information asymmetry in pricing of credit derivatives. · information asymmetry in pricing of...

IntroductionThe informational structure

Pricing under the historical probabilityRisk neutral pricingNumerical example

INFORMATION ASYMMETRY IN PRICING OFCREDIT DERIVATIVES.

Caroline HILLAIRET, CMAP , Ecole PolytechniqueJoint work with Ying JIAO, LPMA , Université Paris VII

Enlargement of Filtrations and Applications to Finance and InsuranceJena, May 31-June 4, 2010

This research is part of the Chair Financial Risks of the Risk Foundation sponsored by Société Générale, the Chair Derivatives of the Futuresponsored by the Fédération Bancaire Française, and the Chair Finance and Sustainable Development sponsored by EDF and Calyon

1/ 36Caroline HILLAIRET, CMAP , Ecole Polytechnique Joint work with Ying JIAO, LPMA , Université Paris VIIInformation asymmetry in Pricing of Credit Derivatives



OUTLINE OF THE TALK

1 INTRODUCTIONIntroductionThe framework

2 THE INFORMATIONAL STRUCTUREFull InformationProgressive and delayed InformationNoisy full Information

3 PRICING UNDER THE HISTORICAL PROBABILITYPricing for the full informationPricing for the progressive and delayed informationPricing for the noisy full information

4 RISK NEUTRAL PRICING

5 NUMERICAL EXAMPLE




IntroductionThe framework

OUTLINE

1 INTRODUCTIONIntroductionThe framework

2 The informational structureFull InformationProgressive and delayed InformationNoisy full Information

3 Pricing under the historical probabilityPricing for the full informationPricing for the progressive and delayed informationPricing for the noisy full information

4 Risk neutral pricing5 Numerical example





INTRODUCTION

There exist two main approaches in the credit risk modelling: thestructural approach and the reduced-form approach.The two approaches are related by the accessibility of information

on the underlying asset value process (Duffie-Lando 2001,Jeanblanc-Valchev 2005, Coculescu-Geman-Jeanblanc 2006,Guo-Jarrow-Zeng 2008), delayed or noisy observation of the underlyingprocess,on the default threshold (El Karoui 1999, Giesecke-Goldberg 2008),constant or random default barrier.

AIM study the impact of the information concerning the default threshold inthe credit analysis, in addition to the partial observation of the underlyingasset process.





INTRODUCTION

(Ω,A, P) probability space.(Xt)t≥0 positive continous-time process : asset value of the firm.F = (Ft = σ(Xs, s ≤ t))t≥0 satisfying the usual conditionsDefault threshold L, random variable in A.Default time τ

τ = inft : Xt ≤ L where X0 > L

with the convention that inf ∅ = +∞

structural approach : L is a constant or a deterministic function L(t),then τ is an F-stopping time.reduced-form approach : L is unknown and the default arrives in amore surprising way.





We introduce the decreasing process X∗ defined as

X∗t = infXs, s ≤ t.

Then τ = inft : X∗t = L.

If L is independent of F∞:denote the distribution function of L by FL(x) := P(L ≤ x), then

P(τ > t|F∞) = P(X∗t > L|F∞) = FL(X∗

t ).

Note that the (H)-hypothesis is satisfied, that is,

P(τ > t|F∞) = P(τ > t|Ft).

We may recover the Cox-process model.





OUR FRAMEWORKThe manager of the firm has full information concerning the underlyingasset of the firm and he chooses the default barrier.⇒ L is a random variable set by the manager of the firm.

The investors on the market have different levels of information on thefundamental process (Xt)t≥0 and on the default threshold L.

Full information knowledge of (Xt)t≥0 and L (manager of the firm)Noisy full information knowledge of (Xt)t≥0 + noisy signal on L +default time observableProgressive information knowledge of (Xt)t≥0 + default time observableDelayed information delayed information on (Xt)t≥0 + default timeobservable





PRICING FRAMEWORK

different level of information ⇒

different filtration (Ht)t

different pricing measure Q

Evaluate a credit-sensitive derivative claim of maturity T :the value process at time t < τ ∧ T is given by

Vt = RtEQ

[CR−1

T 11τ>T +

∫ T

t11τ>uR−1

u dGu + Zτ 11τ≤TR−1τ

∣∣∣Ht

](1)

whereC (FT-measurable) represents the payment at the maturity T (if τ ≥ T)G (F-adapted) represents the dividend paymentZ (F-predictable) represents the recovery payment at the default time τ

R is the discount factor process.8/ 36

Caroline HILLAIRET, CMAP , Ecole Polytechnique Joint work with Ying JIAO, LPMA , Université Paris VIIInformation asymmetry in Pricing of Credit Derivatives



Full InformationProgressive and delayed InformationNoisy full Information

OUTLINE

1 IntroductionIntroductionThe framework

2 THE INFORMATIONAL STRUCTUREFull InformationProgressive and delayed InformationNoisy full Information







FULL INFORMATION

We assume that the filtration F is generated by a Brownian motion B.The manager knows the threshold L ω-wise from the beginning. Thus hisinformation is given as the initial enlargement of the filtration F with respectto L :

ASSUMPTION ( HS)

GM = (GMt )t≥0 with GM

t := Ft ∨ σ(L).

We assume that P(L ∈ ·|Ft)(ω) ∼ P(L ∈ ·) for all t for P almost all ω ∈ Ω.

Assumption (HS) is satisfied if L is independent of F∞.Assumption (HS) is equivalent to: there exists a unique probabilitymeasure PL, equivalent to P, identical to P on F∞, and under which forany t ≥ 0, Ft and σ(L) are independent (Grorud-Pontier 1998).





Let p be the density of probability : PL ∈ dx|Ft = pt(x)PL ∈ dx.dpt(x) = pt(x)ρM

t (x)dBt (Jacod 1985).

EPL [dP

dPL |GMt ] = pt(L).

EXAMPLE

Example in a finite time horizon T < T ′ :

L = ls11]0,a[(XT′) + li11]a,+∞[(XT′), li < ls

PROPOSITION

We define the probability density (assumed to be a (GM, P) martingale)

YM(L) = E(−∫ .

0ρM

s (L)(dBs − ρMs (L)ds)).

Any (F, Q)-local martingale is an (GM , YM(L)Q)-local martingale.11/ 36

Caroline HILLAIRET, CMAP , Ecole Polytechnique Joint work with Ying JIAO, LPMA , Université Paris VIIInformation asymmetry in Pricing of Credit Derivatives




THE PROGRESSIVE INFORMATION

This is the standard information structure in the credit risk analysis.Investors know at each time t whether or not default has occured. Thus theirinformation is given as the progressive enlargement of filtration of F withrespect to τ :

ASSUMPTION ( HN )

G = (Gt)t≥0 with Gt = Ft ∨ σ(τ ≤ s, s ≤ t).

REMARK

If L is independent of F∞, the standard (H)-hypothesis is satisfied:every (F, P) local martingale is also a (G, P) local martingale.





THE DELAYED INFORMATION

FDt :=

F0 if t ≤ δ(t),Ft−δ(t) if t > δ(t),

constant delay time model : δ(t) = δ

discrete observation model : δ(t) = t − t(m)i , t(m)

i ≤ t < t(m)i+1 where

0 = t(m)0 < t(m)

1 < · · · < t(m)m = T are the only discrete dates on which

the information is renewed.The delayed information is the progressive enlargement of filtration of Lwith respect to FD:

ASSUMPTION ( HD)

GD = (GDt )t≥0 with GD

t = FDt ∨ σ(τ ≤ s, s ≤ t).





NOISY FULL INFORMATION

The investor observes the value of the firm (Xt)t and receives a noisy signal(Ls = f (L, εs))s≥0 on the threshold L.

ASSUMPTION ( HN )

GI = (GIt )t≥0 with GI

t = ∩u>t(Fu ∨ σ(f (L, εs) ∨ σ(τ ≤ s, s ≤ u))

• f : R2 → R is a given measurable function.• ε = εt, t ≥ 0 is independent of F∞ ∨ σ(L).• P(L ∈ ·|Ft)(ω) ∼ P(L ∈ ·) for all t for P almost all ω ∈ Ω.

EXAMPLE

Example in a finite time horizon T < T ′ :Ls = L + Wg(T′−s) with W an independent Brownian motion andg : [0, T ′] → [0,∞[ a strictly increasing bounded function with g(0) = 0





F It := ∩u>t(Fu ∨ σ(f (L, εs)s ≤ u))

GI is the progressive enlargement of filtration of FI with respect to τ .

ρIt := EP[ρM

t (L)|F It ]

PROPOSITION

We define the probability density (assumed to be a (FI , P) martingale)

Y I = E(−∫ .

0ρI

s(dBs − ρIsds)).

Then any (F, Q)-local martingale is an (FI , Y IQ)-local martingale.

The following relations hold:GM ⊃ GI ⊃ G ⊃ GD.




Pricing for the full informationPricing for the progressive and delayed informationPricing for the noisy full information

OUTLINE



3 PRICING UNDER THE HISTORICAL PROBABILITYPricing for the full informationPricing for the progressive and delayed informationPricing for the noisy full information






AIM Compute the price of the contingent claim (C, G, Z) with maturity Tgiven different sources of information :

Vt = RtEP

[CR−1

T 11τ>T +

∫ T

t11τ>uR−1

u dGu + Zτ11τ≤TR−1τ

∣∣∣Ht

]

−→ P is the historical probability measure

−→ (Ht)t≥0 describes the accessible information for the investors.(Ht)t≥0 = (GM

t )t≥0 for the full information,(Ht)t≥0 = (GI

t )t≥0 for a noisy full information.(Ht)t≥0 = (Gt)t≥0 for the progressive information.(Ht)t≥0 = (GD

t )t≥0 for the delayed information.





GMt = Ft ∨ σ(L) for the manager (full information). We assume (HS).

LEMMA

For any θ ≥ t and any positive Fθ ⊗ B(R)-measurable function φθ(·), onehas

EP[φθ(L)11τ>θ | GMt ] =

1pt(L)

EP[φθ(x)pθ(x)11X∗

θ>x | Ft]x=L

where pt(x)(ω) =dPL

tdPL (ω, x), PL

t (ω, dx) being a regular version of theconditional law of L given Ft and PL being the law of L.





PROOF

For θ ≥ t, using the fact thatFθ and σ(L) are independent under PL

EPL [ dPdPL |GM

t ] = pt(L)

PL is identical to P on F∞

We have

EP[φθ(L)11τ>θ | GMt ] = EP[φθ(L)11X∗

θ>L | Ft ∨ σ(L)]

= pt(L)−1EPL [φθ(L)pθ(L)11X∗

θ>L|Ft ∨ σ(L)]

= pt(L)−1EPL [φθ(x)pθ(x)11X∗

θ>x|Ft]x=L

= pt(L)−1EP[φθ(x)pθ(x)11X∗

θ>x|Ft]x=L.





PROPOSITION

We define FMt (x) := pt(x)11X∗

t >x. The value process of the contingent claim(C, G, Z) given the full information (GM

t )t≥0 is

VMt = 11τ>t

VMt (L)

pt(L)

where

VMt (L) = RtEP

[CR−1

T FMT (x)+

∫ T

tFM

s (x)R−1s dGs−

∫ T

tZsR−1

s dFMs (x)

∣∣∣∣Ft

]

x=L.





PRICING FOR THE PROGRESSIVE INFORMATION

Gt = Ft ∨ σ(τ ≤ s, s ≤ t)

PROPOSITION

For any θ ≥ t and any Fθ-measurable random variable φθ, one has

EP[φθ11τ>θ | Gt] = 11τ>tEP[φθSθ | Ft]

St.

where St := P(τ > t | Ft). The value process given the progressiveinformation flow G is

Vt = 11τ>tRtSt

EP

[R−1

T ST C +

∫ T

tR−1

u SudGu −∫ T

tR−1

u ZudSu

∣∣∣∣Ft

].





PRICING FOR THE DELAYED INFORMATION

GDt = FD

t ∨ σ(τ ≤ s, s ≤ t)

PROPOSITION

For any θ ≥ t and any Fθ-measurable random variable φθ, one has

EP[φθ11τ>θ | GDt ] = 11τ>t

EP[φθSθ|FDt ]

EP[St|FDt ]

The value process for a delay-informed investor is

VDt =

11τ>t

E[St|FDt ]

EP

[ RtRT

STC +

∫ T

t

RtRu

SudGu −∫ T

t

RtRu

ZudSu

∣∣∣∣FDt].





PRICING FOR THE NOISY FULL INFORMATION

GIt = F I

t ∨ σ(τ ≤ s, s ≤ t) with F It = ∩u>t(Fu ∨ σ(Ls, s ≤ u)).

LEMMA

We assume (HN) with Lt = L + εt, εt being a continuous process withbackwardly independent increments and whose marginal has density qt.Then, for any θ ≥ t and for any Fθ-measurable φθ, one has

EP[φθ11τ>θ|F It ] =

∫EP[φθpθ(l)11X∗

θ>l|Ft]qt(Lt − l)PL(dl)∫

Rpt(l)qt(Lt − l)PL(dl) .





PROOF (1)

Let Aθ ∈ Fθ and h a bounded measurable function. Using the independenceof Fθ ∨ σ(L) and ε, we have

EP

(h(L)11Aθ

|F It)

= EP (h(L)11Aθ|Ft ∨ σ(Lt) ∨ σ((εt − εs), s ≤ t)))

= EP (h(L)11Aθ|Ft ∨ σ(L + εt))

Then for U ∈ B(R2),

P ((L, L + εt) ∈ U|Ft) =

∫

R211U(l, x)qT−t(x − l)PL

t (dl)dx.

ThusEP[h(L)|F I

t ] =

∫h(l)qt(Lt − l)PL(dl)∫

Rpt(l)qt(Lt − l)PL(dl)





PROOF (2)

If θ > t, we use the following successive conditional expectations

EP

`

φθ11τ>θ|Ft ∨ σ(L + εt)´

= EP

`

EP

`

φθ11τ>θ|Ft ∨ σ(L + εt) ∨ σ(L)´

|Ft ∨ σ(L + εt)´

.

Using the fact that ε is independent to FT ∨ σ(L), we have

EP

(φθ11τ>θ|Ft ∨ σ(L + εt) ∨ σ(L)

)= EP

(φθ11X∗

θ>l|Ft ∨ σ(L)

)=: ht(L)

where ht(L) = 1pt(L) [EP(φθpθ(l)11X∗

θ>l|Ft)]l=L corresponds to the full

information. Therefore

EP[φθ11τ>θ|F It ] =

∫EP[φθpθ(l)11X∗

θ>l|Ft]qt(Lt − l)PL(dl)∫

Rpt(l)qt(Lt − l)PL(dl) .





PROPOSITION

We assume (HN) with Lt = L + εt, εt being a continuous process withbackwardly independent increments and whose marginal has density qt. Thevalue process for the noisy full information flow GI is given by

V It =

11τ>t∫R

FMt (l)qt(Lt − l)PL(dl)

∫VM

t (l)qt(Lt − l)PL(dl)

where VM and FM are defined for the full information.

FMt (x) := pt(x)11X∗

t >x.

VMt (L) = RtEP

[CR−1

T FMT (x)+

∫ T

tFM

s (x)R−1s dGs−

∫ T

tZsR−1

s dFMs (x)

∣∣∣∣Ft

]

x=L.




OUTLINE




4 RISK NEUTRAL PRICING

5 Numerical example




RISK NEUTRAL PROBABILITIES

We assume that a pricing probability Q is given with respect to thefiltration F of the fundamental process X (for example, Q such that X isan (F, Q) local martingale).We want to focus on the change of probability measures due to thedifferent sources of informations and on its impact on the pricing ofcredit derivatives, ⇒ without loss of generality, we take the historicalprobability P to be the benchmark pricing probability Q on F and G.The pricing probability for the manager is QM where dQM

dQ= YM(L)

with YM(L) = E(−∫ .

0 ρMs (L)(dBs − ρM

s (L)ds))The pricing probability for the noisy full information is QI wheredQI

dQ= Y I with Y I = E(−

∫ .

0 ρIs(dBs − ρI

sds))We also take Q as the pricing probability for the delayed information




RISK NEUTRAL PRICING FOR THE FULL INFORMATION

PROPOSITION

1) Define FQM

t (l) = 11X∗

t >l. Then the value process of a credit sensitiveclaim (C, G, Z) for the manager’s full information under the risk neutralprobability measure QM is given by

VQM

t = 11τ>tRt EP

[CR−1

T FQM

T (x)

+

∫ T

tFQM

s (x)R−1s dGs −

∫ T

tZsR−1

s dFQM

s (x) |Ft]

x=L.




2) Let ε be a continuous process with backwardly independent incrementssuch that the probability law of εt has a density qt(·) w.r.t. the Lebesguemeasure. Let µt,θ be the probability law of εθ − εt. Then the value processfor the insider’s noisy full information under QI is given by

VQI

t =11τ>t∫

RFM


∫VQ

I


where

VQI

t (l) = RtEP

[CR−1

T FIt,T(u, l) +

∫ T

tFI

t,θ(u, l)R−1θ dGθ

−∫ T

tR−1

θ ZθdFIt,θ(u, l)|Ft

]u=Lt

,

FIt,θ(u, l) = E

( ∫ θ

t

∫ρI

s(u + y)µt,θ(dy)dBs

)−1FM

θ (l).




OUTLINE




4 Risk neutral pricing5 NUMERICAL EXAMPLE




BINOMIAL MODEL

Let L be a (Ft)t≥0-independent random variable taking two valuesli, ls ∈ R, li ≤ ls such that

P(L = li) = α, P(L = ls) = 1 − α (0 < α < 1).

We suppose that the asset values process X satisfies the Black Scholesmodel :

dXtXt

= µdt + σdBt, t ≥ 0.

For t ≥ 0 and h, l > 0,

EP(11X∗

t >l − 11X∗

t+h>l|Ft) = 11X∗

t >l

(Φ

(−Yt − νhσ√

h)

+ e2νσ−2YtΦ(−Yt + νh

σ√

h))

=: 11X∗

t >lΦt,h(l)where Φ is the standard Gaussian cumulative distribution function,

Yt = νt + σBt + ln X0

l and ν = µ − 12σ2




VALUE PROCESS OF A DEFAULTABLE BOND

Numerical values in the following simulations : li = 1, ls = 3, α = 12 .

0 0.2 0.4 0.6 0.8 10.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

managerprogressivedelayednoisy

0 0.2 0.4 0.6 0.8 11

2

3

4

5

6

7

8firm value

dynamic price of the defaultable bond firm value

FIGURE: L = li




0 0.2 0.4 0.6 0.8 10.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1managerprogressivedelayednoisy

0 0.2 0.4 0.6 0.8 13

4

5

6

7

8

9firm value


FIGURE: L = ls




We compare the dynamic price of the defaultable bond, for the samescenario of the firm value but depending on the level of the threshold fixedby the manager.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.4

0.5

0.6

0.7

0.8

0.9

1

manager infmanager supprogressivebruite infbruite sup

0 0.2 0.4 0.6 0.8 14

4.5

5

5.5

6

6.5

7

7.5

8

valeur de firm


FIGURE: L = ls or L = li




REFERENCES

Coculescu, D., Geman, H., Jeanblanc, M., 2006. Valuation of Defaultsensitive claims under imperfect Information, Finance and Stochastics.Corcuera, J.M., Imkeller, P., Kohatsu-Higa, A., Nualart, D., 2004.Additional utility of insiders with imperfect dynamical information,Finance and Stochastics, 8, 437-450.Duffie, D., Lando, D. 2001. Term Structures of Credit Spreads withIncomplete Accounting Information, Econometrica, 69, 633-664.Giesecke K, Goldberg L. R., 2008. The Market Price of credit risk : theimpact of Asymmetric Information.Grorud, A., Pontier, M., 1998. Insider Trading in a Continuous TimeMarket Model, International Journal of Theorical and Applied FinanceGuo, X., Jarrow, R., Zeng, Y., 2008. Credit risk with incompleteinformation, forthcoming Mathematics of Operations Research. 1,331-347.


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