Modelling of successive default events
Nicole El Karoui, CMAP, Ecole PolytechniqueMonique Jeanblanc, Universite d’Evry, FranceYing Jiao, Ecole Superieure d’Ingenieur Leonard de Vinci
Princeton, Credit Risk, May 2008
Financial support from Fondation du Risque and Federation des Banques Francaises 1
The aim of this talk is
• to give a general framework for multi-defaults modelling
• to obtain the dynamics of derivative products in a multinamesetting.
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The basic tool is the conditional law of the default(s) with respect to areference filtration
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Single Name
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Notation
• G is the global market filtration,• τ is a default time,• Ht = 11τ≤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,
11{τ>t}E[Y G|Gt] = 11{τ>t}E[Y G11{τ>t}|Ft]
P(τ > t|Ft)a.s.
on 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.vsαt(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-continuousmartingale
Under the above hypotheses, we can compute Gt-conditionalexpectations on the set {τ ≤ t}A ”weak version” of J-hypothesis consists of the existence of thedensity only for 0 ≤ t ≤ θ. This is useful for before-default studies, butnot sufficient for the after-default ones.
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Density and F-martingales
We assume J-hypothesis and introduce the cadlag (super) martingales
• Conditional survival process (St = P(τ > t|Ft), t ≥ 0) (Azemasuper-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) ofSt(θ)
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Decompositions of Survival process S (super-martingale)
• The Doob-Meyer decomposition of S is St = MFt − AF
t with
– the F-martingale
MFt = −
∫ t
0
(αt(u) − αu(u))du = St − S0 +∫ t
0
αu(u)du
– the increasing process AFt =
∫ t
0αu(u) du
• The multiplicative decomposition isSt = LF
t DFt
where
– The F-martingale LF is given as dLFt = e
∫ t0 λF
sdsdMFt
– The decreasing process DF is DFt = exp
(− ∫ t
0λF
sds)
where
λFt = αt(t)
Ston {St > 0}.
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Links with the classical intensity approach
• The G-intensity is the G-adapted process λG such that(11{τ≤t} −
∫ t
0λG
s ds, t ≥ 0) is a G-martingale.
• Under the weak version of J-Hypothesis,
λGt = 11{τ>t}λF
t = 11{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 informationto 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 = Xt11{τ>t} + Xt(τ)11{τ≤t}where Xt and Xt(θ) are Ft-measurable.
The process MX is a G-martingale if its decomposition as
MXt := Xt11{τ>t} + Xt(τ)11{τ≤t}
satisfies
• (XtSt +∫ t
0Xs(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 :XtL
Ft − ∫ t
0(Xs − Xs(s))LF
sλFs ds is an F-martingale
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Immersion hypothesis
Immersion holds if (and only if) any F-martingale is aG-martingale.
Under immersion hypothesis,
• αt(θ) = αt∧θ(θ)
• S is a non-increasing process
• LF is a constant
• the processMX
t := Xt11{τ>t} + Xt(τ)11{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
0λF
sds ≥ Θ} whereΛ 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 martingalessuch that
∫∞0
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-classicalpart)
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F-martingale representation and HJM framework
Model the family of F-martingales St(θ) in the HJM framework
SupposedSt(θ)St(θ)
= Φt(θ)dMt, t, θ ≥ 0
where M is a continuous multi-dimensional F-martingale, thenSt(θ) = St(t) exp
(− ∫ θ
tλt(u)du
)where λt(θ) is the forward intensity
and
* St(θ) = S0(θ) exp(∫ t
0Φs(θ)dMs − 1
2
∫ t
0|Φs(θ)|2d〈M〉s
);
* St = exp(− ∫ t
0λF
sds +∫ t
0Φs(s)dMs − 1
2
∫ t
0|Φs(s)|2d〈M〉s
).
* λt(θ) = λ0(θ) −∫ t
0ϕs(θ)dMs +
∫ t
0ϕs(θ)Φs(θ)d〈M〉s.
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For any θ ≥ 0, assume that λ0(θ) is a family of positive probabilitydensities.
Let b(θ) be a given family of non-negative F-adapted processes. Define
ϕt(θ) = −bt(θ)λ0(θ) exp(∫ t
0
bs(θ)dWs − 12
∫ t
0
bs(θ)2ds
)and let
αt(θ) = λt(θ) exp(−∫ t
0
λt(v)dv
)where λt(θ) = λ0(θ) −
∫ θ
0ϕs(θ)dWs +
∫ t
0ϕs(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
2θ2 〈M〉t
)where M is an F-martingale.
• Exponential law S0(θ) = P(τ > θ) = exp(−θM0).
• Probability condition : Mt + 12θ 〈M〉t ≥ 0
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Comparison with interest rate modelling
• Zero-coupon B(t, T ) = E[e−∫ T
trsds|Ft].
Short rate rt = −∂T |T=t log B(t, T ).
• Defaultable zero-coupon without actualization
E[11{τ>T}|Gt] = 11{τ>t}E[ST /St|Ft] := 11{τ>t}Bτ (t, T ).
Intensity λFt = −∂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 densityprocess (α0
t (θ), t ≥ 0) constant in time after θ
2. Let ZGt = Zt11{τ>t} + Zt(τ)11{τ≤t} a positive (G, P0)-martingale
with expectation 1
3. Define dP = ZGT dP0 on GT . The RN density of P w.r.t. P0 on Ft is
ZFt = ZtSt +
∫ t
0Zs(s)α0
s(s)ds.
4. Then the density process of τ under P is
αPt (θ) = α0
θ(θ)Zt(θ)ZF
t
for θ < t
αPt (θ) =
E[Zθ(θ)α0θ(θ)|Ft]
ZFt
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∫∞0
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
on Ft is ZFt =
∫∞0
qt(u)f(u)du and the density process of τ
under P is
αPt (θ) = qt(θ)f(θ)(ZF
t )−1
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Two ordered default times
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 theordered 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 usingG
(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 )
= 11{τ>t}P(τ > t, σ > θ|Ft)
P(τ > t|Ft)+ 11{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,σt (u1, u2)
• Note that ατ,σt (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[11{τ>t}YT (τ, σ) | Gt
]=
E[ ∫∞
tdu1
∫∞u1
du2 YT (u1, u2) ατ,σT (u1, u2)|Ft
]∫∞t
du1
∫∞u1
du2 ατ,σt (u1, u2)
• on {τ ≤ t < σ},
E[11{τ>t}YT (τ, σ) | Gt
]=
E[ ∫∞
tdu2 YT (u1, u2)α
τ,σT (u1, u2)|Ft
]∫∞t
du2 ατ,σt (u1, u2)
∣∣∣u1=τ
• on {σ ≤ t},
E[11{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 (Schonbucher-Schubert)
• Cox process model for τ1 and τ2 :
τi = inf{t : Λit ≥ Θi}
C is the survival copule of Θ1, Θ2
• Marginal survival process Sit = P(τi > t|F∞) = e−Λi
t , immersionhypothesis satisfied for F and G
i.
• The joint survival probability is obtained from
P(τ1 > θ1, τ2 > θ2|F∞) = C(S1θ1
, S2θ2
).
ThereforeS1,2
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 + θ22)
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a survival c.d.f of two exponential r.v. with unit parameter linkedby a Clayton copula.
• Diffuse the copula function as a martingale : ∀t, θ1, θ2 ≥ 0, let
St(θ1, θ2) = exp(− (θ2
1M1t + θ2
2M2t
) 12 − At
)where
At =18
∫ t
0
1 + X12s
X23s
d〈X〉s and Xs = θ21M
1s + θ2
2M2s
where M1, M2 positive F-martingales s.t. 〈M1, M2〉t > 0.
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Two ordered default times
Exponential diffusion model
• A two-dimensional exponential example:
exp(− θ1M
1t − θ2M
2t − 1
2θ21〈M1〉t − 1
2θ22〈M2〉t − θ1θ2
(〈M1, M2〉t + a
))M1, M2 positive F martingales s.t. 〈M1, M2〉t ≥ 0
• At t = 0, S0(θ1, θ2) = exp(−θ1M10 − θ2M
20 − a θ1θ2).
• Dependence at t > 0 characterized by 〈M1, M2〉t• Probability condition
M1t M2
t − 〈M1, M2〉t > a > 0
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Several Defaults, Applications to pricing
Several Defaults, Applications to pricing
• Generalization to n successive defaults σ1 ≤ · · · ≤ σn by a recursivemethod
• Representation of conditional expectation with respect toG(1,···,n)
t = Ft ∨Hσ1t ∨ · · · ∨ Hσn
t
Let Yt(u1, · · · , un) be a family of r.v. Ft ⊗ B(Rn)-measurable wheret, u1, · · · , un ≥ 0. Then
E[YT (σ1, · · · , σn)|G(1,···,n)
t
]=
n∑i=0
11{σi≤t<σi+1} qit(T, σ1, · · · , σi, YT )
where qit(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 theunderlying portfolio :
E[11{σk>T}YT |G(1,···,n)
t
]=
k−1∑i=0
11{σi≤t<σi+1}qit,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 11{τi≤t} and key term tocalculate :
E[(K − lT )+|G(1,···,n)
t
]=
∫ K
−∞du E
[11{σ�u�+1>T}|G(1,···,n)
t
]=
∫ K
−∞du
�u�∑i=0
11{σi≤t<σi+1}qit,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 recoveryδ and premium κ and Xt(κ) = 11t≤τ2∧τ1Xt(κ) + 11τ2<t≤τ1Xt(κ) its price.Let
St(u, v) = P(τ1 > u, τ2 > v|Ft) = S0(u, v) +∫ t
0
gs(u, v)dWs
Then:
dXt(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 ) dWt
]with
σt(T ) = − 1St(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
dXt(κ) =(− δ(t)λ1|2(t, τ2) + κ + Xt(κ)λ1|2(t, τ2)
)dt + σ
1|2t (T ) dWt
where
λ1|2(t, s) = − αt(t, s)∂2St(t, s)
σ1|2t (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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Several Defaults, Applications to pricing
Perspectives
A general framework for portfolio of defaultable names :
• explicit model studies for the joint density process
• application to the pricing
• calibration of parameters
• dynamic hedging
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