Representation theorem

Credit Risk IV

T.R. Bielecki, M. Jeanblanc and M. Rutkowski

MLV, M2 janvier 2007

IV. Hazard process Approach

1. General case 2. (H)-Hypothesis

3. Representation theorem 4. Partial information

General case

The model

Two kinds of information :

the information from the asset’s prices, denoted as (F_t, t ≥ 0)

the information from the default time τ modeled by the filtration H_t generated by the default process H_t = 11_τ≤t.

We denote by G_t ^def= F_t ∨ H_t.

Key lemma

Any G_t-random variable is equal, on the set {τ > t}, to an F_t-measurable random variable.

We denote by F_t = P(τ ≤ t|F_t) the conditional law of τ given the information F_t, and G_t = 1 − F_t.

Let X be an F_T-measurable integrable r.v. Then, E(X11_{T <τ}|G_t) = 11_{{τ >t}}E(X11_{{τ >T}}|F_t)

E(11_{{τ >t}}|F_t) = 11_{{τ >t}}e^ΓtE(Xe^−ΓT|F_t).

where Γ_t ^def= −ln(1 − F_t) = −lnG_t

Let h be an F-predictable process. Then,

Γ _T

Martingales

We assume for simplicity that F is continuous.

(i) The process L_t ^def= (1 − H_t)e^Γt is a G-martingale.

(ii) The process M_t ^def= H_t − Γ_t∧τ is a G -martingale as soon as F (or Γ) is increasing.

The submartingale F admits a decomposition as F = Z + A where Z is a martingale and A a predictable increasing process.

(iii) The process

M_t ^def= H_t −

_t∧τ

dA_u 1 − F

Proofs: The process L_t = (1 − H_t)e^Γt is a G-martingale.

From the key lemma, for t > s

E(L_t|G_s) = E(11_{{τ >t}}e^Γt|G_s) = 11_{{τ >s}}e^ΓsE(11_{{τ >t}}e^Γt|F_s)

= 11_{{τ >s}}e^ΓsE(E(11_{{τ >t}}|Ft)e^Γt|Fs) = 11_{{τ >s}}e^ΓsE(e^−Γte^Γt|Fs)

= 11_{{τ >s}}e^Γs = L_s

The process M_t = H_t − Γ_t∧τ is a G -martingale as soon as F (or Γ) is increasing.

From integration by parts formula :

dL_t = (1 − H_t)e^ΓtdΓ_t − e^ΓtdH_t and the process M_t = H_t − Γ(t ∧ τ) can be written

M_t ^def=

]0,t]

dH_u −

]0,t]

(1 − H_u)dΓ_u = −

]0,t]

e^−ΓudL_u and is a G-martingale since L is G-martingale.

The process

M_t = H_t −

_t∧τ

0

dA_u 1 − F_u is a G-martingale.

Let s < t. We give the proof in two steps, using the Doob-Meyer decomposition of F as F_t = Z_t + A_t.

First step: we prove

E(H_t|G_s) = H_s + 11_s<τ 1

1 − F_sE(A_t − A_s|F_s) Indeed,

E(H_t|G_s) = 1 − P(t < τ|G_s) = 1 − 11_s<τ 1

1 − F_sE(1 − F_t|F_s)

= 1 − 11_s<τ 1

1 − F_sE(1 − Z_t − A_t|F_s)

= 1 − 11_s<τ 1

1 − F_s(1 − Z_s − A_s − E(A_t − A_s|F_s))

= 1 − 11_s<τ 1

1 − F (1 − F_s − E(A_t − A_s|F_s))

In a second step, we prove that, setting Λ_t = _t

0(1 − H_s)_1−F^dAs

s , E(Λ_t∧τ|Gs) = Λ_s∧τ + 11_s<τ 1

1 − F_sE(A_t − A_s|Fs) From the key formula,

E(Λ_t∧τ|G_s) = Λ_s∧τ11_τ≤s + 11_s<τ 1

1 − F_sE

_∞

s

Λ_t∧udF_u|F_s

= Λ_s∧τ11_τ≤s + 11_s<τ 1

1 − F_sE

_t

s

Λ_udF_u +

_∞

t

Λ_tdF_u|F_s

= Λ_s∧τ11_τ≤s + 11_s<τ 1

1 − F_sE

_t

s

Λ_udF_u + Λ_t(1 − F_t)|F_s

We now use IP formula, using that Λ is bounded variation and continuous d(Λ_t(1 − F_t)) = −Λ_tdF_t + (1 − F_t)dΛ_t = −Λ_tdF_t + dA_t

hence

_t

s

Λ_udF_u + Λ_t(1 − F_t) = −Λ_t(1 − F_t) + Λ_s(1 − F_s) + A_t − A_s + Λ_t(1 − F_t)

= Λ_s(1 − F_s) + A_t − A_s

From

E(Λ_t∧τ|G_s) = Λ_s∧τ11_τ≤s + 11_s<τ 1

1 − F_sE

_t

s

Λ_udF_u + Λ_t(1 − F_t)|F_s

and _t

s

Λ_udF_u + Λ_t(1 − F_t) = Λ_s(1 − F_s) + A_t − A_s it follows that

E(Λ_t∧τ|G_s) = Λ_s∧τ11_τ≤s + 11_s<τ 1

1 − F_sE(Λ_s(1 − F_s) + A_t − A_s|F_s)

= Λ_s∧τ + 11_s<τ 1

1 − F_sE(A_t − A_s|Fs) .

If A is absolutely continuous wrt the Lebesgue measure, there exists an F-adapted process γ, called the intensity such that the process

H_t −

_t∧τ

0

γ_udu = H_t −

_t

0

(1 − H_u)γ_udu is a G-martingale. The process γ satisfies

γ_t = lim

h→0

1 h

P(t < τ < t + h|F_t) P(t < τ|F_t)

Let V and R be F-predictable processes. The process V_t = V_t11_{t<τ} + R_τ11_{τ≤t}

is a G-martingale if and only if the process V_te^−Γt +

_t

0

R_ue^−ΓudΓ_u is an F-martingale

Proof: The direct part comes from the fact that if V is a G-martingale, then EQ(V_t|Ft) is an F-martingale. The converse is an immediate

application of the Key Lemma

Let P be the ex-dividend price process of a claim which delivers R_τ at

default time and pays a cumulative coupon C till the default time, i.e. the discounted cum-dividend process

β_tP_t + 11_{τ≤t}β_τR_τ +

_t∧τ

0

β_udC_u

is a G-martingale. Let P_t be the predefault price of the process P, i.e., ˜P is F-predictable and P_t = 11_{t<τ}P_t. Then the process

P_t^∗ = α_tP_t +

_t

0

α_sdC_s +

_t

0

R_uα_u dΓ_u is an F-martingale.

Conversely, if V is an F-predictable process such that the process α_tV_t + _t

0 α_sdC_s + _t

0 R_uα_u dΓ_u is an F-martingale, then (the discounted cum-dividend) process

β_tV_t11_{t<τ} + 11_{τ≤t}β_τR_τ +

_t∧τ

0

β_udC_u is a G-martingale.

Proof: This is an application of the previous Lemma .

Computation in a restricted filtration

Let F ⊂ F and G_t = F_t ∨ H_t. From

F_t = P(τ ≤ t|F_t) we deduce

F_t = P(τ ≤ t|Ft) = E(F_t|Ft)

The computation of the intensity is more difficult, the F- intensity in the

(H) Hypothesis

Complete model case

Let S be a semi-martingale on (Ω,G,P) such that there exists a unique probability Q, equivalent to P on F_T, where F_t = F_t^S = σ(S_s, s ≤ t) such that (S_t = S_tR_t,0 ≤ t ≤ T) is an F^S-martingale under the probability Q.

We assume that there exists a probability ˜Q, equivalent to P on GT such that (S_t,0 ≤ t ≤ T) is a G-martingale under the probability ˜Q.

Then, any (F,Q)-martingale is a (G,Q)-martingale and the Q F

Definition and Properties of immersion

We shall now examine the immersion property (or (H)-hypothesis) which reads:

(H) Every F square-integrable martingale is a G square integrable martingale.

This hypothesis implies that the F-Brownian motion remains a Brownian motion in the enlarged filtration and that every F-local martingale is a G-local martingale .

Assume that G = F ∨ H, where F is an arbitrary filtration and H is

generated by the process H_t = 11_{τ≤t}. Then the following conditions are equivalent to the hypothesis (H).

(i) For any t ∈ IR₊, we have

P(τ ≤ t| F_t) = P(τ ≤ t | F∞).

(ii) For any t ∈ IR₊, the σ-fields F∞ and Gt are conditionally independent given F_t under P, that is,

EP(ξ η | F_t) = EP(ξ | F_t)EP(η | F_t)

for any bounded, F -measurable random variable ξ and bounded,

Change of a probability measure

Kusuoka shows, by means of a counter-example, that the hypothesis (H) is not invariant with respect to an equivalent change of the underlying

probability measure, in general.

Let Q be a probability measure equivalent to P on (Ω,G_t) for every t ∈ IR₊, with the associated Radon-Nikod´ym density process η. If the density process η is F-adapted then we have

Q(τ ≤ t| F_t) = P(τ ≤ t| F_t)

for every t ∈ IR₊. Hence, the hypothesis (H) is also valid under Q and the F-intensities of τ under Q and under P coincide.

Proof:

Q(τ ≤ t| F_t) = EP(η_t11_{τ≤t} | F_t)

E_P(η_t | F_t) = EP(η_t11_{τ≤t} | F∞) E_P(η_t | F_∞)

= EP(η_∞11_{τ≤t} | F∞)

E_P(η_∞ | F_∞) = P(τ ≤ t| F∞).

Stochastic Barrier Suppose that

P(τ ≤ t|F_∞) = 1 − e^−Γt

where Γ is an arbitrary continuous strictly increasing F-adapted process.

There exists a random variable Θ, independent of F_∞, with exponential law of parameter 1, such that τ ^law= inf {t ≥ 0 : Γ_t > Θ}. In fact Θ ^def= Γ_τ.

Proof: : Suppose that

P(τ ≤ t|F∞) = 1 − e^−Γt

where Γ is an arbitrary continuous strictly increasing F-adapted process.

Let us set Θ ^def= Γ_τ. Then

{t < Θ} = {t < Γ_τ} = {C_t < τ},

where C is the right inverse of Γ, so that Γ_Ct = t. Therefore P(Θ > u|F_∞) = e^−ΓCu = e^−u.

We have thus established the required properties, namely, the probability law of Θ and its independence of the σ-field F_∞. Furthermore,

τ = inf{t : Γ_t > Γ_τ} = inf{t : Γ_t > Θ}.

Representation theorem

Kusuoka establishes the following representation theorem. Under (H), any G-square integrable martingale admits a representation as the sum of a stochastic integral with respect to the Brownian motion and a stochastic integral with respect to the discontinuous martingale M.

Suppose that hypothesis (H) holds under P and that any F-martingale is continuous. Then, the martingale M_t^h = E_P(h_τ| G_t), where h is an

F-predictable process such that E(h_τ) < ∞, admits the following decomposition

M_t^h = m^h₀ +

_t∧τ

0

e^Γudm^h_u +

]0,t∧τ]

(h_u − J_u) dM_u,

where m^h is the continuous F-martingale m^h_t = EP

∞ 0

h_udF_u | F_t ,

J_t = e^Γt(m^h_t − _t

0 h_udF_u) and M is the discontinuous G-martingale

Proof: : We know that

M_t^h = E(h_τ| G_t)

= 11_{τ≤t}h_τ + 11_{{τ >t}}e^ΓtE ^∞

t

h_udF_u F_t

= 11_{τ≤t}h_τ + 11_{{τ >t}}e^Γt

m^h_t − _t

0

h_udF_u

=

_t

0

h_udH_u + 11_{{τ >t}}J_t .

From the facts that Γ is an increasing process m^h a continuous martingale

Partial information

As pointed out by Jamshidian, “one may wish to apply the general theory perhaps as an intermediate step, to a subfiltration that is not equal to the default-free filtration. In that case, F rarely satisfies hypothesis (H)”.

Information at discrete times Assume that

dV_t = V_t(µdt + σdW_t), V₀ = v

i.e., V_t = ve^σ(Wt+νt) = ve^σXt. The default time is assumed to be the first hitting time of α with α < v, i.e.,

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

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

Ft = σ(V_t1,· · ·, V_tn, t_i ≤ t)

The process F_t = P(τ ≤ t|F_t) F is continuous and increasing in [t_i, t_i+1[ but is not increasing.

Lemma 0.1 The process ζ defined by ζ_t =

i,ti≤t

∆F_ti.

is an F-martingale.

The Doob-Meyer decomposition of F is

From

P(inf

s≤tX_s > z) = Φ(ν, t, z), where

Φ(ν, t, z) = N

νt − z

√t

− e^2νzN

z + νt

√t

, for z < 0, t > 0,

= 0, for z ≥ 0, t ≥ 0, Φ(ν,0, z) = 1, for z < 0

we obtain (we skip the parameter ν in the definition of Φ) for t₁ < t < t₂ and X_t1 > a

Another example, related with Parisian stopping times is presented in C¸ etin et al.

Delayed information

Guo et al. suggested to start form a structural model with delayed

information, i.e. the reference filtration is Ft = σ(S_u, u ≤ t − δ). In that case, (H) hypothesis is not satisfied.

Intensity approach

In the so-called intensity approach, the default time τ is a G-stopping time. The intensity is defined as any non-negative process λ, such that

M_t ^def= H_t −

_t∧τ

0

λ_sds

is a G-martingale.

The existence of the intensity relies on the fact that H is a sub-martingale and can be written as M + A where M is a martingale M and A a

predictable increasing process. The increasing process A is such that A_t11_t≥τ = A_τ11_t≥τ.

The intensity exists only if τ is a totally inaccessible stopping time.

We emphasize that, in that setting the intensity is not well defined after

If the process Y_t = E

X exp

− _T

0 λ_udu |G_t is continuous at time τ, then, setting L_t = 11_{t<τ}e^Γt

E(X11_{{T <τ}}|G_t) = 11_{t<τ}E

X exp

−

_T

t

λ_udu

|G_t

= L_tY_t

If Y is not continuous

E(X11_{{T <τ}}|G_t) = L_tY_t − E(∆Y_τ11_{τ <T}|G_t).

It can be mentioned that the continuity of the process depends on the choice of λ after time τ.

If the process Y is not continuous, then setting U_t = L_tY_t = 11_t<τ exp

_t

0

λ_sds

E

X exp

−

_T

0

λ_udu

G_t

we have U_T = X11_{{T <τ}} and

dU_t = L_t−dY_t + Y_t−dL_t + d[L, Y ]_t = L_t−dY_t + Y_t−dL_t + ∆L_t∆Y_t and

E(U_T|G_t) = E(X11_{{T <τ}}|G_t) = U_t − E(∆Y_τ11_{τ <T}|G_t).

Then, for any X ∈ G_T :

E(X11_{T <τ}|G_t) = 11_{τ >t}

e^ΛtE(e^−ΛTX|G_t) − E(e^Λτ ∆Y_τ11_{τ <T}|G_t) where Y_t = E(X exp (−Λ_T)|Gt) and Λ_t = _t

0 λ_udu

CDS Price, General case

The ex-dividend price of a credit default swap, with a rate process κ and a protection payment δ_τ at default, equals, for every t ∈ [s, T],

S_t(κ) = E_Q

11_{t<τ≤T}δ_τ G_t − E_Q

11_{t<τ}

_τ∧T

t

κ_sds G_t ,

= 11_{t<τ} 1

G_t E_Q

−

_T

t

δ_u dG_u +

_∞

t

dG_u

_u∧T

t

κ_v dv F_t

.

We now assume that (H) hypothesis holds between F and G, that is F-martingales are G-martingales. Then, F is increasing and the process

M_t = H_t −

_t∧τ

0

γ_u du, with γ_tdt = ^dF_G^t

t is a G-martingale.

The dynamics of the ex-dividend price S_t(κ) are

dS_t(κ) = −S_t−(κ)dM_t+(1−H_t)B_tG⁻¹_t dm_t+(1−H_t)(r_tS_t(κ)+κ−δ_tγ_t)dt,

where m is the (Q,F)-martingale given by

m_t = E_Q

_T

0

B_u⁻¹δ_uG_uγ_u du − κ

_T

0

B_u⁻¹G_u duF_t

.

Hedging defaultable claims

Our aim is to hedge

Y = 11_{T≥τ}Z_τ + 11_{{T <τ}}X.

using two CDS with maturities T_i, rates κ_i and protection payment δⁱ. We assume r = 0. Let ζ_tⁱ defined as

mⁱ_t = E_Q

_T

0

δ_uⁱ G_uγ_u du − κ_i

_T

0

G_u du F_t

, dmⁱ_t = ζ_tⁱdW_t

Assume that there exist F-predictable processes φ¹, φ² such that

2

i=1

φⁱ_t

δ_tⁱ − S_tⁱ(κ_i)

= Z_t − y_t,

2

i=1

φⁱ_tζ_tⁱ = ζ_t,

where y is given by

y_t = 1

G_t E_Q

−

_T

t

Z_u dG_u + G_TX F_t

. Let φ⁰_t = V_t(φ) − ₂

i=1 φⁱ_tS_tⁱ(κ_i), where the process V (φ) is given by dV_t(φ) =

2

i=1

φⁱ_t

dS_tⁱ(κ_i) + dD_tⁱ

with the initial condition V₀(φ) = E_Q(Y ). Then the self-financing trading