Two default times

Monique Jeanblanc Evry University

with N. El Karoui, Y. Jiao

T. Bielecki, D. Coculescu, Y. Le Cam, M. Rutkowski, S. Valchev Risk Day, March 19, LSE LONDON

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Credit Risk Model

• A reference filtration F and a random time τ are given

• H is the filtration generated by the process H_t = 11_τ≤t

• G_t = F_t ∨ H_t

The filtration G enjoys the Minimal assumption: if A ∈ G_t, there exists A ∈ F_t such that

A ∩ {t < τ} = A ∩ {t < τ}

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It follows that, if (Y_t, t ≥ 0) is a G-adapted process, there exists an F-adapted process (Y_t, t ≥ 0) such that

Y_t11_t<τ = Y_t11_t<τ . The survival hazard process is denoted by S:

P(τ > t|F_t) = S_t

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• for any integrable F_T-measurable r.v. X

E(11_{{T <τ}}X | G_t) = 11_{t<τ} 1

S_t E(S_TX | F_t) .

• for any positive predictable process h

E(h_τ11_{{τ <T}}|G_t) = h_τ11_{{τ <t}} − 11_{{τ >t}} 1 S_t E(

_T

t

h_udS_u|F_t).

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The process (S_t, t ≥ 0) is a F-supermartingale and admits a Doob-Meyer decomposition

S_t = M_t^τ − A^τ_t

where M^τ is an F-martingale, and A^τ is an F-predictable increasing process.

The process

H_t − Λ_t∧τ = H_t −

_t∧τ

0

dA^τ_s S_s−

is a G-martingale.

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If A is absolutely continuous w.r.t. Lebesgue measure, H_t − Λ_t∧τ = H_t −

_t∧τ

0

a^τ_s

S_s− ds and λ_t = _S^aτt

t− is called the intensity.

In a general setting, the knowledge of the intensity does not allow to compute the conditional expectations

E(11_{{T <τ}}X | G_t)

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Martingale density

For any T, the process S_t(T) ^def= P(τ > T|Ft) is an F-martingale.

We assume that there exists a family of strictly positive processes (α_t(u), t ≥ 0) such that

S_t(T) =

_∞

T

α_t(u)du

For any u, the process (α_t(u), t ≥ 0) is an F-martingale, and can be interpreted as

α_t(u)du = P(τ ∈ du|F_t)

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In the case where F is a Brownian filtration generated by W d_tα_t(u) = ψ_t(u)dW_t

which, from Ventzell theorem, implies that (if S is continuous) dS_t = Ψ_tdW_t − α_t(t)dt

where Ψ_t = _∞

t ψ_t(u)du.

This equality gives the Doob-Meyer decomposition of S. In particular, we obtain that

H_t −

_t∧τ

0

α_s(s) S_s ds is a G-martingale.

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In a general setting, if the family of processes α exists, and if S is continuous, the process

H_t − Λ_t∧τ where

dΛ_t = α_t(t)

S_t dt = α_t(t) _∞

t α_t(u)dudt , is a G-martingale.

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We can now extend the previous result. Let X ∈ F_T and h a Borel function. Then,

E(Xh(T ∧ τ)|G_t)11_{{τ >t}} = E _t^∞ Xh(u ∧ T)α_T(u)du|F_t

S_t 11_{{τ >t}}

E[Xh(T ∧ τ)|Gt]11_{τ≤t} = E

Xh(s)α_T(s)

α_t(s) Ft

s=τ11_{τ≤t}

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Intuitively, with Y = Xh(T ∧ τ)

E(Y |G_t)11_{τ≤t} = 11_{τ≤t}f_t(τ)

where f_t(u) is a family of F-adapted processes. It follows that, for u < t f_t(u) = E(Y |Ft, τ = u) = E(Y 11_τ∈du|Ft)

P(τ ∈ du|F_t)

= E(Xh(u)11_τ∈du|F_t)

P(τ ∈ du|F_t = E(Xh(u)P(τ ∈ du|F_T)|F_t) P(τ ∈ du|F_t)

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More generally, if (Y (t, u), t ≥ 0) is a family of F-adapted processes, then

E(Y (T, τ)|G_t)11_{{τ >t}} = E _t^∞ Y (T, u)α_T(u)du|F_t

S_t 11_{{τ >t}}

E[Y (T, τ)|G_t]11_{τ≤t} = E

Y (T, T)α_T(s)

α_t(s) F_t

s=τ11_{τ≤t}

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Particular case: Immersion property (or (H) hypothesis) The filtration F is said to be immersed in G if F-martingales are G-martingales. This condition is equivalent to

S_t = P(τ > t|F_t) = P(τ > t|F_∞) = P(τ > t|F_s),∀t, s; t ≤ s . In that case, the process S is non-increasing, and H_t − Λ_t∧τ is a G-martingale, where Λ_t = −lnS_t.

There is equivalence between F is immersed in G and for any u ≥ 0, the martingale (α_t(u), t ≥ 0) is constant after u.

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COX process model: Immersion

Let λ be an F-adapted non-negative process, Θ a random variable with exponential law, independent of F∞ and

τ ^def= inf{t :

_t

0

λ_sds ≥ Θ}

In that case,

S_t = exp − _t

0

λ_udu

,

and immersion property holds. In particular, P(τ > θ|Ft) = exp

− _θ

0

λ_udu

= S_θ, ∀θ < t .

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Imperfect information: Immersion Let τ = inf {t, X_t ≤ b (t)} .

where X_t = Ψ (B_t, t) where B is a Brownian motion and Ψ(, t) is invertible. The filtration F is the natural filtration of the observed process is Y

dY_t = µ₁ (Y_t, t)dt + σ (Y_t, t)dB_t + s(Y_t, t) dW_t Y₀ = y₀ .

It can be proved that F is immersed in F ∨ H and numerical approximation of the process S are done.

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Partial Information: No immersion Assume that

V_t = ve^σ(Wt+νt) = ve^σXt and let, for α < v

τ = inf{t : V_t ≤ α} = inf{t : X_t ≤ a}

The reference filtration F is the filtration of the observations of V at discrete times t₁,· · ·t_n where t_n ≤ t < t_n+1, i.e.,

F_t = σ(V_t1,· · ·, V_tn, t_i ≤ t) The process S_t = P(τ > t|F_t) is not decreasing.

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Two default times: no Immersion

Let τ_i, i = 1,2 two default times, H_tⁱ = 11_τi≤t, Hⁱ the natural filtration of Hⁱ. Let F = H², G = F ∨ H¹.

Let G(t, s) = P(τ₁ > t, τ₂ > s) and S_t⁽¹⁾ = P(τ₁ > t|F_t).

Then,

dS_t⁽¹⁾ = ∂₂G(t, t)

∂₂G(0, t) − G(t, t) G(0, t)

dM_t²

+ H_t²∂₁h⁽¹⁾(t, τ₂) + (1 − H_t²)∂₁G(t, t) G(0, t)

dt

where M² is the F-martingale M_t² = H_t² −

_t∧τ2

0

P(τ₂ ∈ ds)

P(τ₂ > s) and h⁽¹⁾(t, s) = _∂^∂2G(t,s)

2G(0,s)

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Generalized Cox Processes

Let Θ be independent of F∞ and V ∈ F∞, with V > 0.

τ = inf{t : Λ_t ≥ ΘV } Then,

S_t(T) = P(τ > T|F_t) = P(Λ_T > ΘV |F_t) = P exp −Λ_T

V ≥ exp−Θ F_t

,

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Let us denote Ψ_t = exp−Λ_t/V = ^∞

t ψ_sds, with ψ_s = λ_s

V exp− _s

0

λ_u V du, and define, for any pair (s, t)

γ(s, t) = E(ψ_s| F_t) . We have for any T and t :

S_t(T) =

_∞

T

γ(s, t)ds =

_∞

T

α_t(s)η(ds) where η([0, t]) = P(τ ≤ t) is the law of the variable τ and α_t(s) = γ(s, t)/γ(s,0).

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Two default times

Let τ₁, τ₂ two default times and σ₁ < σ₂ the ordered defaults. We suppose that the conditional probability admits a density

P(σ₁ > u₁, σ₂ > u₂|F_t) =

_∞

u1

dv₁

_∞

u2

dv₂ p_t(v₁, v₂)

In particular _∞

u

p_t(u, v)dv = α⁽¹⁾_t (u) where α⁽¹⁾_t (u) is the density of P(σ₁ > u|F_t).

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Let H_t¹ = 11_σ1≤t and H¹ its natural filtration. Then P(σ₂ > t|Ft ∨ H¹_t) = 11_σ1>t + 11_σ1<t

_∞

t p_t(σ₁, v)dv α⁽¹⁾_t (σ₁)

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The process P(σ² > θ|F_t ∨ H¹_t) admits a density:

P(σ² > θ|F_t ∨ H_t¹) =

_∞

θ

α⁽²⁾_t (s)ds where

α⁽²⁾_t (θ) = 11_σ1>t

_∞

t du p_t(u, θ) _∞

t du _∞

u dvp_t(u, v) + 11_σ1≤t p_t(σ₁, θ) α⁽¹⁾_t (σ₁)

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The intensity processes can be computed: the process 11_σ2≤t − _t∧σ2

0 λ⁽²⁾_s ds where

λ⁽²⁾_t = 11_[σ1,σ2](t) p_t(σ₁, t) _∞

t p_t(σ₁, v)dv is a martingale

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As a general result, we get, for G = H ∨ H¹ ∨ H² E(Xh(σ₁, σ₂)|G_t)11_t<σ1 = 11_t<σ1 E(_∞

t du_{1 ∞}

u1 du₂Xh(u₁, u₂)p_T(u₁, u₂)|F_t) _∞

t du_{1 ∞}

t du₂ p_t(u₁, u₂)

E(Xh(σ₁, σ₂)|G_t)11_σ1≤t<σ2 = 11_σ1≤t<σ2 E( _∞

u1∨t du₂Xh(u₁, u₂)p_T(u₁, u₂)|F_t)

_∞

t∨u1 du₂ p_t(u₁, u₂)

u1=σ1

E(Xh(σ₁, σ₂)|G_t)11_σ2≤t = 11_σ2≤tE(Xh(u₁, u₂)p_T(u₁, u₂)|F_t) p_t(u₁, u₂)

u1=σ1,u2=σ2

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Enlargement of filtration formula

In general, τ is not an honest time. However, it is possible to prove that any F-martingale is a G-semi-martingale. If X is a F martingale, if S admits a density, then

X_t = X_t +

_t∧τ

0

dX, M^τ _s S_s− +

_t

t∧τ

g(τ, dv),

= X_t +

_t∧τ

0

_∞

s

η(dv)dX, α^v _s S_s− +

_t

t∧τ

g(τ, dv)

where X is a G-martingale and

g(u, dv) = dα_·(u), X _v α_v(u)

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t ≥ as

S_t = L_tD_t

where L is a local martingale and D a predictable non-increasing process.

From S_t = M_t^τ −A^τ_t and S_t = L_tD_t it follows from Yoeurp theorem that dS_t = L_t−dD_t + D_tdL_t

hence

dD_t = − 1

L_t⁻ dA^τ_t = −D_t− dA^τ_t

S_t⁻ = −D_t− dΛ_t and D_t = E(−Λ)_t. If Λ is continuous D_t = exp(−Λ_t).

Of course, if immersion property holds, L ≡ 1.

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The price of a DZC with maturity T is (we assume null interest rate) D(t, T) = P(τ > T|Gt) = 11_{τ >t} 1

S_tE(S_T|Ft) ^def= 11_{τ >t}D(t, T )

Using the multiplicative decomposition S_t = D_tL_t, and assuming that (L_t, t ≤ T) is a martingale, the price of the defaultable payoff X11_{T <τ} is

E(X11_{T <τ}|Gt) = 11_t<τ 1

S_tE(XS_T|Ft) = 11_t<τ 1

D_t E(XD_T|Ft) where dP|_Ft = L_tdP|_Ft

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To end the talk, the sentence

Le livre n’est pas termin´e.

La fin n’a pas ´et´e ´ecrite, elle n’a jamais ´et´e trouv´ee.

M. Duras, L’´et´e 80.

has to be changed into

Le travail n’est pas termin´e.

La fin n’a pas ´et´e ´ecrite, elle n’a pas encore ´et´e trouv´ee.

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