[PPT] - Modelling of successive default events Nicole El Karoui, CMAP, Ecole PowerPoint Presentation

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Modelling of successive default events

Nicole El Karoui, CMAP, Ecole Polytechnique Monique Jeanblanc, Universit´ e d’´ Evry, France Ying Jiao, Ecole Sup´ erieure d’Ing´ enieur L´ eonard de Vinci Princeton, Credit Risk, May 2008

Financial support from Fondation du Risque and F´ ed´ eration des Banques Fran¸ caises 1

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The aim of this talk is

to give a general framework for multi-defaults modelling
to obtain the dynamics of derivative products in a multiname

setting.

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The basic tool is the conditional law of the default(s) with respect to a reference filtration

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Notation

G is the global market filtration,
τ is a default time,
Ht = 1

1τ≤t is the default processes,

H is the natural filtration of H, with H ⊂ G,
F is a reference filtration, with F ⊂ G .

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On the set {τ > t}: ”before-default” (F, G, τ) satisfy the minimal assumption (MA) if ∀ t ≥ 0 and ZG ∈ Gt, ∃ ZF ∈ Ft such that ZG ∩ {τ > t} = ZF ∩ {τ > t}.

If Gτ := F ∨ H, then (F, Gτ, τ) enjoys MA.
Under MA, for any G∞-measurable (integrable) r.v. Y G,

1 1{τ>t}E[Y G|Gt] = 1 1{τ>t} E[Y G1 1{τ>t}|Ft] P(τ > t|Ft) a.s.

n the set A := {ω : P(τ > t|Ft)(ω) > 0}.

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On the set {τ ≤ t}: ”after-default”

1. We assume that G = Gτ = F ∨ H
2. J-Hypothesis (Jacod for enlargement of filtration purpose):

We assume that there exists a family of Ft ⊗ B(R+) r.vs αt(θ) such that P(τ ∈ dθ|Ft) = αt(θ)dθ dθ ⊗ dP − a.s. and for any θ the process αt(θ), t ≥ 0 is a right-continuous martingale Under the above hypotheses, we can compute Gt-conditional expectations on the set {τ ≤ t} A ”weak version” of J-hypothesis consists of the existence of the density only for 0 ≤ t ≤ θ. This is useful for before-default studies, but not sufficient for the after-default ones.

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Density and F-martingales We assume J-hypothesis and introduce the c` adl` ag (super) martingales

Conditional survival process (St = P(τ > t|Ft), t ≥ 0) (Az´

ema super-martingale)

Conditional probability process (St(θ) = P(τ > θ|Ft), t ≥ 0).

Note that, for t ≤ θ, one has St(θ) = E[Sθ|Ft],

Family of martingales (for θ ∈ R): the densities (αt(θ), t ≥ 0) of

St(θ)

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Decompositions of Survival process S (super-martingale)

The Doob-Meyer decomposition of S is St = M F

t − AF t with

– the F-martingale M F

t = −

t (αt(u) − αu(u))du = St − S0 + t αu(u)du – the increasing process AF

t =

t

0 αu(u) du

The multiplicative decomposition is

St = LF

t DF t

where – The F-martingale LF is given as dLF

t = e t

0 λF sdsdM F

t

– The decreasing process DF is DF

t = exp

−

t

0 λF sds

where

λF

t = αt(t) St

n {St > 0}.

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Links with the classical intensity approach

The G-intensity is the G-adapted process λG such that

(1 1{τ≤t} − t

0 λG s ds, t ≥ 0) is a G-martingale.

Under the weak version of J-Hypothesis,

λG

t = 1

1{τ>t}λF

t = 1

1{τ>t} αt(t) St− a.s..

For any θ ≥ t,

αt(θ) = E[λG

θ |Ft]

a.s.. Note that the intensity approach does not contain enough information to study the after-default case (i.e. for θ < t).

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A characterization of G-martingales Any Gt-measurable r.v. X can be written as X = Xt1 1{τ>t} + Xt(τ)1 1{τ≤t} where Xt and Xt(θ) are Ft-measurable. The process M X is a G-martingale if its decomposition as M X

t

:= Xt1 1{τ>t} + Xt(τ)1 1{τ≤t} satisfies

(XtSt +

t

0 Xs(s)αs(s) ds, t ≥ 0) is an F-martingale

For any θ, (Xt(θ)αt(θ), t ≥ θ) is an F-martingale

Remark: The first condition is equivalent to : XtLF

t −

t

0(Xs − Xs(s))LF sλF s ds is an F-martingale

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Immersion hypothesis Immersion holds if (and only if) any F-martingale is a G-martingale. Under immersion hypothesis,

αt(θ) = αt∧θ(θ)
S is a non-increasing process
LF is a constant
the process

M X

t

:= Xt1 1{τ>t} + Xt(τ)1 1{τ≤t} is a G-martingale if (a) Xt(θ) is a F-martingale on [θ, ∞). (b) Xt − t

0(Xs − Xs(s))λF s ds is an F-martingale

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Toy Exemple: Cox process model Let τ = inf{t : Λt := t

0 λF sds ≥ Θ} where

Λ is an F-adapted increasing process, Λ0 = 0, limt→∞ Λt = +∞ Θ is a G-measurable r.v. independent of F∞, Θi ∼ exp(1). F is immersed in G = F ∨ H. The conditional distribution of τ is ⎧ ⎨ ⎩ P(τ > θ| Ft) = E[e−Λθ| Ft], for θ > t P(τ > θ| Ft) = e−Λθ, for θ ≤ t and the density is ⎧ ⎨ ⎩ αt(θ) = E[λθe−Λθ| Ft], for θ > t = λθe−Λθ, for θ ≤ t

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Modelling the density process Two possible solutions:

Model St(θ) := P(τ > θ|Ft) and then take derivatives w.r.t. θ
Model the density αt(θ) as a family of strictly positive martingales

such that ∞ αs(θ)dθ = 1 Remarks :

for fixed θ, both processes are positive F martingales
reference to the interest models
distinction between θ ≥ t (classical part) and θ < t (non-classical

part)

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F-martingale representation and HJM framework Model the family of F-martingales St(θ) in the HJM framework Suppose dSt(θ) St(θ) = Φt(θ)dMt, t, θ ≥ 0 where M is a continuous multi-dimensional F-martingale, then St(θ) = St(t) exp

−

θ

t λt(u)du

where λt(θ) is the forward intensity

and * St(θ) = S0(θ) exp t

0 Φs(θ)dMs − 1 2

t

0 |Φs(θ)|2dMs

;

* St = exp

−

t

0 λF sds +

t

0 Φs(s)dMs − 1 2

t

0 |Φs(s)|2dMs

.

* λt(θ) = λ0(θ) − t

0 ϕs(θ)dMs +

t

0 ϕs(θ)Φs(θ)dMs.

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For any θ ≥ 0, assume that λ0(θ) is a family of positive probability densities. Let b(θ) be a given family of non-negative F-adapted processes. Define ϕt(θ) = −bt(θ)λ0(θ) exp t bs(θ)dWs − 1 2 t bs(θ)2ds

and let

αt(θ) = λt(θ) exp

−

t λt(v)dv

where λt(θ) = λ0(θ) −

θ

0 ϕs(θ)dWs +

t

0 ϕs(θ)Φs(θ)ds.

Then the family (αt(θ), t ≥ 0) is a density process.

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Examples of martingale density process

Compatibility between martingale and probability properties

– the r.v. St(θ) is [0, 1]-valued – for any t, the map θ → St(θ) is non-increasing

A Generalized exponential model: ∀ t, θ ≥ 0, let

St(θ) = exp

− θMt − 1

2 θ2 Mt

where M is an F-martingale.
Exponential law S0(θ) = P(τ > θ) = exp(−θM0).
Probability condition : Mt + 1

2θ Mt ≥ 0

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Comparison with interest rate modelling

Zero-coupon B(t, T) = E[e−

T

t rsds|Ft].

Short rate rt = −∂T |T =t log B(t, T).

Defaultable zero-coupon without actualization

E[1 1{τ>T }|Gt] = 1 1{τ>t}E[ST /St|Ft] := 1 1{τ>t}Bτ(t, T). Intensity λF

t = −∂T |T =t log Bτ(t, T).

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A general construction of density process. I.

1. Start with P0 with immersion hypothesis, and τ with density

process (α0

t(θ), t ≥ 0) constant in time after θ

2. Let ZG

t = Zt1

1{τ>t} + Zt(τ)1 1{τ≤t} a positive (G, P0)-martingale with expectation 1

3. Define dP = ZG

T dP0 on GT . The RN density of P w.r.t. P0 on Ft is

ZF

t = ZtSt +

t

0 Zs(s)α0 s(s)ds.

4. Then the density process of τ under P is

αP

t (θ)

= α0

θ(θ)Zt(θ)

ZF

t

for θ < t αP

t (θ)

= E[Zθ(θ)α0

θ(θ)|Ft]

ZF

t

for θ ≥ t.

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A general construction of density process. II.

1. Start with P0 with τ independent of F∞ with density f
2. Let q∞(u) a family of F∞-measurable r.v. such that

∞ q∞(u)f(u)du = 1

3. Define dP = q∞(τ)dP0 on G∞.
4. Then, setting qt(u) = E0(q∞(u)|Ft), the RN density of P w.r.t. P0
n Ft is ZF

t =

∞ qt(u)f(u)du and the density process of τ under P is αP

t (θ)

= qt(θ)f(θ)(ZF

t )−1

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Two ordered default times

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Two ordered default times

Notation Two G-stopping times: τ = τ (1) := min(τ1, τ2) and σ = τ (2) := max(τ1, τ2). Before-default and after-default analysis extended naturally to the

rdered defaults
Filtrations: H(1) for τ and H(2) for σ respectively. Let

G(1) = F ∨ H(1) and G(2) = F ∨ H(1) ∨ H(2)= G(1) ∨ H(2).

On the set {t < τ}, it suffices to apply directly the previous studies
On the sets {τ ≤ t < σ} and {σ ≤ t} a recursive procedure using

G(1) as the reference filtration and G(2) as the global filtration

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Two ordered default times

G(1) -conditional survival probability of σ

The G(1)-conditional survival probability of σ is

Sσ|G(1)

t

(θ) = P(σ > θ|G(1)

t

) = 1 1{τ>t} P(τ > t, σ > θ|Ft) P(τ > t|Ft) + 1 1{τ≤t} ∂sP(σ > θ, τ > s|Ft) ∂sP(τ > s|Ft)

s=τ
We assume that there exists ατ,σ such that

P(τ > θ1, σ > θ2|Ft) = ∞

θ1

du1 ∞

θ2

du2 ατ,σ

t

(u1, u2)

Note that ατ,σ

t

(u1, u2) = 0, ∀u1 ≥ u2.

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Two ordered default times

Computation of G(2)-conditional expectations Explicit formulas on the three sets:

on {t < τ},

E

1

1{τ>t}YT (τ, σ) | Gt

=

E ∞

t

du1 ∞

u1 du2 YT (u1, u2) ατ,σ T (u1, u2)|Ft

∞

t

du1 ∞

u1 du2 ατ,σ t

(u1, u2)

on {τ ≤ t < σ},

E

1

1{τ>t}YT (τ, σ) | Gt

= E

∞

t

du2 YT (u1, u2)ατ,σ

T (u1, u2)|Ft

∞

t

du2 ατ,σ

t

(u1, u2)

u1=τ
on {σ ≤ t},

E

1

1{τ>t}YT (τ, σ) | Gt

= E
YT (u1, u2) ατ,σ

T (u1, u2) | Ft

ατ,σ

t

(u1, u2)

u1=τ u2=σ

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Two ordered default times

A standard example (Sch¨

nbucher-Schubert)
Cox process model for τ1 and τ2 :

τi = inf{t : Λi

t ≥ Θi}

C is the survival copule of Θ1, Θ2

Marginal survival process Si

t = P(τi > t|F∞) = e−Λi

t, immersion

hypothesis satisfied for F and Gi.

The joint survival probability is obtained from

P(τ1 > θ1, τ2 > θ2|F∞) = C(S1

θ1, S2 θ2).

Therefore S1,2

t

(θ1, θ2) = E

C(S1

θ1, S2 θ2)|Ft

.

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Two ordered default times

A copula diffusion example

Joint survival probability

S0(θ1, θ2) = P(τ1 > θ1, τ2 > θ2) = exp

− (θ2

1 + θ2 2)

1 2

a survival c.d.f of two exponential r.v. with unit parameter linked by a Clayton copula.

Diffuse the copula function as a martingale : ∀t, θ1, θ2 ≥ 0, let

St(θ1, θ2) = exp

−
θ2

1M 1 t + θ2 2M 2 t

1

2 − At

where

At = 1 8 t 1 + X

1 2

s

X

2 3

s

dXs and Xs = θ2

1M 1 s + θ2 2M 2 s

where M 1, M 2 positive F-martingales s.t. M 1, M 2t > 0.

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Two ordered default times

Exponential diffusion model

A two-dimensional exponential example:

exp

− θ1M 1

t − θ2M 2 t − 1

2θ2

1M 1t − 1

2θ2

2M 2t − θ1θ2

M 1, M 2t + a
M 1, M 2 positive F martingales s.t. M 1, M 2t ≥ 0
At t = 0, S0(θ1, θ2) = exp(−θ1M 1

0 − θ2M 2 0 − a θ1θ2).

Dependence at t > 0 characterized by M 1, M 2t
Probability condition

M 1

t M 2 t − M 1, M 2t > a > 0

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Several Defaults, Applications to pricing

Generalization to n successive defaults σ1 ≤ · · · ≤ σn by a recursive

method

Representation of conditional expectation with respect to

G(1,···,n)

t

= Ft ∨ Hσ1

t

∨ · · · ∨ Hσn

t

Let Yt(u1, · · · , un) be a family of r.v. Ft ⊗ B(Rn)-measurable where t, u1, · · · , un ≥ 0. Then E

YT (σ1, · · · , σn)|G(1,···,n)

t

=

n

i=0

1 1{σi≤t<σi+1} qi

t(T, σ1, · · · , σi, YT )

where qi

t(T, s1, · · · , si, YT ) is a ratio of Ft conditional expectations,

σ0 = 0 and σn+1 = ∞.

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Several Defaults, Applications to pricing

Pricing of the portfolio credit derivatives

kth-to-default swap depends on the kth default time of the

underlying portfolio : E

1

1{σk>T }YT |G(1,···,n)

t

=

k−1

i=0

1 1{σi≤t<σi+1}qi

t,Q(T, σ1, · · · , σi, YT S(k) T )

where S(k)

T

= P(σk > T|Ft).

For a CDO tranche, total loss lT = n

i=1 1

1{τi≤t} and key term to calculate : E

(K − lT )+|G(1,···,n)

t

=

K

−∞

du E

1

1{σ⌊u⌋+1>T }|G(1,···,n)

t

=

K

−∞

du

⌊u⌋

i=0

1 1{σi≤t<σi+1}qi

t,Q

T, σ1, · · · , σi, S⌊u⌋

T

.

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Several Defaults, Applications to pricing

Dynamics of CDSs prices Let X(κ) be the price of a CDS written on the default τ1, with recovery δ and premium κ and Xt(κ) = 1 1t≤τ2∧τ1 Xt(κ) + 1 1τ2<t≤τ1 Xt(κ) its price. Let St(u, v) = P(τ1 > u, τ2 > v|Ft) = S0(u, v) + t gs(u, v)dWs Then: d Xt(κ) = 1 St(t, t)

δ(t)∂1St(t, t) + κSt(t, t) −
∂1St(t, t) + ∂2St(t, t)

Xt(κ)

dt

− T

t

δ(u)αt(u, t) + κ ∂2St(u, t)
du
dt + σt(T) d

Wt

with

σt(T) = − 1 St(t, t) T

t

δ(u) ∂1gt(u, t) + κ1gt(u, t)
du + gt(t, t)

Xt(κ)

.

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Several Defaults, Applications to pricing

d Xt(κ) =

− δ(t)λ1|2(t, τ2) + κ +

Xt(κ)λ1|2(t, τ2)

dt + σ1|2

t

(T) d Wt where λ1|2(t, s) = − αt(t, s) ∂2St(t, s) σ1|2

t

(T) = 1 ∂2St(t, τ2)(At(τ2) − Xt(κ)∂2gt(t, τ2)), At(s) = − T

t

δ(u)∂12gt(u, s)du − κ T

t

∂2gt(u, s) du.

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