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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
IntroductionThe informational structure
Pricing under the historical probabilityRisk neutral pricingNumerical example
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
2/ 36Caroline HILLAIRET, CMAP , Ecole Polytechnique Joint work with Ying JIAO, LPMA , Université Paris VIIInformation asymmetry in Pricing of Credit Derivatives
IntroductionThe informational structure
Pricing under the historical probabilityRisk neutral pricingNumerical 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
3/ 36Caroline HILLAIRET, CMAP , Ecole Polytechnique Joint work with Ying JIAO, LPMA , Université Paris VIIInformation asymmetry in Pricing of Credit Derivatives
IntroductionThe informational structure
Pricing under the historical probabilityRisk neutral pricingNumerical example
IntroductionThe framework
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.
4/ 36Caroline HILLAIRET, CMAP , Ecole Polytechnique Joint work with Ying JIAO, LPMA , Université Paris VIIInformation asymmetry in Pricing of Credit Derivatives
IntroductionThe informational structure
Pricing under the historical probabilityRisk neutral pricingNumerical example
IntroductionThe framework
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.
5/ 36Caroline HILLAIRET, CMAP , Ecole Polytechnique Joint work with Ying JIAO, LPMA , Université Paris VIIInformation asymmetry in Pricing of Credit Derivatives
IntroductionThe informational structure
Pricing under the historical probabilityRisk neutral pricingNumerical example
IntroductionThe framework
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.
6/ 36Caroline HILLAIRET, CMAP , Ecole Polytechnique Joint work with Ying JIAO, LPMA , Université Paris VIIInformation asymmetry in Pricing of Credit Derivatives
IntroductionThe informational structure
Pricing under the historical probabilityRisk neutral pricingNumerical example
IntroductionThe framework
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
7/ 36Caroline HILLAIRET, CMAP , Ecole Polytechnique Joint work with Ying JIAO, LPMA , Université Paris VIIInformation asymmetry in Pricing of Credit Derivatives
IntroductionThe informational structure
Pricing under the historical probabilityRisk neutral pricingNumerical example
IntroductionThe framework
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
IntroductionThe informational structure
Pricing under the historical probabilityRisk neutral pricingNumerical example
Full InformationProgressive and delayed InformationNoisy full Information
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
9/ 36Caroline HILLAIRET, CMAP , Ecole Polytechnique Joint work with Ying JIAO, LPMA , Université Paris VIIInformation asymmetry in Pricing of Credit Derivatives
IntroductionThe informational structure
Pricing under the historical probabilityRisk neutral pricingNumerical example
Full 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).
10/ 36Caroline HILLAIRET, CMAP , Ecole Polytechnique Joint work with Ying JIAO, LPMA , Université Paris VIIInformation asymmetry in Pricing of Credit Derivatives
IntroductionThe informational structure
Pricing under the historical probabilityRisk neutral pricingNumerical example
Full InformationProgressive and delayed InformationNoisy full Information
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
IntroductionThe informational structure
Pricing under the historical probabilityRisk neutral pricingNumerical example
Full InformationProgressive and delayed InformationNoisy full Information
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.
12/ 36Caroline HILLAIRET, CMAP , Ecole Polytechnique Joint work with Ying JIAO, LPMA , Université Paris VIIInformation asymmetry in Pricing of Credit Derivatives
IntroductionThe informational structure
Pricing under the historical probabilityRisk neutral pricingNumerical example
Full InformationProgressive and delayed InformationNoisy full Information
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).
13/ 36Caroline HILLAIRET, CMAP , Ecole Polytechnique Joint work with Ying JIAO, LPMA , Université Paris VIIInformation asymmetry in Pricing of Credit Derivatives
IntroductionThe informational structure
Pricing under the historical probabilityRisk neutral pricingNumerical example
Full InformationProgressive and delayed InformationNoisy full Information
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
14/ 36Caroline HILLAIRET, CMAP , Ecole Polytechnique Joint work with Ying JIAO, LPMA , Université Paris VIIInformation asymmetry in Pricing of Credit Derivatives
IntroductionThe informational structure
Pricing under the historical probabilityRisk neutral pricingNumerical example
Full InformationProgressive and delayed InformationNoisy full Information
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.
15/ 36Caroline HILLAIRET, CMAP , Ecole Polytechnique Joint work with Ying JIAO, LPMA , Université Paris VIIInformation asymmetry in Pricing of Credit Derivatives
IntroductionThe informational structure
Pricing under the historical probabilityRisk neutral pricingNumerical example
Pricing for the full informationPricing for the progressive and delayed informationPricing for the noisy full information
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
16/ 36Caroline HILLAIRET, CMAP , Ecole Polytechnique Joint work with Ying JIAO, LPMA , Université Paris VIIInformation asymmetry in Pricing of Credit Derivatives
IntroductionThe informational structure
Pricing under the historical probabilityRisk neutral pricingNumerical example
Pricing 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.
17/ 36Caroline HILLAIRET, CMAP , Ecole Polytechnique Joint work with Ying JIAO, LPMA , Université Paris VIIInformation asymmetry in Pricing of Credit Derivatives
IntroductionThe informational structure
Pricing under the historical probabilityRisk neutral pricingNumerical example
Pricing for the full informationPricing for the progressive and delayed informationPricing for the noisy full 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.
18/ 36Caroline HILLAIRET, CMAP , Ecole Polytechnique Joint work with Ying JIAO, LPMA , Université Paris VIIInformation asymmetry in Pricing of Credit Derivatives
IntroductionThe informational structure
Pricing under the historical probabilityRisk neutral pricingNumerical example
Pricing for the full informationPricing for the progressive and delayed informationPricing for the noisy full information
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.
19/ 36Caroline HILLAIRET, CMAP , Ecole Polytechnique Joint work with Ying JIAO, LPMA , Université Paris VIIInformation asymmetry in Pricing of Credit Derivatives
IntroductionThe informational structure
Pricing under the historical probabilityRisk neutral pricingNumerical example
Pricing for the full informationPricing for the progressive and delayed informationPricing for the noisy full information
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.
20/ 36Caroline HILLAIRET, CMAP , Ecole Polytechnique Joint work with Ying JIAO, LPMA , Université Paris VIIInformation asymmetry in Pricing of Credit Derivatives
IntroductionThe informational structure
Pricing under the historical probabilityRisk neutral pricingNumerical example
Pricing for the full informationPricing for the progressive and delayed informationPricing for the noisy full information
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
].
21/ 36Caroline HILLAIRET, CMAP , Ecole Polytechnique Joint work with Ying JIAO, LPMA , Université Paris VIIInformation asymmetry in Pricing of Credit Derivatives
IntroductionThe informational structure
Pricing under the historical probabilityRisk neutral pricingNumerical example
Pricing for the full informationPricing for the progressive and delayed informationPricing for the noisy full information
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].
22/ 36Caroline HILLAIRET, CMAP , Ecole Polytechnique Joint work with Ying JIAO, LPMA , Université Paris VIIInformation asymmetry in Pricing of Credit Derivatives
IntroductionThe informational structure
Pricing under the historical probabilityRisk neutral pricingNumerical example
Pricing for the full informationPricing for the progressive and delayed informationPricing for the noisy full information
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) .
23/ 36Caroline HILLAIRET, CMAP , Ecole Polytechnique Joint work with Ying JIAO, LPMA , Université Paris VIIInformation asymmetry in Pricing of Credit Derivatives
IntroductionThe informational structure
Pricing under the historical probabilityRisk neutral pricingNumerical example
Pricing for the full informationPricing for the progressive and delayed informationPricing for the noisy full information
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)
24/ 36Caroline HILLAIRET, CMAP , Ecole Polytechnique Joint work with Ying JIAO, LPMA , Université Paris VIIInformation asymmetry in Pricing of Credit Derivatives
IntroductionThe informational structure
Pricing under the historical probabilityRisk neutral pricingNumerical example
Pricing for the full informationPricing for the progressive and delayed informationPricing for the noisy full information
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) .
25/ 36Caroline HILLAIRET, CMAP , Ecole Polytechnique Joint work with Ying JIAO, LPMA , Université Paris VIIInformation asymmetry in Pricing of Credit Derivatives
IntroductionThe informational structure
Pricing under the historical probabilityRisk neutral pricingNumerical example
Pricing for the full informationPricing for the progressive and delayed informationPricing for the noisy full information
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.
26/ 36Caroline HILLAIRET, CMAP , Ecole Polytechnique Joint work with Ying JIAO, LPMA , Université Paris VIIInformation asymmetry in Pricing of Credit Derivatives
IntroductionThe informational structure
Pricing under the historical probabilityRisk neutral pricingNumerical example
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 PRICING
5 Numerical example
27/ 36Caroline HILLAIRET, CMAP , Ecole Polytechnique Joint work with Ying JIAO, LPMA , Université Paris VIIInformation asymmetry in Pricing of Credit Derivatives
IntroductionThe informational structure
Pricing under the historical probabilityRisk neutral pricingNumerical 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
28/ 36Caroline HILLAIRET, CMAP , Ecole Polytechnique Joint work with Ying JIAO, LPMA , Université Paris VIIInformation asymmetry in Pricing of Credit Derivatives
IntroductionThe informational structure
Pricing under the historical probabilityRisk neutral pricingNumerical example
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.
29/ 36Caroline HILLAIRET, CMAP , Ecole Polytechnique Joint work with Ying JIAO, LPMA , Université Paris VIIInformation asymmetry in Pricing of Credit Derivatives
IntroductionThe informational structure
Pricing under the historical probabilityRisk neutral pricingNumerical example
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
t (l)qt(Lt − l)PL(dl)
∫VQ
I
t (l)qt(Lt − l)PL(dl)
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).
30/ 36Caroline HILLAIRET, CMAP , Ecole Polytechnique Joint work with Ying JIAO, LPMA , Université Paris VIIInformation asymmetry in Pricing of Credit Derivatives
IntroductionThe informational structure
Pricing under the historical probabilityRisk neutral pricingNumerical example
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
31/ 36Caroline HILLAIRET, CMAP , Ecole Polytechnique Joint work with Ying JIAO, LPMA , Université Paris VIIInformation asymmetry in Pricing of Credit Derivatives
IntroductionThe informational structure
Pricing under the historical probabilityRisk neutral pricingNumerical 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
32/ 36Caroline HILLAIRET, CMAP , Ecole Polytechnique Joint work with Ying JIAO, LPMA , Université Paris VIIInformation asymmetry in Pricing of Credit Derivatives
IntroductionThe informational structure
Pricing under the historical probabilityRisk neutral pricingNumerical example
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
33/ 36Caroline HILLAIRET, CMAP , Ecole Polytechnique Joint work with Ying JIAO, LPMA , Université Paris VIIInformation asymmetry in Pricing of Credit Derivatives
IntroductionThe informational structure
Pricing under the historical probabilityRisk neutral pricingNumerical example
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
dynamic price of the defaultable bond firm value
FIGURE: L = ls
34/ 36Caroline HILLAIRET, CMAP , Ecole Polytechnique Joint work with Ying JIAO, LPMA , Université Paris VIIInformation asymmetry in Pricing of Credit Derivatives
IntroductionThe informational structure
Pricing under the historical probabilityRisk neutral pricingNumerical example
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
dynamic price of the defaultable bond firm value
FIGURE: L = ls or L = li
35/ 36Caroline HILLAIRET, CMAP , Ecole Polytechnique Joint work with Ying JIAO, LPMA , Université Paris VIIInformation asymmetry in Pricing of Credit Derivatives
IntroductionThe informational structure
Pricing under the historical probabilityRisk neutral pricingNumerical example
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.
36/ 36Caroline HILLAIRET, CMAP , Ecole Polytechnique Joint work with Ying JIAO, LPMA , Université Paris VIIInformation asymmetry in Pricing of Credit Derivatives