SLIDE 1
Density models for credit risk
Nicole El Karoui, Ecole Polytechnique, France Monique Jeanblanc, Universit´ e d’´ Evry; Institut Europlace de Finance Ying Jiao, Universit´ e Paris VII Workshop on stochastic calculus and finance, July 2009, Hong-Kong
Financial support from La Fondation du Risque and F´ ed´ eration Bancaire Fran¸ caise 1
SLIDE 2 Density Hypothesis
Density Hypothesis
Let (Ω, A, F, P) be a filtered probability space. A strictly positive and finite random variable τ (the default time) is given. Our goals are
- to show how the information contained in the reference filtration F
can be used to obtain information on the law of τ,
- to investigate the links between martingales in the different
filtrations that will appear.
2
SLIDE 3
Density Hypothesis
We assume the following density hypothesis: there exists a non-atomic non-negative measure η such that, for any time t, there exists an Ft ⊗ B(R+)-measurable function (ω, θ) → αt(ω, θ) which satisfies P(τ ∈ dθ|Ft) = αt(θ)η(dθ), P − a.s. The conditional distribution of τ is characterized by the survival probability defined by Gt(θ) := P(τ > θ|Ft) = ∞
θ
αt(u)η(du) Let Gt := Gt(t) = P(τ > t|Ft) = ∞
t
αt(u)η(du) Observe that the set At := {Gt > 0} contains a.s. the event {τ > t}.
3
SLIDE 4 Density Hypothesis
The family αt(.) is called the conditional density of τ w.r.t. η given Ft. Note that
- Gt(θ) = E(Gθ|Ft) for any θ ≥ t
- the law of τ is P(τ > θ) =
∞
θ
α0(u)η(du)
∞ αt(u)η(du) = 1
- For an integrable FT ⊗ σ(τ) r.v. YT (τ), one has, for t ≤ T:
E(YT (τ)|Ft) = E( ∞ YT (u)αT (u)η(du)|Ft)
- The default time τ avoids F-stopping times, i.e., P(τ = ϑ) = 0 for
every F-stopping time ϑ.
Gt(θ) := P(τ > θ|Ft) = ∞
θ
αt(u)η(du) 4
SLIDE 5 Density Hypothesis
The family αt(.) is called the conditional density of τ w.r.t. η given Ft. Note that
- Gt(θ) = E(Gθ|Ft) for any θ ≥ t
- taking, w.l.g., α0(u) = 1, the law of τ is P(τ > θ) =
∞
θ
η(du)
∞ αt(u)η(du) = 1
- For an integrable FT ⊗ σ(τ) r.v. YT (τ), one has, for t ≤ T:
E(YT (τ)|Ft) = E( ∞ YT (u)αT (u)η(du)|Ft)
- The default time τ avoids F-stopping times, i.e., P(τ = ϑ) = 0 for
every F-stopping time ϑ.
Gt(θ) := P(τ > θ|Ft) = ∞
θ
αt(u)η(du) 5
SLIDE 6
Density Hypothesis
By using the density, we adopt an additive point of view to represent the conditional probability of τ Gt(θ) = ∞
θ
αt(u)η(du) In the default framework, the “intensity” point of view is often preferred, and one uses a multiplicative representation as Gt(θ) = exp(− θ λt(u)η(du)) where λt(u) = −∂u ln Gt(u) is the “forward intensity”.
6
SLIDE 7
Computation of conditional expectations
Computation of conditional expectations
Let D = (Dt)t≥0 be the smallest right-continuous filtration such that τ is a D-stopping time, and let G = F ∨ D. Any Gt-measurable r.v. HG
t may be represented as
HG
t = HF t 1{τ>t} + Ht(τ)1
1{τ≤t} where HF
t is an Ft-measurable random variable and Ht(τ) is Ft ⊗ σ(τ)
measurable HF
t = E[HG t 1{τ>t}|Ft]
Gt a.s. on At; = 0 if not .
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SLIDE 8
Computation of conditional expectations
Computation of conditional expectations
Let D = (Dt)t≥0 be the smallest right-continuous filtration such that τ is a D-stopping time, and let G = F ∨ D. Any Gt-measurable r.v. HG
t may be represented as
HG
t = HF t 1{τ>t} + Ht(τ)1
1{τ≤t} where HF
t is an Ft-measurable random variable and Ht(τ) is Ft ⊗ σ(τ)
measurable HF
t = E[HG t 1{τ>t}|Ft]
Gt a.s. on At; = 0 if not .
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SLIDE 9 Computation of conditional expectations
Immersion property In the particular case where αt(θ) = αθ(θ), ∀θ ≤ t
Gt = 1 − t αt(θ)η(dθ) = 1 − t αT (θ)η(dθ) = P(τ > t|FT ) a.s. for any T ≥ t and P(τ > t|Ft) = P(τ > t|F∞). This last equality is equivalent to the immersion property (i.e. F martingales are G-martingales). Conversely, if immersion property holds, then P(τ > t|Ft) = P(τ > t|F∞) hence, the process G is decreasing and the conditional survival functions Gt(θ) are constant in time on [θ, ∞), i.e., Gt(θ) = Gθ(θ) for t > θ.
Gt := P(τ > t|Ft) = ∞
t
αt(u)η(du) 9
SLIDE 10
Dynamic point of view and density process
Dynamic point of view and density process
Regular Version of Martingales One of the major difficulties is to prove the existence of a universal c` adl` ag martingale version of this family of densities.
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SLIDE 11 Dynamic point of view and density process
F-decompositions of the survival process G
- The Doob-Meyer decomposition of the super-martingale G is
given by Gt = 1 + M F
t −
t αu(u)η(du) where M F is the c` adl` ag square-integrable martingale defined as M F
t = −
t
∞ αu(u)η(du)|Ft] − 1.
- Let ζF := inf{t : Gt− = 0}. We define λF
t := αt(t) Gt− for any t ≤ ζF and
let λF
t = λF t∧ζF for any t > ζF. The multiplicative decomposition of
G is given by Gt = LF
t e− t
0 λF sη(ds)
where dLF
t = e t
0 λF sη(ds)dM F
t .
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SLIDE 12 Dynamic point of view and density process
F-decompositions of the survival process G
- The Doob-Meyer decomposition of the super-martingale G is
given by Gt = 1 + M F
t −
t αu(u)η(du) where M F is the c` adl` ag square-integrable martingale defined as M F
t = −
t
∞ αu(u)η(du)|Ft] − 1.
- Let ζF := inf{t : Gt− = 0}. We define λF
t := αt(t) Gt− for any t ≤ ζF and
let λF
t = λF t∧ζF for any t > ζF. The multiplicative decomposition of
G is given by Gt = LF
t e− t
0 λF sη(ds)
where dLF
t = e t
0 λF sη(ds)dM F
t .
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SLIDE 13 Dynamic point of view and density process
Proof: 1) First notice that ( t
0 αu(u)η(du), t ≥ 0) is an F-adapted
continuous increasing process. By the martingale property of (αt(θ), t ≥ 0), for any fixed t, Gt = ∞
t
αt(u)η(du) = E[ ∞
t
αu(u)η(du)|Ft], a.s.. From the properties of the density, 1 − Gt = t
0 αt(u)η(du) and
M F
t := −
t (αt(u) − αu(u))η(du) = E[ ∞ αu(u)η(du)|Ft] − 1. 2) By definition of LF
t and 1), we have
dLF
t = e t
0 λF sη(ds)dGt + e
t
0 λF sη(ds)λF
t Gtη(dt) = e t
0 λF sη(ds)dM F
t ,
which implies the result. △
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SLIDE 14 Dynamic point of view and density process
Relationship with the G-intensity Definition: Let τ be a G-stopping time. The G-compensator ΛG of τ is the G-predictable increasing process such that (1 1{τ≤t} − ΛG
t , t ≥ 0) is a
G-martingale. The G-compensator is stopped at τ, i.e., ΛG
t = ΛG t∧τ.
ΛG coincides, on the set {τ ≥ t}, with an F-predictable process ΛF, i.e. ΛG
t 1{τ≥t} = ΛF t 1{τ≥t}.
- the G-compensator ΛG of τ admits a density given by
λG
t = 1
1{τ>t}λF
t = 1
1{τ>t} αt(t) Gt− . In particular, τ is a totally inaccessible G-stopping time.
- For any t < ζF and T ≥ t, we have αt(T) = E[λG
T |Ft].
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SLIDE 15 Dynamic point of view and density process
Relationship with the G-intensity Definition: Let τ be a G-stopping time. The G-compensator ΛG of τ is the G-predictable increasing process such that (1 1{τ≤t} − ΛG
t , t ≥ 0) is a
G-martingale. The G-compensator is stopped at τ, i.e., ΛG
t = ΛG t∧τ.
ΛG coincides, on the set {τ ≥ t}, with an F-predictable process ΛF, i.e. ΛG
t 1{τ≥t} = ΛF t 1{τ≥t}.
- the G-compensator ΛG of τ is absolutely continuous w.r.t. η with
λG
t = 1
1{τ>t}λF
t = 1
1{τ>t} αt(t) Gt− . In particular, τ is a totally inaccessible G-stopping time.
- For any t < ζF and T ≥ t, we have αt(T) = E[λG
T |Ft].
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SLIDE 16 Dynamic point of view and density process
Relationship with the G-intensity Definition: Let τ be a G-stopping time. The G-compensator ΛG of τ is the G-predictable increasing process such that (1 1{τ≤t} − ΛG
t , t ≥ 0) is a
G-martingale. The G-compensator is stopped at τ, i.e., ΛG
t = ΛG t∧τ.
ΛG coincides, on the set {τ ≥ t}, with an F-predictable process ΛF, i.e. ΛG
t 1{τ≥t} = ΛF t 1{τ≥t}.
- the G-compensator ΛG of τ is absolutely continuous w.r.t. η with
λG
t = 1
1{τ>t}λF
t = 1
1{τ>t} αt(t) Gt− . In particular, τ is a totally inaccessible G-stopping time.
- For any t < ζF and T ≥ t, we have αt(T) = E[λG
T |Ft].
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SLIDE 17 Dynamic point of view and density process
Proof: 1) The G-martingale property of (1 1{τ≤t} − t
0 λG s η(ds), t ≥ 0) is
equivalent to the G-martingale property of (1 1{τ>t}e
t
0 λG sη(ds) = 1
1{τ>t}e
t
0 λF sη(ds), t ≥ 0)
This follows from E[1 1{τ>t}e
t
0 λF sη(ds)|Gs]
= 1 1{τ>s} E[1 1{τ>t}e
t
0 λF sη(ds)|Fs]
Gs = 1 1{τ>s} E[Gte
t
0 λF sη(ds)|Fs]
Gs = 1 1{τ>s} LF
s
Gs , where the last equality follows from the F-local martingale property of
- LF. Moreover, the continuity of the compensator ΛG implies that τ is
totally inaccessible.
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SLIDE 18 Dynamic point of view and density process
2) By the martingale property of density, for any T ≥ t, αt(T) = E[αT (T)|Ft]. Applying 1), we obtain αt(T) = E
GT − |Ft
T |Ft],
∀t < ζF, hence, the value of the density can be partially deduced from the intensity. △
18
SLIDE 19 Dynamic point of view and density process
G-martingale characterization A c` adl` ag process Y G is a G-martingale if and only if there exist an F-adapted c` adl` ag process Y and an Ft ⊗ B(R+)-optional process Yt(.) such that Y G
t = Yt1{τ>t} + Yt(τ)1
1{τ≤t} and that
t
0 Ys(s)αs(s)η(ds), t ≥ 0) is an F-local martingale;
- (Yt(θ)αt(θ), t ≥ θ) is an F-martingale on [θ, ζθ).
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SLIDE 20
Dynamic point of view and density process
Any F-martingale Y F is a G-semimartingale. Moreover, it admits the decomposition Y F
t = M Y,G t
+ AY,G
t
where M Y,G is a G-martingale and AY,G := At1{τ>t} + At(τ)1 1{τ≤t} is an optional process with finite variation given by At = t d[Y F, G]s Gs− and At(θ) = t
θ
d[Y F, α(θ)]s αs(θ) .
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SLIDE 21 Dynamic point of view and density process
Girsanov theorem Let ZG
t = zt1{τ>t} + zt(τ)1
1{τ≤t} be a positive G-martingale with ZG
0 = 1 and let ZF t = ztGt +
t
0 zt(u)αt(u)η(du) be its F projection.
Let Q be the probability measure defined on Gt by dQ = ZG
t dP.
Then, αQ
t (θ) = αt(θ) zt(θ) ZF
t ,
∀t ∈ [θ, ζθ); and: (i) the Q-conditional survival process is defined on [0, ζF) by GQ
t = Gt
zt ZF
t
(ii) the (F, Q)-intensity process is λF,Q
t
= λF
t
zt(t) zt− , η(dt)- a.s.; (iii) LF,Q is the (F, Q)-local martingale LF,Q
t
= LF
t
zt ZF
t
exp t (λF,Q
s
− λF
s)η(ds), t ∈ [0, ζF)
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SLIDE 22 Dynamic point of view and density process
Girsanov theorem Let ZG
t = zt1{τ>t} + zt(τ)1
1{τ≤t} be a positive G-martingale with ZG
0 = 1 and let ZF t = ztGt +
t
0 zt(u)αt(u)η(du) be its F projection.
Let Q be the probability measure defined on Gt by dQ = ZG
t dP.
Then, αQ
t (θ) = αt(θ) zt(θ) ZF
t ,
∀t ∈ [θ, ζθ); and: (i) the Q-conditional survival process is defined on [0, ζF) by GQ
t = Gt
zt ZF
t
(ii) the (F, Q)-intensity process is λF,Q
t
= λF
t
zt(t) zt− , η(dt)- a.s.; (iii) LF,Q is the (F, Q)-local martingale LF,Q
t
= LF
t
zt ZF
t
exp t (λF,Q
s
− λF
s)η(ds), t ∈ [0, ζF)
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SLIDE 23 Dynamic point of view and density process
Girsanov theorem Let ZG
t = zt1{τ>t} + zt(τ)1
1{τ≤t} be a positive G-martingale with ZG
0 = 1 and let ZF t = ztGt +
t
0 zt(u)αt(u)η(du) be its F projection.
Let Q be the probability measure defined on Gt by dQ = ZG
t dP.
Then, αQ
t (θ) = αt(θ) zt(θ) ZF
t ,
∀t ∈ [θ, ζθ); and: (i) the Q-conditional survival process is defined on [0, ζF) by GQ
t = Gt
zt ZF
t
(ii) the (F, Q)-intensity process is λF,Q
t
= λF
t
zt(t) zt− , η(dt)- a.s.; (iii) LF,Q is the (F, Q)-local martingale LF,Q
t
= LF
t
zt ZF
t
exp t (λF,Q
s
− λF
s)η(ds), t ∈ [0, ζF)
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SLIDE 24 Dynamic point of view and density process
Girsanov theorem Let ZG
t = zt1{τ>t} + zt(τ)1
1{τ≤t} be a positive G-martingale with ZG
0 = 1 and let ZF t = ztGt +
t
0 zt(u)αt(u)η(du) be its F projection.
Let Q be the probability measure defined on Gt by dQ = ZG
t dP.
Then, αQ
t (θ) = αt(θ) zt(θ) ZF
t ,
∀t ∈ [θ, ζθ); and: (i) the Q-conditional survival process is defined on [0, ζF) by GQ
t = Gt
zt ZF
t
(ii) the (F, Q)-intensity process is λF,Q
t
= λF
t
zt(t) zt− , η(dt)- a.s.; (iii) LF,Q is the (F, Q)-local martingale LF,Q
t
= LF
t
zt ZF
t
exp t (λF,Q
s
− λF
s)η(ds), t ∈ [0, ζF)
24
SLIDE 25 Dynamic point of view and density process
Proof: For any t ∈ [0, ζF), the Q-conditional probability can be calculated by GQ
t = Q(τ > t|Ft) = E[1
1{τ>t}ZG
t |Ft]
ZF
t
= zt Gt ZF
t
and, for any θ ≤ t, Q(τ ≤ θ|Ft) = EP[1 1{τ≤θ}ZG
t |Ft]
ZF
t
= EP[1 1{τ≤θ}zt(τ)|Ft] ZF
t
= θ
0 zt(u)αt(u)η(du)
ZF
t
. The density process is then obtained by taking derivatives. Finally, we use λF,Q
t
= αQ
t (t)/GQ t− and LF,Q t
= GQ
t e t
0 λF,Q s
η(ds).
25
SLIDE 26
Dynamic point of view and density process
Given a density process, it is possible to construct a random variable τ such that P(τ > θ|Ft) = Gt(θ) as we present now: Let (Ω, A, F, P) and τ a random variable with law P(τ > t) = G0(t) = ∞
t
η(du), independent of F∞, constructed on an extended probability space, and α the given density process. Define dQ|Gt = QG
t dP|Gt
with QG
t = 1
1t<τ 1 Gt ∞
t
αt(u)η((du) + 1 1τ≤tαt(τ) . Then, Q is a probability which coincides with P on Ft and under Q, αQ = α. (Note that this result was obtained by Grorud and Pontier in a finite horizon case)
26
SLIDE 27
Dynamic point of view and density process
Given a density process, it is possible to construct a random variable τ such that P(τ > θ|Ft) = Gt(θ) as we present now: Let (Ω, A, F, P) and τ a random variable with law P(τ > θ) = G0(θ) = ∞
θ
η(du), independent of F∞, constructed on an extended probability space, and α the given density process. Define dQ|Gt = QG
t dP|Gt
with QG
t = 1
1t<τ 1 Gt ∞
t
αt(u)η((du) + 1 1τ≤tαt(τ) . Then, Q is a probability which coincides with P on Ft and under Q, αQ = α. (Note that this result was obtained by Grorud and Pontier in a finite horizon case)
27
SLIDE 28
Dynamic point of view and density process
Given a density process, it is possible to construct a random variable τ such that P(τ > θ|Ft) = Gt(θ) as we present now: Let (Ω, A, F, P) and τ a random variable with law P(τ > t) = G0(t) = ∞
t
η(du), independent of F∞, constructed on an extended probability space, and α the given density process. Define dQ|Gt = ZG
t dP|Gt
with ZG
t = 1
1t<τ 1 Gt ∞
t
αt(u)η(du) + 1 1τ≤tαt(τ) . Then, Q is a probability which coincides with P on Ft and under Q, αQ = α. (Note that this result was obtained by Grorud and Pontier in a finite horizon case)
28
SLIDE 29
Dynamic point of view and density process
Given a density process, it is possible to construct a random variable τ such that P(τ > θ|Ft) = Gt(θ) as we present now: Let (Ω, A, F, P) and τ a random variable with law P(τ > t) = G0(t) = ∞
t
η(du), independent of F∞, constructed on an extended probability space, and α the given density process. Define dQ|Gt = ZG
t dP|Gt
with ZG
t = 1
1t<τ 1 Gt ∞
t
αt(u)η(du) + 1 1τ≤tαt(τ) . Then, Q is a probability which coincides with P on Ft and under Q, αQ = α. (Note that this result was obtained by Grorud and Pontier in a finite horizon case)
29
SLIDE 30
Dynamic point of view and density process
The change of probability measure generated by the two processes zt = (LF
t )−1,
zt(θ) = αθ(θ) αt(θ) provides a model where the immersion property holds true, and where the intensity processes does not change. More generally, the only changes of probability measure for which the immersion property holds with the same intensity process are generated by a process z such that (ztLF
t , t ≥ 0) is an uniformly integrable
martingale.
30
SLIDE 31
Dynamic point of view and density process
Assume that immersion property holds under P. 1) Let the Radon-Nikod´ ym density (ZG
t , t ≥ 0) be a pure jump
martingale with only one jump at time τ. Then, the (F, P)-martingale (ZF
t , t ≥ 0) is the constant martingale equal to 1. Under Q, the intensity
process is λF,Q
t
= λF
t
zt(t) zt , η(dt)-a.s., and the immersion property still holds. 2) The only changes of probability measure compatible with immersion property have Radon-Nikod´ ym densities that are the product of a pure jump positive martingale with only one jump at time τ, and a positive F-martingale.
31
SLIDE 32
Modelling of density process
Modelling of density process
32
SLIDE 33
Modelling of density process
HJM framework and short rate models To model the family of density processes we make references to the classical interest rate models. We suppose in what follows that F is a Brownian filtration.
See D.C. Brody and L. Hugston (2001) for related approach 33
SLIDE 34 Modelling of density process
Suppose that for any θ ≥ 0, the bounded martingale (Gt(θ), t ≥ 0) satisfies dGt(θ) = Zt(θ)dWt where (Zt(θ), t ≥ 0) is an F-predictable process. If the process zt(θ) such that Zt(θ) = θ
0 zt(u)η(du) is bounded by an integrable process, then
- 1. dαt(θ) = −zt(θ)dWt.
- 2. The martingale part in the Doob-Meyer decomposition of G is
given by M F
t = 1 −
t
0 Zs(s)dWs.
See D.C. Brody and L. Hugston (2001) for related approach 34
SLIDE 35 Modelling of density process
Proof: 1) is obvious by definition. 2) is obtained by using previous results and integration by part, in fact, W F
t = 1 −
t η(du) t
u
zs(u)dWs = 1 − t dWs s zs(u)η(du)
Observe in addition that Zt(0) = 0 since Gt(0) = 1 for any t ≥ 0, which implies 2)
See D.C. Brody and L. Hugston (2001) for related approach 35
SLIDE 36
Modelling of density process
We can also consider (Gt(θ), t ≥ 0) in the classical HJM models where its dynamics is given in multiplicative form. We also deduce the dynamics of the forward rate, in both forward and backward forms. The density can then be calculated as αt(θ) = λt(θ)Gt(θ).
See D.C. Brody and L. Hugston (2001) for related approach 36
SLIDE 37 Modelling of density process
For any t, θ ≥ 0, let Ψ(t, θ) = Zt(θ)
Gt(θ) and define ψ(t, θ) by
Ψ(t, θ) = θ
0 ψ(t, u)η(du). Recall the forward rate λt(θ) of τ given by
λt(θ) = −∂θ ln Gt(θ). Then
t
0 Ψ(s, θ)dWs − 1 2
t
0 |Ψ(s, θ)|2ds
t
0 ψ(s, θ)dWs +
t
0 ψ(s, θ)Ψ(s, θ)∗ds;
t
0 λF sη(ds) +
t
0 Ψ(s, s)dWs − 1 2
t
0 |Ψ(s, s)|2ds
dGt(θ) = Zt(θ)dWt 37
SLIDE 38 Modelling of density process
For any t, θ ≥ 0, let Ψ(t, θ) = Zt(θ)
Gt(θ) and define ψ(t, θ) by
Ψ(t, θ) = θ
0 ψ(t, u)η(du). Recall the forward rate λt(θ) of τ given by
λt(θ) = −∂θ ln Gt(θ). Then
t
0 Ψ(s, θ)dWs − 1 2
t
0 |Ψ(s, θ)|2ds
t
0 ψ(s, θ)dWs +
t
0 ψ(s, θ)Ψ(s, θ)∗ds;
t
0 λF sη(ds) +
t
0 Ψ(s, s)dWs − 1 2
t
0 |Ψ(s, s)|2ds
dGt(θ) = Zt(θ)dWt 38
SLIDE 39 Modelling of density process
For any t, θ ≥ 0, let Ψ(t, θ) = Zt(θ)
Gt(θ) and define ψ(t, θ) by
Ψ(t, θ) = θ
0 ψ(t, u)η(du). Recall the forward rate λt(θ) of τ given by
λt(θ) = −∂θ ln Gt(θ). Then
t
0 Ψ(s, θ)dWs − 1 2
t
0 |Ψ(s, θ)|2ds
t
0 ψ(s, θ)dWs +
t
0 ψ(s, θ)Ψ(s, θ)∗ds;
t
0 λF sη(ds) +
t
0 Ψ(s, s)dWs − 1 2
t
0 |Ψ(s, s)|2ds
dGt(θ) = Zt(θ)dWt 39
SLIDE 40 Modelling of density process
For any t, θ ≥ 0, let Ψ(t, θ) = Zt(θ)
Gt(θ) and define ψ(t, θ) by
Ψ(t, θ) = θ
0 ψ(t, u)η(du). Recall the forward rate λt(θ) of τ given by
λt(θ) = −∂θ ln Gt(θ). Then
t
0 Ψ(s, θ)dWs − 1 2
t
0 |Ψ(s, θ)|2ds
t
0 ψ(s, θ)dWs +
t
0 ψ(s, θ)Ψ(s, θ)∗ds;
t
0 λF sη(ds) +
t
0 Ψ(s, s)dWs − 1 2
t
0 |Ψ(s, s)|2ds
dGt(θ) = Zt(θ)dWt 40
SLIDE 41
Modelling of density process
Proof: The process Gt(θ) is the solution of the equation dGt(θ) Gt(θ) = Ψ(t, θ)dWt, ∀ t, θ ≥ 0. Hence 1), from which we deduce immediately 2) by differentiation w.r.t. θ. Now, note that by 1), ln Gt = − t λ0(s)η(ds) + t Ψ(s, t)dWs − 1 2 t |Ψ(s, t)|2ds
41
SLIDE 42
Modelling of density process
Woreover, we have by 2) that t λs(s)η(ds) = t λ0(s)η(ds) − t η(ds) s ψ(u, s)dWu + t η(ds) s ψ(u, s)Ψ(u, s)∗du = t λ0(s)η(ds) − t (Ψ(u, t) − Ψ(u, u))dWu +1 2( t |Ψ(u, t)|2 − t |Ψ(u, u)|2)du. Observe in addition that by definition of the forward rate λt(θ) and then, we have λs(s) = λF
s, which implies 3). Finally, 4) is a direct result
from 2). As a conditional survival probability, Gt(θ) is decreasing on θ, which is equivalent to that λt(θ) is positive. This condition is similar as for the zero coupon bond prices.
42
SLIDE 43
Modelling of density process
In the second approach, we can borrow short rate models for the F-intensity λF of τ, and then obtain the dynamics of conditional probability Gt(T) for T ≥ t. The monotonicity condition of Gt(T) on T is equivalent to the positivity condition on λF.
43
SLIDE 44 Modelling of density process
Example: The non-negativity of λ is satisfied if
- for any θ, the process ψ(θ)Ψ(θ) is non negative, or if ψ(θ) is
non negative;
- for any θ, the local martingale ζt(θ) = λ0(θ) −
t
0 ψs(θ)dWs is
a Dol´ eans-Dade exponential of some martingale, i.e., is solution of ζt(θ) = λ0(θ) + t ζs(θ)bs(θ)dWs , that is, if − t
0 ψs(θ)dWs =
t
0 bs(θ)ζs(θ)dWs. Here the initial condition
is a positive constant λ0(θ).
λt(θ) = λ0(θ) − t
0 ψ(s, θ)dWs +
t
0 ψ(s, θ)Ψ(s, θ)∗ds
44
SLIDE 45 Modelling of density process
Hence, we set ψt(θ) = −bt(θ)ζt(θ) = −bt(θ)λ0(θ) exp t bs(θ)dWs − 1 2 t b2
s(θ)ds
- where λ0 is a positive intensity function and b(θ) is a non-positive
F-adapted process. Then, the family αt(θ) = λt(θ) exp
θ λt(v) dv
where λt(θ) = λ0(θ) − t ψs(θ) dWs + t ψs(θ)Ψs(θ) ds satisfies the required assumptions.
λt(θ) = λ0(θ) − t
0 ψ(s, θ)dWs +
t
0 ψ(s, θ)Ψ(s, θ)∗ds
45
SLIDE 46
Modelling of density process
Other examples
46
SLIDE 47 Modelling of density process
Example: “Cox-like” construction. Here
- λ is a non-negative F-adapted process, Λt =
t
0 λsds
- Θ is a given r.v. independent of F∞ with unit exponential law
- V is a F∞ -measurable non-negative random variable
- τ = inf{t : Λt ≥ ΘV }.
For any θ and t, Gt(θ) = P(τ > θ|Ft) = P(Λθ < ΘV |Ft) = P
V ≥ e−Θ
Let us denote exp(−Λt/V ) = 1 − t
0 ψsds, with
ψs = (λs/V ) exp − s (λu/V ) du, and define γt(s) = E (ψs| Ft). Then, αt(s) = γt(s)/γ0(s).
47
SLIDE 48
Modelling of density process
Backward construction of the density Let ϕ(·, α) be a family of densities on R+, depending of some parameter and X ∈ F∞ a random variable. Then ∞ ϕ(u, X)du = 1 and we can chose αt(u) = E(α∞(u)|Ft) = E(ϕ(u, X)|Ft)
48
SLIDE 49 Modelling of density process
Other example Let β(θ) be a family of non negative F-martingales with expectation equal to 1 dβt(θ) = βt(θ)σt(θ)dWt and set αt(θ) = βt(θ) Lβ
t
where Lβ
t =
∞ βt(θ)η(dθ). Then, α is a family of densities under Q with dQ = Lβ
t dP and
dαt(θ) = αt(θ)(σt(θ) − σt)(dWt − σtdt) where σt =
1 Lβ
t
∞ βt(θ)σt(θ)η(dθ). Note the similarity with normalized and non-normalized filter. See also Brody and Hughston.
49
SLIDE 50 Multidefaults
Multidefaults
Computation of prices in case of multidefaults is now easy
- In a first step, one orders the default
- Computation before the first default are done in the reference
filtration
- Between the first and the second default, one takes as new reference
filtration the filtration generated by the first default and the previous reference filtration, as explained previously for the ”after default” computations
- and we continue till the end
50
SLIDE 51
Multidefaults
Let τ = τ1 ∧ τ2 andσ = τ1 ∨ τ2 and assume that Gt(θ1, θ2) := P(τ > θ1, σ > θ2|Ft) = ∞
θ1
∞
θ2
αt(u, v)dη(u, v). The F-density of τ is given by ατ|F
t
(θ1) = ∞
θ1
αt(θ1, v)η2(dv), a.s.. For any θ2, t ≥ 0, the G(1)-density of σ is given by ασ|G(1)
t
(θ2) = 1 1{τ>t} ∞
t
αt(u, θ2)η1(du) Gτ|F
t
(t) + 1 1{τ≤t} αt(τ, θ2) ατ|F
t
(τ) , a.s..
51
SLIDE 52
Multidefaults
Begin at the beginning, and go on till you come to the end. Then, . . . Lewis Carroll, Alice’s Adventures in Wonderland.
52
SLIDE 53
Multidefaults
Begin at the beginning, and go on till you come to the end. Then, stop. Lewis Carroll, Alice’s Adventures in Wonderland.
53
SLIDE 54
Multidefaults
THANK YOU FOR YOUR ATTENTION
54
