρ_t,σ_t(Z_t^j,It(t), j∈ Jt)



, t∈R+.

Then,W^k=B^k−β^kλ.

Proof. For0 ≤t ≤s≤s⁰ <∞, for any boundedB_tmeasurable functionF and measurable bounded functionf, we compute

E[F f(τI)1{τi≤t<τj,i∈I,j∈J}(B_s^k0 −B_s^k)]

= E[F f(τI)1{τ_i≤t,i∈I}1{t<τ_j,j∈J}E[(B^k_s0−B^k_s)|Bt∨σ(τN)]]

= E[F f(τI)1{τi≤t,i∈I}1{t<τj,j∈J}(_α¹2 t

Rs⁰

sςudu) ¯m^k_t].

Ifk∈I,m¯^k_t ∈ Bt∨σ(τI). Ifk /∈I,

E[F f(τI)1{τi≤t,i∈I}1{t<τj,j∈J}(_α¹2 t

Rs⁰

sςudu) ¯m^kt]

= E[F f(τI)1{τi≤t,i∈I}(_α(t)¹ Rs⁰

sςudu)E[1{t<τj,j∈J}

¯ m^k_t

α(t)|Bt∨σ(τI)]]

= E[F f(τI)1{τi≤t,i∈I}(_α(t)¹ Rs⁰ sςudu) E[1{Z^j,I_t (t)<^m¯

j t α(t)−λ^IP

i∈I

¯ mit

α(t),j∈J}(_α(t)^m¯kt −λ^IP

i∈I

¯ mⁱ_t

α(t)+λ^IP

i∈I

¯ mⁱ_t

α(t))|Bt∨σ(τ_I)]]

= E[F f(τI)1{τi≤t,i∈I}(_α(t)¹ Rs⁰

sςudu)E[1{Z_t^j,I(t)<ξ_j,j∈J}(ξk+λ^IP

i∈I

¯ mⁱ_t α(t))]]

= E[F f(τI)1{τi≤t,i∈I}(_α(t)¹ Rs⁰

sςudu)b^k_J((Z_t^j,I(t), j∈J), λ^IP

i∈I

¯ mⁱ_t α(t))]

= E[F f(τ_I)1{τi≤t,i∈I}1{t<τj,j∈J}(_α(t)¹ Rs⁰ sςudu)^b

k

J((Z_t^j,I(t),j∈J), λ^IP

i∈I

¯ mit α(t)) Φ_{J,ρI ,σI}(Z_t^j,I(t),j∈J) ] This, combined with the formula (2.9), implies

E[(B_s^k0−B_s^k)|Gt] = ( 1 α(t)

Z s⁰ s

ςudu)×



1{k∈It}

¯ m^kt

α(t)+1{k /∈It}

b^k_J

t((Z_t^j,It(t), j∈ Jt), λ^ItP

i∈It

¯ mⁱ_t α(t)) ΦJt,ρt,σt(Zt^j,It(t), j∈ Jt)



.

TheGdrift ofB^kis obtained as the differential of the above with respect to Lebesgue measure, i.e.

ς(t) α(t)



1{k∈I_t}

¯ m^k_t

α(t)+1{k /∈I_t}

b^k_Jt((Zt^j,It(t), j∈ Jt), λ^ItP

i∈It

¯ mⁱ_t α(t)) ΦJt,ρ_t,σ_t(Z_t^j,It(t), j∈ Jt)



dt.

3. Reduced DGC Model

We now study the invariance properties of the DGC model. In this perspective, the market information before the default event of the bank or of its counterparty is modeled by the filtrationF= (Ft)t≥0, where

F_t=∩_s>t B_s∨ _

i∈N^?

σ(τi∧s)

, (3.1)

augmented so as to satisfy the usual conditions.

Because of the multivariate density property of the family of(τ^j, j∈N^?)with respect to the filtrationB(same proof as Theorem 2.1), the computations we have made inGin the previous section can be made similarly inF. In particular, the following splitting formula holds (cf. (2.9)): for anyt >0andI⊆N^?,writingJ =N^?\I,

{τi≤t < τj :i∈I, j∈J} ∩ Ft={τi≤t < τj:i∈I, j∈J} ∩ Bt∨σ(τI). (3.2)

Moreover, the so-called condition (H’) holds, i.e. the processesB^k, k∈N,areFsemimartingales, and the random times τj, j∈N^?,areFtotally inaccessible stopping times, as stated in the following lemma. Fort >0, let

I_t^?={i∈N^?:τi≤t}, J_t^?=N^?\ I_t^?, ρ^?_t =ρ^It?, σ^?_t =σ^I?t. Lemma 3.1. For anyk∈N, the processW¯_t^k =B_t^k−Rt

0β¯^k_sds, t≥0is anFlocal martingale, where β¯^kt = ς(t)

α(t)



1{k∈I_t^?}

¯ m^kt

α(t)+1{k /∈I^?_t}

b^k_J?

t((Z_t^j,It?(t), j∈ J_t^?), λ^It?P

i∈I_t^?

¯ mⁱ_t α(t)) ΦJ_t^?,ρ^?_t,σ^?_t(Z_t^j,It?(t), j∈ J_t^?)



, t∈R+.

Forj ∈N^?,τj is anFtotally inaccessible stopping time and the processdM¯_t^j =d1^τj≤t−¯γ^j_tdt, t >0,is anFlocal martingale, where

¯

γ^j_t =1{t<τj}

h˙j(t) α(t)ψ_J^j?

t,ρ^?_t,σ^?_t Z_t^j,I?t(t), j∈ J_t^?

, t∈R+. (3.3)

The family of processesW¯^k, k ∈NandM¯^j, j∈N^?,has the martingale representation property in the filtrationF. 3.1. The Azéma supermartingale

Our next aim is to compute the Azéma supermartingale of the random timeτ−1∧τ0in the filtrationF, i.e., E[1{t<τ−1∧τ0}|Ft], t≥0.

Lemma 3.2. The Azéma supermartingale of the random timeτ−1∧τ0in the filtrationFis given by

St=ΦJ_t^?∪{−1,0},ρ^?_t,σ_t^?(Z_t^j,It?(t), j∈ J_t^?∪ {−1,0})

ΦJ_t^?,ρ^?_t,σ^?_t(Z_t^j,I?t(t), j∈ J_t^?) , t≥0. (3.4) In particular, the Azéma supermartingaleSis positive.

Proof. For any boundedBtmeasurable functionsFand measurable bounded functionf, we compute (cf. (2.12)) E[F f(τI)1{τi≤t<τj,i∈I,j∈J}1{t<τ−1∧τ0}]

= E[F f(τI)1{τ_i≤t:i∈I}E[1{t<τ_j:j∈J}1{t<τ−1∧τ₀}|B_t∨σ(τI)]]

= E[F f(τI)1{τ_i≤t:i∈I}Φ_{J∪{−1,0},ρ}^I,σ^I(Z_t^j,I(t) :j∈J∪ {−1,0})]

= E[F f(τI)1{τ_i≤t<τ_j:i∈I,j∈J}

ΦJ∪{−1,0},ρI ,σI(Z_t^j,I(t):j∈J∪{−1,0}) Φ_{J,ρI ,σI}(Z_t^j,I(t):j∈J) ]

= E[F f(τI)1{τi≤t<τj:i∈I,j∈J}

Φ_J?

t∪{−1,0},ρ?t,σ?t(Z^j,I

t?

t (t):j∈J_t^?∪{−1,0}) Φ_J?

t,ρ?t,σ?t(Z^j,I

?t

t (t):j∈J_t^?) ],

where conditioning and the tower rule are used in the next-to-last identity. With the formula (3.2), we conclude

E[1{t<τ−1∧τ0}|Ft] = ΦJ_t^?∪{−1,0},ρ^?_t,σ_t^?(Z_t^j,It?(t) :j∈ J_t^?∪ {−1,0}) ΦJ_t^?,ρ^?_t,σ_t^?(Z_t^j,It?(t) :j∈ J_t^?) .

Letν= _S¹ S^c, whereS^cdenotes the continuous martingale component of the(F,Q)Azéma supermartingaleS.

Lemma 3.3. We have

dνt = P

j∈J_t−^?∪{−1,0}ψ_J^j?

t−∪{−1,0},ρ^?_t,σ^?_t Z^j,I

? t−

t (t) :j∈ J_t^?∪ {−1,0}

dζt^j,It?

−P

j∈J_t−^? ψ_J^j?

t−,ρ^?_t,σ^?_t Z^j,I

? t−

t (t) :j∈ J_t^? dζt^j,It?, where, forI⊆N^?,ζ_t^j,Idenotes the martingale part of

−_α(t)¹ dm^j_t+_(|I|−1)%+1^% P

i∈I 1 α(t)dmⁱ_t

inF.

Proof. To obtaindS_t^c(which is then divided bySt), it suffices to apply Itô calculus to the expression (3.4) ofSon every random interval whereI_t−^? is constant. Note that, knowingtis in such an interval,τ_It−^? is inFt. Also note that the jumps ofSttriggered by the jumps ofI_t−^? have no impact here, becauseS^cis a continuous local martingale.

3.2. Freductions ofβ^k, γ^j

Lemma 3.4. The triplet(τ−1∧τ0,F,G)satisfies the condition (B).

Proof. To check the condition (B), by the monotone class theorem, we only need consider the elementaryGpredictable processes of the formU =νf(τ−1∧s, τ0∧s)1(s,t],for anFsmeasurable random variableF and a Borel functionf. SinceU1(0,τ]=F f(s, s)1(s,t]1(0,τ],we may takeU⁰=F f(s, s)1(s,t]in the condition (B).

Next we consider the reduction of the processesβ^k, γ^jin the filtrationF. Notice that, fort < τ−1∧τ0, It=I_t^?, Jt=J_t^?∪ {−1,0}.

Therefore, the following lemma holds.

Lemma 3.5. TheFreduction ofγ^j, j∈N^?,is

eγt^j=1{t<τj}

h˙j(t) α(t)ψ_J^j?

t∪{−1,0},ρ^?_t,σ_t^? Zt^j,It?(t), j∈ J_t^?∪ {−1,0}

, t∈R+. (3.5)

Similarly, theFreduction ofβ^k, k∈N,is

βe_t^k= ς(t) α(t)×

1{k∈I_t^?}

¯ m^k_t α(t)+ 1{k /∈I_t^?}

b^k_J?

t∪{−1,0}((Z_t^j,I?t(t), j∈ J_t^?∪ {−1,0}), λ^It?P

i∈I_t^?

¯ mⁱ_t α(t)) ΦJ_t^?∪{−1,0},ρ^?_t,σ_t^?(Zt^j,It?(t), j∈ J_t^?∪ {−1,0})



, t∈R+.

Note that the processesγ^j,eγ^j andβ^k,βe^kare càdlàg. The next result shows that the processβ^k(and consequentlyβ)e is linked withβ¯^kthrough the processν.

Lemma 3.6. Fork∈N,

Z t 0

βes^kds= Z t

0

β¯s^kds+hB^k, νit, t∈[0, τ−1∧τ0].

Proof. Notice thatB^kis a continuous process. By the Jeulin–Yor formula (see e.g. Dellacherie, Maisonneuve, and Meyer (1992, no 77 Remarques b))),

B^kt − Z t

0

β¯s^kds− hB^k, νit, t∈[0, τ−1∧τ0], defines aGlocal martingale. But, acccording to Theorem 2.4, the drift ofB^kinGisRt

0β^ksds, t≥0. We conclude that Z t

0

βes^kds= Z t

0

βs^kds= Z t

0

β¯s^kds+hB^k, νit

fort∈[0, τ−1∧τ0].

Knowing theFreductionsβe^kandeγ^jofβ^kandγ^j, in view of the martingale representation property inF,accounting also for the avoidance ofτ−1∧τ0 andτj, j ∈ N^?, the strategy for constructing an invariance probability measure P becomes clear. It is enough to find a probability measurePequivalent toQonFT (given a constantT >0) such that the (F,P)drift ofB^k, k∈N,isβe^kand the(F,P)compensator ofτ^j, j∈N^?, has the density processeγ^j.

To implement this idea, the following estimates will be useful.

Lemma 3.7. There exists a constantC >0such that hνit≤C(X

i∈N

sup

0<s≤t

|mⁱs|²+ 1)t (3.6)

and for0≤r≤tandj∈N^?

eγ^j_r∨γ¯_r^j≤C(X

i∈N

sup

0<s≤t

|mⁱ_s|+ 1), (3.7)

eγ^jrln(γer^j∨¯γ^jr)≤CX

i∈N

sup

0<s≤t

(|mⁱs|+ 1) ln(|mⁱs|+ 1).

Proof. Applying Lemma A.2 to the formula (3.3) and noting that the functionα,continuous and positive, is bounded away from 0 on[0, T],we obtain, for positive constantsCthat may change from place to place,

hνit≤C Z t

0

(X

I⊆N

X

j∈N\I

|Z_s^j,I(s)|+ 1)²ds≤C(X

I⊆N

X

j∈N\I

sup

0<s≤t

|Z_s^j,I(s)|+ 1)²t,

which yields (3.6). Applying Lemma A.2 to the formulas (3.3) forγ¯^jand (3.5) foreγ^j, we obtain the first line in (3.7)), whence the second line follows from

eγ_r^jln(eγ_r^j∨γ¯_r^j)≤C(max

i∈N sup

0<s≤t

|mⁱ_s|+ 1) ln C(max

i∈N sup

0<s≤t

|mⁱ_s|+ 1)

= max

i∈N sup

0<s≤t

C(|mⁱ_s|+ 1) ln(C|mⁱ_s|+ 1).

Notice that the processes¯γ^jare positive. Consider theFlocal martingaleµ=ν+P

j∈N^?(^eγ

j

−

¯

γ^j₋ −1)M¯^j. Lemma 3.8. The Doléans-Dade exponentialE(µ)is a true(F,Q)martingale.

Proof. Following Lepingle and Mémin (1978, Theorem III.1), we consider P

j∈N^?

(1 + (^eγ

j τj−

¯

γ^j_τj− −1)) ln(1 + (^eγ

j τj−

¯

γ^j_τj−−1))−(^eγ

j τj−

¯

γ^j_τj− −1)

1{τj≤t},

and itsFpredictable dual projection

Mt = P

j∈N^?

Rt 0

(1 + (^eγ

j s−

¯

γ_s−^j −1)) ln(1 + (^eγ

j s−

¯

γ^j_s− −1))−(^eγ

j τj−

¯ γ^j_s− −1)

¯ γ_s−^j ds

= P

j∈N^?

Rt 0

eγ_s−^j (ln(γe_s−^j )−ln(¯γ_s−^j ))−(γe_τ^j_j−−¯γ_s−^j ) ds.

Combining Lemma 3.7 and Lemma A.3, we prove thate^12hνit+Mt isQintegrable for a sufficiently smallt =t0 > 0.

According to Lepingle and Mémin (1978, Theorem III.1), we conclude thatE[E(µ)t0] = 1. The same argument applied with the conditional probabilityQ[·|Ft0]instead ofQproves thatE[E(µ)2t0|F_t0] = 1. Iterating, we arrive atE[E(µ)kt0] = 1 for any integerk >0.

3.3. The invariance probability measure

We have proved thatE(µ)is an(F,Q)true martingale. We can then define a new probability measureP=E(µ).Qon FT.

Theorem 3.1. The probability measurePis an invariance probability measure for the DGC model(τ−1∧τ0,F,G,Q)on the horizon[0, T],for any constantT >0.

Proof. By the Girsanov theorem, the intensity ofτi,i∈N^?,inFunderPiseγⁱ, while the drift ofB^k, k∈N,inFunder Pis( ¯β^kλ+hB^k, νi).

Given a constantT >0, let us prove that the probability measurePsuch that _d^dP

Q =E(µ)is an invariance probability measure for the quadruplet(τ−1∧τ0,F,G,Q). According to Crépey and Song (2017, Corollary C.1), we only need to consider the locally bounded(F,P)local martingalesPin the condition (A). We write

cW^k=B^k−β¯^kλ− hB^k, νi k∈N, and Mc^j=1[τ^j,∞)−eγ^jλ, k∈N, j ∈N^?. Thanks to Lemma 3.6, the stopped process

(cW^k)^{τ−1∧τ0−}= (B^k−β^kλ)^τ−1∧τ0 = (W^k)^τ−1∧τ0 is a(G,Q)local martingale. The(G,Q)local martingale property of

(Mc^j)^{τ−1∧τ0−}= (1[τ^j,∞)−γ^jλ)^τ−1∧τ0 = (M^j)^τ−1∧τ0 (cf. Lemma 3.5) is clear.

As we did in Lemma 3.1 under the probabilityQ, it can be proven that the family of processescW^k, k∈N,andMc^j, j∈N^?,has the martingale representation property in the filtrationFunderP. Hence any(F,P)local martingalePis an stochastic integral inFof the processesWc^kand ofMc^junder the probability measureP. The natural idea is to say, then, P^{τ−1∧τ0−}is the stochastic integral inGof the processes(W^k)^τ−1∧τ0and of(M^j)^τ−1∧τ0under the probabilityQ, so that P^{τ−1∧τ0−}itself is a(G,Q)local martingale. However, knowing the discussion in Jeulin and Yor (1979) about “faux amis”

regarding enlargement of filtration and stochastic integrals, we have to be careful. More precisely, we need to distinguish between the stochastic integral in the sense of semimartingales and the stochastic integral in the sense of local martingales, recalling from Émery (1980) (cf. also Delbaen and Schachermayer (1994, Theorem 2.9)) that a stochastic integral in the sense of semimartingales with respect to a local martingale need not be a local martingale.

We can argue as follows. We consider separately the cases of continuous and purely discontinuousP. WhenP is a continuous(F,P)local martingale,Pis the(F,P)stochastic integral, in the sense of local martingales, of anFpredictable (n+ 2)dimensional processH = (H^k, k ∈N)with respect to the(n+ 2)dimensional process(Wc^k, k ∈N). Since the matrix(^dhcWk,cWk

0it

dt , k, k⁰ ∈N)is uniformly positive-definite, for everyk∈N,H^kis individually(F,P)integrable with respect to the one dimensional Brownian motionWc^kin the sense of local martingales (see Jacod and Shiryaev (2003, Chapter III, Section 4)). Moreover, ashH^kcW^k, νiexists underP(notingH^kcW^kis continuous), by the Girsanov theorem (see He, Wang, and Yan (1992, Theorem 12.13)),hH^kWc^k, νiis of locally integrable total variation underQ. HenceH^k is integrable with respect tohcW^k, νiunderQ. Moreover the(F,Q)martingale part ofcW^kis an(F,Q)Brownian motion, hence the(F,Q)integrability ofH^kagainst this martingale part reduces to the a.s. finiteness of(H^k)²λ,which holds underPand therefore underQ. In sum,H^kis(F,Q)integrable with respect tocW^kin the sense of semimartingales. As the hypothesis (H’) holds betweenF⊆GunderQ(see after (3.2)), Jeulin (1980, Proposition 2.1) implies thatH^k is(G,Q) integrable with respect toWc^kin the sense of semimartingales. By Jeanblanc and Song (2013, Lemma 2.1), the stochastic integrals in the sense of the(F,Q)semimartingales and in the sense of the(G,Q)semimartingales are the same, hence P =P

k∈NH^kWc^k also holds in the sense of(G,Q)semimartingales. By Lemma 3.6,(cW^k)^τ−1∧τ0 = (W^k)^τ−1∧τ0is a(G,Q)local martingale. Applying He, Wang, and Yan (1992, Theorem 9.16), we conclude that, in fact,H^kis(G,Q)

integrable with respect to(Wc^k)^τ−1∧τ0 in the sense of local martingales. Hence P^{τ−1∧τ0−} =X

k∈N

(H^kWc^k)^{τ−1∧τ0−} =X

k∈N

(H^kW^k)^τ−1∧τ0

is a(G,Q)local martingale.

Consider now the case ofP purely discontinuous. Without loss of generality we suppose that the locally bounded processPis in fact bounded. Then,Pis the(F,P)stochastic integral (in the sense of local martingale) of anFpredictable ndimensional processK = (K^j, j ∈ N^?)with respect to thendimensional process(Mc^j, j ∈ N^?). The processes Mc^j, j ∈ N^?,have disjoint jump times with jump amplitude 1. This implies thatK^j is integrable with respect to Mc^j individually. Moreover, asPis bounded, the random variablesKτ_j, j∈N^?,are bounded, henceKitself is bounded (cf.

He et al. (1992, Theorem 7.23)). As a consequence,Kis automatically(G,Q)integrable with respect to(Mc^j, j∈N^?) in the sense of local martingale. By Jeanblanc and Song (2013, Lemma 2.1) again,

P^{τ−1∧τ0−}= X

j∈N^?

(K^jMc^j)^{τ−1∧τ0−} = X

j∈N^?

(K^jM^j)^τ−1∧τ0,

which is a(G,Q)local martingale.

3.4. Alternative Proof of the Condition (A)

Theorem 3.1 yields an explicit construction of the invariance probability measurePin the DGC model. If we only want to establish the condition (A), i.e. the existence ofP, a shorter proof is available based on the sufficiency condition of Crépey and Song (2017, Theorem 5.1).

Lemma 3.9. The(G,Q)intensityγof the random timeτ−1∧τ0is given by γ=1[0,τ−1∧τ0](γ⁻¹+γ⁰).

Proof. This follows from, for example, Crépey and Song (2016, Lemma 6.2).

Theorem 3.2. The condition (A) holds in the DGC model(τ−1∧τ0,F,G,Q).

Proof. Given a constant horizonT >0, according to Crépey and Song (2017, Theorem 5.1), we only need to prove the exponential integrability ofRτ∧T

0 γsds, which can be done similarly to the proof of Lemma 3.8.

4. Wrong Way Risk

As visible in (2.11), the default intensities of the surviving names spike at defaults in the DGC model. This is very much related to the departure from the immersion property in this model, i.e. the fact that the invariance probability measure Pis not equal to the pricing measureQonFT.This ’wrong way risk’ feature (cf. Crépey and Song (2016)) makes the DGC model appropriate for dealing with counterparty risk on credit derivatives, notably portfolios of CDS contracts traded between a bank and its counterparty, respectively labeled as−1and0,and bearing on reference firmsi= 1, . . . , n.

To illustrate this numerically, in this concluding section of the paper, we study the valuation adjustment accounting for counterparty and funding risks (total valuation adjustment TVA) embedded in one CDS between a bank and its counterparty on a third reference firm.

In Figure 1, the left graph shows the TVA computed as a function of the correlation parameter%in a DGC model of the three credit names (hencen= 1): the bank, its counterparty and the reference credit name of the CDS. The different curves correspond to different levels of credit spreadλ¯of bank: the higher¯λ, the higher the funding costs for the bank, resulting in higher TVAs. All the TVA numbers are computed by a Monte Carlo scheme dubbed “FT scheme of order 3”

in Crépey and Nguyen (2016, Section 6.1). FT refers to Fujii and Takahashi (2012a,b). The numerical parameters are set as in Crépey and Nguyen (2016, Section 6.1), to which we refer the reader for a complete description of the CDS contract,

of the FT numerical scheme and of other numerical experiments involving CDS portfolios (as opposed to a single contract here).

The right panel of Figure 1 shows the analog of the left graph, but in a fake DGC model, where we deliberately ignore the impact of the default of the counterparty in the valuation of the CDS at timeτ−1∧τ0(technically, in the notation of Crépey and Song (2016, Equation (6.7)), we replace (Pe_t^e+∆e^e_t) byPt−in the coefficientfb), in order to kill the wrong-way risk feature of the DGC model. We can see from the figure that, for large%,the corresponding fake TVA numbers are five to ten times smaller than the “true” TVA levels that can be seen in the left panel. In addition of being much smaller for large%,the fake DGC TVA numbers in the right panel are mostly decreasing with%. This shows that the wrong-way risk feature of the DGC model is indeed responsible for the “systemic” increasing pattern observed in the left panel.

0.2 0.4 0.6 0.8 0.99

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

%

TVA

TVA one CDS different correlations

¯λ= 0%

¯λ= 1%

¯λ= 2%

¯λ= 3%

0.2 0.4 0.6 0.8 0.99

0 0.05 0.1 0.15 0.2 0.25

̺

TVA

TVA one CDS different correlations

¯λ= 0%

¯λ= 1%

¯λ= 2%

¯λ= 3%

Figure 1. Left: TVA on one CDS as a function of the correlation parameter%and for different bank credit spreads¯λ,in a DGC model with three names: the bank, its counterparty and the reference credit name of the CDS.Right: Analog results in a fake DGC model without wrong-way-risk.

A. Gaussian Estimates

In this appendix we derive the Gaussian estimates that are used in the proofs of Lemmas 3.7 and 3.8.

Lemma A.1. Given a positive decreasing continuously differentiable functionΓonR+such that Z

R+

t^dΓ(t)dt <∞ and lim

t↑∞t^d−1Γ(t)→0, for some integerd≥0, we writeg(y) =−^Γ_Γ(y)^0(y), G(y) =R∞

y t^dΓ(t)dt.Lety¯≥0andα, >0.

(i)Ifg(y)≥αyfory >y, then¯ G(y)≤ 1

α+

y^d−1Γ(y) for y >y¯∨ r

|d−1|( 1 α² +1

α).

(ii)Ifg(y)≤αyfory >y, then¯ G(y)≥ 1

α−

y^d−1Γ(y) for y >y¯∨ r

|d−1|( 1 α² −1

α).