Multiple Defaults and Counterparty Risks by Density …...Multiple Defaults and Counterparty Risks...

Multiple Defaults and Counterparty Risks byDensity Approach

Ying JIAO

Université Paris 7

This presentation is based on joint works with N. El Karoui, M.Jeanblanc and H. Pham

JENA, June 2010 Multiple Defaults and Counterparty Risks using Density

Introduction

Motivation : study the impact of default events on themarkets and the roles of different information using thedensity approach of default

Applications : modelling of multiple defaults and optimalinvestment with counterparty risks

Main ideas :we need the whole term structure of the conditional density— the intensity hypothesis is not enough !decompose suitably the initial problem in the global filtrationusing the before-default and after-default analysiseach decomposed problem is solved in the referencedefault free filtration


The progressive enlargement of filtration plays an essentialrole in the credit risk modelling

On the market (Ω,A,P): the default information σ(τ ∧ t)and the default-free information F = (Ft)t≥0.

The global information G = (Gt)t≥0 Gt = Ft ∨ σ(τ ∧ t).

Key remark: any Gt -measurable random variable Y Gt is

written in the form

Y Gt = Yt1t≤τ + Yt(τ)1t>τ,

where Yt is Ft - measurable and Yt(θ) is Ft ⊗ B(R+)-measurable.


For single credit name, the pricing of a default sensitiveclaim consists of computing the conditional expectationw.r.t. G under some risk-neutral probability.

Most credit derivatives cease to exist once the defaultoccurs. So classically, the pricing is on the set t < τ.

The idea (e.g. Bielecki-Jeanblanc-Rutkowski) is toestablish an explicit relationship between the G and the F

conditional expectations

Key Lemma (Dellacherie-Meyer, Jeulin-Yor): for anyA-measurable r.v. Y ,

11τ>tE[Y |Gt ] = 11τ>tE[Y 11τ>t|Ft ]

P(τ > t |Ft )a.s. (1)

on the set A = St := P(τ > t |Ft) > 0.


Default density approach

To study the impact of a default on the market and themultiple defaults, we are interested in what happens after adefault, i.e. on t ≥ τ.

Adopting the conditional density of default is suitable forthe “after-default” studies.

Similar to the Jacod’s hypothesis in the initial enlargementof filtration.

Density Hypothesis

For any t ≥ 0, there exists a family of Ft ⊗ B(R+) r.v. αt(θ) s.t.for any bounded Borel function f on R+,

E[f (τ)|Ft ] =

∫ ∞

0f (θ)αt(θ)dθ a.s.


“After-default” pricing

Let YT (θ) be a bounded FT ⊗ B(R+) r.v., T , θ ≥ 0. Thenfor any t ≤ T , on Aθ = αt(θ) > 0,

11τ≤t E [YT (τ)|Gt ] = 11τ≤tE

[

Y (T , θ)αT (θ)∣

∣Ft]

αt(θ)

∣

∣

∣

θ=τ, a.s.

(2)

The “after-default” density αt(θ), t ≥ θ is needed.Under the H-hypothesis, P(τ > t |Ft) = P(τ > t |F∞), so

αt(t) = αT (t), ∀T ≥ t .

The before-default density implies the whole termstructure.In the general case, the “after-default” density requiresmore information and the change of probability measure isuseful.


“After-default” density without H-hypothesis

We begin from a probability measure P where theH-hypothesis holds.

Let (Nt := 11τ≤t − ΛGt , t ≥ 0) be the (G,P)-martingale of

pure jump and assume the intensity hypothesis, i.e.,ΛG

t =∫ t

0 λGs ds =

∫ t∧τ

0 λsds where λ is F-adapted.

The intensity satisfies λt = αt(t)/St .

The density of τ under P is

αt(θ) = E[αθ(θ)|Ft ] = E[λθSθ|Ft ] = E[λθe−Λθ |Ft ] t , θ ≥ 0.

Suppose that F is generated by a Brownian motion W .Then W is also a (G,P)-Brownian motion.


Our aim is to find the “after-default” density under aprobability Q where H-hypothesis is not satisfied.By the martingale representation theorem in G (Kusuoka),any positive martingale Q with expectation 1 can be writtenas the solution of a SDE

dQt = Qt−(ΨtdWt + ΦtdNt), Q0 = 1

where Ψ and Φ, Φ > −1, are G-predictable processes.Using the decomposed form

Ψt = ψt11t≤τ+ψt(τ)11t>τ and Φt = φt11t≤τ+φt(τ)11t>τ,

it follows Qt = qt11t<τ + qt(τ)11t≥τ where

qt = exp(

∫ t

0ψudWu −

12

∫ t

0ψ2

udu −

∫ t

0λuφudu

)

, t ≥ 0

qt(θ) = qθ(θ) exp(

∫ t

θ

ψu(θ)dWu −12

∫ t

θ

ψu(θ)2du)

, t ≥ θ.

with qθ(θ) = qθ(1 + φθ).


The restriction of Q on F is given by

QFt = E[Qt |Ft ] = qtSt +

∫ t

0qt(u)λuSudu.

Let Q be the probability measure defined by dQ = QtdP

on Gt . Then

αQt (θ) =

1QF

t

qt(θ)αθ(θ), t ≥ θ

αQt (θ) =

1QF

tEP[αQ

θ (θ)|Ft ]

=1

QFt

EP[qθ(1 + φθ)αθ(θ)|Ft ], t < θ.


Remarks

The pricing problem is composed into a “before-default”one and a “after-default” one.

The initial problem concerns the global information G. Thetwo decomposed pricing formulas are related to thedefault-free information F. The density process isF-adapted.

The filtration F is more familier to work with.

This general decomposition methodology is useful to dealwith multiple defaults and counterparty risks.


Modelling Successive Defaults


Introduction

The before-default and after-default analysis adaptsnaturally to study the ordered default events in a recursivemanner.

For the pricing of credit portfolio derivatives such as basketdefault swaps and CDOs, it suffices to consider theordered defaults.

Links between the top-down models and bottom-upmodels of multiple credit modelling.


Default information

We consider a family of random times (τ1, · · · , τn) on(Ω,A,P), whose increasing-ordered permutation is

σ1 ≤ σ2 · · · ≤ σn.

Denote by Di = (Dit)t≥0 the filtration associated with σi and

by D(i) = (D(i)t )t≥0 := D1 ∨ · · · ∨ Di .

LetG(i) = (G

(i)t )t≥0 := F ∨ D(i)

and define for convenience G(0) = F.

Market full information : G(n)


Ordered defaults and loss process

In the top-down models, one works with the cumulativeloss of the underlying portfolio defined by

Lt :=

n∑

i=1

11τi≤t.

Loss information: DL = (DLt )t≥0 where DL

t = σ(Ls, s ≤ t)including the loss value and the timing of jumps of the lossprocess.

It holds Lt =∑n

i=1 11σi≤t.

The same information flow as the ordered defaults:

D(n)t = DL

t , t ≥ 0.


Joint density

Joint Density Hypothesis for σ = (σ1, · · · , σn): there existsa family of Ft ⊗ B(Rn

+)-measurable functions(ω,u) → αt(u) where u = (u1, · · · ,un) ∈ Rn

+, such that forany bounded Borel function f : Rn

+ → R,

E[f (σ)|Ft ] =

∫

Rn+

f (u)αt (u)du, t ≥ 0.

Remark: α(u) is null outside the set u1 ≤ · · · ≤ un.

The marginal density of σ(k) := (σ1, · · · , σk ) is given by

α(k)t

(

u(k)

)

=

∫

Rn−k+

αt(u)d(u(>k))

where d(

u(>k)

)

= duk+1 · · · dun.


Pricing of credit portfolio derivatives

Payoff of credit portfolio derivativesk th-to-default swap: YT (σ) = 11σk >T

CDO tranche: YT (σ) = (LT − k)+ = (∑n

i=1 11σi≤T − k)+

Let T ≥ t ≥ 0 and YT (σ) be a positiveFT ⊗B(Rn

+)-measurable function on Ω × Rn+, then

E[YT (σ)|G(n)t ] =

n∑

i=0

11σi≤t<σi+1

∫

]t,∞[ d(u(>i)) E[YT (u)αT (u)|Ft ]∫

]t,∞[ d(u(>i))αt (u)

∣

∣

∣

u(i)=σ(i)

where u(i) = (u1, · · · ,ui) and σ(i) = (σ1, · · · , σi).

The prices depend on the number and the occurrencetiming of defaults.


Loss intensity and successive default intensities

The loss intensity is the GL-adapted process λL such that(Lt −

∫ t0 λ

Lsds, t ≥ 0) is a GL-martingale.

The G(k)-intensity of σk is the G(k)-adapted process λk

such that (11σk≤t −∫ t

0 λksds, t ≥ 0) is a G(k)-martingale.

The G(k)-intensity of σk coincides with its G(n)-intensity, itis null outside the set σk−1 ≤ t < σk and is given as

λkt = 11σk−1≤t<σkλ

k ,Ft

(

σ(k−1)

)

where λk ,Ft

(

u(k−1)

)

is Ft ⊗B(Rk−1+ )-measurable.

The loss intensity is the sum of the intensities of σk , i.e.

λLt =

n∑

k=1

λkt , a.s..


Joint density and successive intensities

Under a probability P where H-hypothesis holds between F

and G(n), then for any θ ∈ Rn+ such that θ1 ≤ · · · ≤ θn and

any 0 ≤ t ≤ θn,

αt(θ) = E

[

n∏

i=1

λi ,Fθi

(

θ(i−1)

)

exp

−

∫ θi

θi−1

λi ,Fu

(

θ(i−1)

)

du

|Ft

]

a.s..

If t > θn, then αt(θ) = αθn(θ).

To obtain the whole term structure of the joint density in thegeneral case, it needs a change of probability measure onG(n) and more information.


Bottom-up vs top-down

The ordered joint density of σ can be deduced from thenon-ordered one of τ = (τ1, · · · , τn). .

Denote by βt(θ), t ≥ 0, θ = (θ1, · · · , θn) ∈ Rn+, the joint

density of τ .

For any θ ∈ Rn+ such that θ1 ≤ · · · ≤ θn,

αt(θ1, · · · , θn) = 11θ1≤···≤θn

∑

Π

βt(

θΠ(1), · · · , θΠ(n)

)

where (Π(1), · · · ,Π(n)) is a permutation of (1, · · · ,n).

In particular, if τ is exchangeable, then

αt(θ1, · · · , θn) = 11θ1≤···≤θn n! βt(θ1, · · · , θn).


Joint density models

Two important points:compatibility between the joint probability property and themartingale propertydescribe the correlation structure in a dynamic manner

An example: how to diffuse a copula model.When t = 0, G0(u) = exp(−(

∑ni=1 u2

i )1/2), which is a c.d.f.of n uni-exponential r.v. linked by a Clayton copula.The conditional probability :

Gt(u) = exp(

−(

n∑

i=1

u2i M i

t )1/2 − At

)

where At = 18

∫ t0

2+X 1/2s

X 3/2s

d 〈X〉s and Xs =∑n

i=1 u2i M i

t

with M i being positive F-martingales such that M i0 = 1 and

⟨

M i ,M j⟩

t > 0 if i 6= j.


Optimal Investment

with Counterparty Risks


Introduction

The counterparty default risks is often modelled in theliterature by the contagion intensity jump (Jarrow-Yu,Crepey-Jeanblanc-Zargari, Herbertsson).

We consider a utility maximization problem where theasset value jumps at the default of the counterparty.

The aim is to analyze the counterparty risk in optimalinvestment and measure the impact of default on tradingstrategy.


Asset model

Asset value S governed by

dSt = St−(µtdt + σtdWt − γtdDt), 0 ≤ t ≤ T ,

where µ, σ > 0 and γ are G-predictable processessatisfying

∫ T0 |µt

σt|2dt +

∫ T0 |σt |

2dt <∞ a.s., Dt = 11τ≤t and−∞ < γt < 1.Since γ < 1, the asset continues to exist after default.St = SF

t 1t<τ + Sdt (τ)1t≥τ , where SF is the price in the

market before default:

dSFt = SF

t

(

µFt dt + σF

t dWt)

, 0 ≤ t ≤ T ,

and Sd (θ) is the price after default at τ = θ:

dSdt (θ) = Sd

t (θ)(

µdt (θ)dt + σd

t (θ)dWt)

, θ < t ≤ T ,

with the initial value Sdθ (θ) = SF

θ (1 − γθ).


Portfolio and wealth process

A trading strategy is a G-predictable process π,representing the proportion of wealth invested in the riskyasset, with corresponding wealth process:

dXt = πtXt−dSt

St−, 0 ≤ t ≤ T .

By writing π in the form: πt = πFt 1t≤τ + πd

t (τ)1t>τ ,

Xt = X Ft 1t<τ + X d

t (τ)1t≥τ ,

where X F is the wealth process in the market beforedefault:

dX Ft = πF

t X Ft

dSFt

SFt, 0 ≤ t ≤ T ,


Portfolio and wealth process after default

X d(θ) is the wealth in the market after default at τ = θ:

dX dt (θ) = πd

t (θ)X dt (θ)

dSdt (θ)

Sdt (θ)

θ < t ≤ T

X dθ (θ) = X F

θ (1 − πFθγθ).

We say that a trading strategy π is admissible, denote by π∈ A, if

∫ T

0|πtσt |

2dt < ∞, and πtγt < 1 a.s.

For π ∈ A, the wealth process remains strictly positive.


The utility maximization problem

Given a utility function U strictly increasing, strictlyconcave and C1 on (0,∞), and satisfying the Inadaconditions U ′(0+) = ∞, U ′(∞) = 0, we consider

V0 = supπ∈A

J0(π) := supπ∈A

E[U(XT )]. (3)

Incomplete market due to jump default process. Existenceand uniqueness of a solution by duality method fromgeneral results of Kramkov-Schachermayer

Problem (3) recently studied by dynamic programmingmethods and BSDE in the G-filtration under H-hypothesisby Lim-Quenez. Similar problem byAnkirchner-Blanchet-Scalliet-Eyraud-Loisel


Derivation of the decomposition

Density approach → Explicit solutions with qualitative andquantitative descriptions of the optimal trading strategy,especially w.r.t. the Merton (default-free) strategy.

By the law of conditional expectations, and under thedensity hypothesis, we get

J0(π) = E[

E[U(XT )|FT ]]

= E[

U(X FT )P[τ > T |FT ] + E[U(X d

T (τ))1τ≤T |FT ]]

= E

[

U(X FT )ST +

∫ T

0U(X d

T (θ))αT (θ)dθ]

,

where ST = P[τ > T |FT ] =∫ ∞

T αT (θ)dθ is the conditionalsurvival probability at T .


Decomposition theorem

Let us introduce the value-function process of the“after-default" optimization problem: for(θ, x) ∈ [0,T ]× (0,∞),

V dθ (x) = ess sup

πd (θ)∈Ad (θ)

E[

U(X d ,xT (θ))αT (θ)

∣

∣Fθ

]

. (4)

Theorem

V0 = supπF∈AF

E

[

U(X FT )ST +

∫ T

0V d

θ (X Fθ (1 − πF

θ γθ))dθ]

. (5)


Two optimization problems in the complete market.For the after-default problem,

Time-inconsistent control problem where the coefficients(µd (θ), σd (θ)) depend on the initial time θdensity term αT (θ) weighting the utility UWe adapt a martingale duality method for solving (4)

For the global before-default problem,a dynamic programming type relation (5)backward resolution from the “after-default" one, bothsolved with respect to F

The optimal trading strategy is given by

πt = πFt 1t≤τ + πd

t (τ)1t>τ , 0 ≤ t ≤ T ,

where πF is an optimal control to (4), and πd (θ) is anoptimal control to V d


θγθ)).


Solution to the after-default utility maximization

The (local) martingale density of the after-default market:

Zt(θ) = exp(

−

∫ t

θ

µdu(θ)

σdu (θ)

dWu−12

∫ t

θ

∣

∣

∣

µdu (θ)

σdu (θ)

∣

∣

∣

2du

)

, θ ≤ t ≤ T .

The optimal value process is

V dθ (x) = E

[

U(

I(

yθ(x)ZT (θ)

αT (θ)

))

αT (θ)∣

∣

∣Fθ

]

.

The corresponding optimal wealth process is equal to:

X d ,xt (θ) = E

[ZT (θ)

Zt(θ)I(

yθ(x)ZT (θ)

αT (θ)

)∣

∣

∣Ft

]

, θ ≤ t ≤ T , (6)

where I = (U ′)−1, and yθ(x) > 0 is Fθ ⊗ B((0,∞))-measurable random variable solution to X d ,x

θ (θ) = x .


The global before-default optimization problem

In the case of CRRA utility functions: U(x) = xp

p for p < 1,p 6= 0,

we solve explicitly

V0 = supπF∈AF

E

[

U(X FT )ST +

∫ T

0V d


θγθ))dθ]

for whichV d

θ (x) = U(x)K pθ

with

Kθ =(

E

[

αT (θ)(ZT (θ)

αT (θ)

)−q∣

∣

∣Fθ

])1q

and q =p

1 − p.


Theorem(1) V0 = U(X0)Y0, where Y is strictly positive, and (Y , φ) is thesmallest (resp. largest) solution when p > 0 (resp. p < 0), inLb

+(F) × L2loc(W ) to the BSDE:

Yt = GT +

∫ T

tf (θ,Yθ, φθ)dθ −

∫ T

tφθdWθ, 0 ≤ t ≤ T , (7)

where

f (t ,Yt , φt) = p ess supπγt<1

[

(µFt Yt + σF

t φt)π −1 − p

2Yt |πσ

Ft |

2

+ K pt

(1 − πγt)p

p

]

,

Remark. We recover Merton value with: GT = 1, Kt = 0.


(2) The optimal strategy is given by:

πFt = arg max

πγt <1

[

(µFt Yt + σF

t φt)π −1 − p

2Yt |πσ

Ft |

2

+ K pt

(1 − πγt)p

p

]

, 0 ≤ t ≤ T ,


Numerical illustrations

Tests with CRRA utility function U(x) = xp

p , p < 1, p 6= 0.

At default: γ constant, τ independent of F and τ ∼ exp(λ):αt(θ) = α(θ) and S(t) = P(τ > t) = e−λt .

For γ > 0 (loss at default), we take µd (θ) = µF θT and

σd(θ) = σF(

2 − θT

)

, θ ∈ [0,T ].For γ < 0 (gain at default), we take µd (θ) = µF

(

2 − θT

)

andσd(θ) = σF(2 − θ

T ), θ ∈ [0,T ],

µF = 0.03, σF = 0.2, T = 1

The after-default optimal strategy: πd(θ) = µd (θ)(1−p)|σd (θ)|2

.

No closed-form solution for the global problem: numericalresolution by Howard algorithm.


Optimal stragegy in function of the jump size γ

Figure: Optimal strategy vs Merton: p = 0.2, λ = 0.01 and 0.3respectively.

−1 −0.5 0 0.5 1−3

−2

−1

0

1

2

3

4

gamma

Pi Merton

PiF with lambda=0.01

PiF with lambda=0.3


References

N. El Karoui, M. Jeanblanc and Y. Jiao (2009), Whathappens after the default: the conditional densityapproach, Stochastic Processes and their Applications,120(7), 1011-1032.

N. El Karoui, M. Jeanblanc and Y. Jiao (2010), Modellingsuccessive default events, working paper.

Y. Jiao (2009), Multiple defaults and contagion risks withglobal and default-free information, working paper.

Y. Jiao and H. Pham (2009), Optimal investment withcounterparty risks: a default-density model approach, toappear in Finance and Stochastics.


Thank you very much for your attention!


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