The mean-variance hedging of a default option
with partial information
Michael Kohlmann†,
∗
Dewen Xiong‡
Abstract. We consider the mean-variance hedging of a defaultable claim in a general
stochastic volatility model. By introducing a new measure Q0 , we derive the martingale
representation theorem with respect to the investors’ filtration G. We present an explicit
form of the optimal-variance martingale measure by means of a stochastic Riccati equation
(SRE). For a general contingent claim, we represent the optimal strategy and the optimal
cost of the mean-variance hedging by means of another backward stochastic differential
equation (BSDE). For the defaultable option, especially when there exists a random recovery
rate we give an explicit form of the solution of the BSDE.
Key words: defaultable risk; mean-variance hedging; stochastic Riccati equation; backward stochastic differential equations; variance-optimal martingale measure
Mathematics Subject Classification (1991): 90A09, 60H30, 60G44
1
Introduction
We discuss the mean-variance hedging of defaultable claims in a stochastic volatility model.
We assume that in the market there exists a risky asset whose price process S is affected
by a random factor θ, which can be interpreted as the index of a financial system, and the
dynamics of (S, θ) are determined by a pair of stochastic differential equations
dSt = St (µ(t, θt , St )dt + σ(t, θt , St )dW1 (t))
dθt = α(t, θt , St )dt + β1 (t, θt , St )dW1 (t) + β2 (t, θt , St )dW2 (t) ,
where (W1 , W2 ) = {(W1 (t), W2 (t)); 0 ≤ t ≤ T } is a 2-dimensional standard Brownian
motion defined on [0, T] and F = (Ft )0≤t≤T is a filtration generated by (W1 , W2 ) satisfying
∗
Supported by National Natural Science Foundation of China under Grant No. 70671069. The second
author thanks University of Konstanz for invitation.
†
Department of Mathematics and Statistics, University of Konstanz, D-78457, Konstanz, Germany.
Email:[email protected]
‡
Department of Mathematics Shanghai Jiaotong University, Shanghai(200240), P.R.China. Email:
[email protected], corresponding author
1
the usual conditions. In this market we assume that the investors know FS = {FtS ; 0 ≤
t ≤ T }, the natural filtration of the price process (partial information). However, they
can not obtain the information of θ directly. There exists a default time τ , such that
τ = inf(t > 0; θt ≤ a) for a constant a, which should be interpreted as the default happening
when the index falls below a certain level. It is easy to see that τ is an F-stopping time,
but not an FS -stopping times. We introduce the associated default process H by setting
Dt = 1τ ≤t for t ∈ R+ . Let D = {Dt ; 0 ≤ t ≤ T } be the filtration generated by the process
D, then the information of the investors is given by G = FS ∨ D so that
FS ⊂ G ⊂ F
In this paper we discuss in a mean variance setting the hedging of a contingent claim
H ∈ L2 (GT ), especially a default claim with random recovery rate, when we have the
information G = (Gt )0≤t≤T .
Similar work has been done in Biagini and Cretarola(2006)[2] , Bielecki, Jeanblanc and
Rutkowski(2004,a) [3] , and Bielecki, Jeanblanc and Rutkowski(2004,b) [4] . However, condition (H) (see e.g. Jeulin (1980)[14] ) in our model no longer holds, i.e., a (P, FS )-martingale
need not be a (P, G)-martingale. Under a change of measure, however, we are able to show
that each FS -martingale is a G-martingale. We derive a martingale representation theorem
under Q0 and continue to treat the problem under this new measure.
By introducing a BSDE which is called stochastic Riccati equation (SRE) in Hu and
Zhou(2003)[10] , Kohlmann and Tang(2003)[15] , and Kohlmann and Zhou(2000)[18] , we describe the variance optimal martingale measure P explicitly. For further properties of the
variance optimal martingale we refer to Delbaen and Schachernayer(1996)[7] , Schweizer(1996)[26] ,
etc. For a general contingent claim H ∈ L2 (GT ), we describe more explicitly the optimal
strategy and the optimal cost of the mean-variance hedging of investors by the solution of an
SRE and another BSDE. This method can also be found in Kohlmann and Tang(2003)[15] ,
Kohlmann and Zhou(2000)[18] , or Bobrovnytska and Schweizer(2004)[5] in Brownian motion
setting.
Therefore it becomes important to solve the BSDE. In this paper we give the explicit
solution of the BSDE for certain contingent claims. By considering ZT0,G = E[Z 0 |GT ] as a
contingent claim and solving the BSDE explicitly, we will see the relationship between the
optimal strategy of the mean-variance hedging of the general contingent claim H and ZT0,G .
We apply these results to solve the BSDE in the case of a defaultable claim with a random
recovery rate.
2
The preliminaries
We begin with a finite time horizon T > 0 and a complete probability space (Ω, F , P ).
Suppose that (W1 , W2 ) = {(W1 (t), W2 (t)); 0 ≤ t ≤ T } is a 2-dimensional standard Brownian
motion on this space defined on [0, T] and F = (Ft )0≤t≤T is a filtration generated by
2
(W1 , W2 ) satisfying the usual conditions. We assume that in this market there exists a
risky asset whose discounted price process S = (St )0≤t≤T is influenced by a random factor
θ = {θt } such as the index of the financial system. We assume that (St , θt ) satisfies the
following stochastic differential equation(SDE):
dSt = St (µ(t, θt , St )dt + σ(t, θt , St )dW1 (t))
(1)
dθt = α(t, θt , St )dt + β1 (t, θt , St )dW1 (t) + β2 (t, θt , St )dW2 (t)
where initial values S0 and θ0 are two deterministic constants. We assume that α(t, x, y),
βi (t, x, y), i = 1, 2, µ(t, x, y) and σ(t, x, y) are all continuous functions such that SDEs (1)
has a unique strong solution for all given constants S0 and θ0 . We also assume that there
exists two positive constants ki , i = 1, 2 such that k1 ≤ σ(·, ·, ·) ≤ k2 and k1 ≤ β2 (·, ·, ·) ≤ k2 .
In this market we assume that the investors know FS = {FtS ; 0 ≤ t ≤ T }, the natural
filtration of the price process. However we have a partial information, as they can not
obtain the information about θ directly. Furthermore, there exists a default time τ , such
as τ = inf(t > 0; θt ≤ a) for a constant a, which implies that if the index of an economic
is too low the default happens. It is clear that τ is an F-stopping time, but not an FS stopping times. We introduce the associated default process H by setting Dt = 1τ ≤t for
t ∈ R+ . Let D = {Dt ; 0 ≤ t ≤ T } be the filtration generated by the process D, then the
information of the investors is given by G = FS ∨ D. Since not each (P, FS )-martingale is a
(P, G)-martingale, the condition (H) (see e.g. Jeulin (1980)[14] ) doesn’t hold. So we need
to change the underlying measure. Let
⎧
µ(t, θt , St )
⎪
⎪
⎨ l1 (t) :=
σ(t, θt , St )
β1 (t, θt , St )
α(t, θt , St )
⎪
⎪
−
l1 (t),
⎩ l2 (t) :=
β2 (t, θt , St ) β2 (t, θt , St )
We assume that l1 (t) and l2 (t) are two uniformly bounded processes. Thus
·
·
0
l1 (u)dW1 (u) −
l2 (u)dW2 (u)
Z := E −
0
0
is a positive square integrable (F, P )-martingale with EZT0 = 1. Thus we can define a new
measure Q0 by
dQ0 = ZT0 ,
dP FT
under Q0
⎧
t
⎪ ⎪
l1 (u)du,
⎨ W1 (t) := W1 (t) +
0 t
⎪
⎪
2 (t) := W2 (t) +
⎩ W
l2 (u)du
0
are two independent F-Brownian motion and (S, θ) can be rewritten as
1 (t)
dSt = St σ(t, θt , St )dW
1 (t) + β2 (t, θt , St )dW
2 (t) .
dθt = β1 (t, θt , St )dW
3
t
1 (u) then M = (Mt )t∈[0,T ] is a continuous square inteσ(u, θu , Su )dW
t
0
σ(u, θu , Su )2 du and FS = FM .
grable (Q , F)-martingale with [M, M ]t = M, M t =
Let Mt :=
0
0
Since [S, S]t ( or [M, M ]t ) is an FS -adapted process and σ(t, θt , St ) is a continuous process, σ(t, θt , St ) is then an FS -predictable process, which implies that the investors know
something about θ, but they don’t know θ exactly, as θ is not FS -adapted. The following
predictable representation property is a slight modification of Theorem 5.17 of Liptser and
Shiryayev(1979)
1 is a (Q0 , FS )-Brownian motion and for every local (Q0 , FS )-martingale m,
Lemma 2.1 W
T
S
ξ(u)2 du < ∞, Q0 -a.s. such
there exists a F -predictable process ξ = {ξ(t)}t∈[0,T ] with
that
0
mt = m0 +
t
0
1 (u).
ξ(u)dW
Remark. From Lemma 2.1, we know that all (Q0 , FS )-martingales are continuous.
Lemma 2.2 Any (Q0 , FS )-martingale is a (Q0 , G)-martingale.
1 }, FS , Q0 ),
1 , denoted by Z 1 ({W
Proof. From Lemma 2.1, the stable subspace generated by W
satisfies
1 }, FS , Q0 ),
H 1 (FS , Q0 ) = Z 1 ({W
where H 1 (FS , Q0 ) := M ∈ Mloc (Q0 , FS ) : EQ0 sup0≤t≤T |Mt | < ∞ . On the other hand,
1 is a (Q0 , F)-Brownian motion which is FS -adapted, thus G-adapted. Therefore, for all
W
0 ≤ s < t ≤ T,
EQ0 [W1 (t)|Gs ] = EQ0 EQ0 [W1 (t) Fs ]
Gs
1 (s)
Gs
= EQ0 W
1 (s),
=W
1 is also a (Q0 , G)-martingale. By Theorem (9.30,a) of Jacod(1979)[11] ,
which means that W
M(Q0 , FS ) ⊂ M(Q0 , G), i.e., all (Q0 , FS )-martingale is (Q0 , G)-martingale. We now assume that τ is a F-stopping time, but not an FS -stopping time. We assume
that
Assumption 2.3 There exists a positive FS -predictable process λ = {λt ; 0 ≤ t ≤ T } such
that
t∧τ
λs ds
Nt := Dt −
0
is a
(Q0 , G)-martingale.
λ is called the intensity or hazard rate of τ with respect to FS .
4
Remark. The process Λt :=
t
0
λu du is called as FS -martingale Hazard process under
Q0 . From Lemma 4.9 of Jeanblanc and Rutkowski(2000)[12] and Lemma 2.2, we see that
Ft := Q0 (1τ ≤t |FtS ) is an increasing process. Since all (Q0 , FS )-martingales are continuous,
by Proposition 4.12 of Jeanblanc and Rutkowski(2000)[12] , we have
Ft = 1 − e−Λt = 1 − e−
t
0
λu du
.
The following lemma is cited from Jeulin(1980)[14]
Lemma 2.4 For any bounded G-predictable process H, there exists a bounded FS -predictable
process J such that
Ht (ω)1t≤τ = Jt (ω)1t≤τ .
Corollary 2.5 For all G-predictable N -integrable process H, there exists an FS -predictable
process J such that
Hu dNu = Ju dNu .
The following lemma is an adaptation of Proposition 3.4 of Jeanblanc and Rutkoski(2000)[12]
to the situation here:
Lemma 2.6 Let U = (Ut )t∈[0,T ] be a FS -predictable process such that Uτ is a Q0 -integrable
random variable, then
EQ0 (Uτ Gt ) = zt 1t<τ + Uτ 1t≥τ ,
where z is the solution of the following stochastic differential equation
t
s
λ
du
u
e0
dms +
(zs − Us )λs ds
zt = m0 +
0
]0,t]
where mt is a right-continuous version of the uniformly integrable (Q0 , FS )-martingale
{EQ0 (Uτ FtS )}t∈[0,T ] .
Corollary 2.7 Under the assumptions of Lemma 2.6, let YtU := EQ0 (Uτ Gt ) (we consider
1 -integrable process θ such
its right-continuous version), then there exists a FS -predictable W
that (Z U , θ) is the unique solution of the following backward stochastic differential equation
U )dN ,
1 (t) + (Ut − Yt−
t ≤ τ,
dYtU = θ(t)dW
t
U
= Uτ .
Yτ
τ ∧T
δ(u)du is Q0 -integrable, let
Remark. For some FS -predictable process δ such that
0
T ∧τ
t
δ
δ
δ(u)du and Zt := EQ0
δ(u)du
Gt , for all t ∈ [0, T ], we have
Ut :=
0
0
Ztδ = ztδ 1t<τ + Uτδ 1t≥τ ,
5
τ ∧T
where ztδ := eΓt EQ0 ( 0
δ(u)du1t<τ |FtS ) can be computed in the following way
t
τ ∧T
Γt
δ
zt =
δ(u)du + e EQ0 (
δ(u)du1t<τ |FtS )
0
τ
T
0 t
Γt
δ(u)du + e
δ(u)du|FtS ) + EQ0 (1{T <τ }
δ(u)du|FtS )
EQ0 (1{t<τ ≤T }
=
t
T s t
T
0 t
S
S
Γt
δ(u)du + e
(
δ(u)du)dFs Ft + EQ0 (1 − FT )
δ(u)du
Ft
EQ0
=
t
t
Tt
0 t
δ(u)du + eΓt EQ0
(1 − Fu )δ(u)du
FtS
=
t T
t
0 t
S
Γt
−Γu
−Γu
δ(u)du + e
e
δ(u)du
Ft −
e
δ(u)du .
EQ0
=
0
0
T
0
Let m(t) := EQ0 0 e−Γu δ(u)du
FtS , then m is a uniformly integrable (Q0 , FS )-martingale,
we consider its right-continuous version, thus
t
t
δ(u)du + eΓt m(t) −
e−Γu δ(u)du
ztδ =
0
t0
s
t
Γu
δ
e dm(u) +
{zs −
δs ds}λs ds.
= m(0) +
0
0
0
And similarly, there exists an FS -predictable process denoted by θ such that (Z δ , θ) is the
unique solution of the following BSDE
⎧
t
δ =θ 1
δ
⎪
d
W
(t)
+
δ(u)du
−
Y
t ≤ T,
dY
⎨
t t≤τ
1
t
t− dNt ,
0
τ ∧T
⎪
⎩ YTδ
=
δ(u)du .
0
Finally, the following martingale representation theorem is adopted from Proposition
2.2 of Kusuoka(1999)[19] or Proposition 3.5 of Jeanblanc and Rutkowski(2000)[12] .
Theorem 2.8 Under Assumption 2.3, let ξ be a GT -measurable Q0 -integrable random variable and let Ntξ := EQ0 (ξ|Gt ) for t ∈ [0, T ]. Then the (Q0 , G)-martingale N ξ admits the
following decomposition
t
ξ
ξ
ξ
γ1 (u)dW1 (u) +
γ2ξ (u)dNu
Nt = N0 +
0
]0,t]
for t ∈ [0, T ], where γ1ξ and γ2ξ are two G-predictable processes.
3
G-variance-optimal martingale measure
We now consider the variance optimal martingale measure for the filtration G.
Definition 3.1 A probability measure Q is called an equivalent local martingale measure (ELMM) for (S, G) if it is a probability measure Q ∼ P with Q |G0 = P |G0 and
such that S is a local (Q, G)-martingale.
6
Denote by Me (G) the set of all equivalent martingale measures for (S, G), and
dQ
2
2
Me (G) := Q ∈ Me (G);
∈ L (P ) .
dP
In our model, M2e (G) = ∅, since Q0 ∈ M2e (G). Therefore there exists a measure P ∈ M2e (G)
called the variance-optimal martingale measure (VOMM) such that
dP dQ ≤
∀Q ∈ M2e (F).
dP 2 ,
dP 2
M (P )
M (P )
We now consider the density process of P with respect to (P, G), the RCLL version of
dP Gt , 0 ≤ t ≤ T .
the strictly positive martingale Zt = EP
dP
Let ZT0,G = E[ZT0 |GT ], we consider the following backward stochastic differential equation(BSDE) under the new measure Q0
⎧
1 (t) + at− γ2 (t)dNt + at− (γ1 (t))2 dt
⎪
⎨ dat = −at− γ1 (t)dW
1
(2)
⎪
⎩ aT = Z 0,G .
T
A solution of (2) is a triplet (a, γ1 , γ2 ) satisfying (2) and that a is a strictly positive RCLL
(Q0 , G)-semimartingale with a > 0, γ1 and γ2 are G-predictable M 0 -integrable processes.
Lemma 3.2 The BSDE (2) has a solution (a, γ1 , γ2 ) and the density process of G-optimalvariance equivalent martingale measure P with respect to (P, G) is then given by
P
0,G
γ2 (u)dNu ),
Z = Z E(
]0,·]
where Z 0,G is the density process of Q0 with respect to (P, G), i.e., Zt0,G = E[Zt0 |Gt ].
Proof. Let Z = EP [ZTP |Gt ] , 0 ≤ t ≤ T , by Lemma 2.2 of Delbaen and Schachermayer(1996)[7] ,
it can be represented in the following form
Z = Z0 + ζdS
= Z0 E ( ζdS)
On the other hand, let Z P ,Q0 = EQ0 ( dP |Gt ), then Z P ,Q0 is a positive
with ζ := ζ/Z.
t
dQ0
0
(Q , G)-martingale, thus there exists a pair of G-predictable processes (ϕ1 , ϕ2 ) such that
·
0
1 +
ϕ1 dW
ϕ2 dN ) .
Z P ,Q = E (−
0
]0,·]
7
P ,Q is a local (Q0 , G)-martingale, thus
As S is also a local (P, G)-martingale, so SZ
Q0 ,G S, Z P ,Q0 = 0 and ϕ = 0 and Z P ,Q0 = E (
ϕ2 dN ). Setting
1
0
]0,·]
a :=
Z P ,Q
Z
0
and applying Itô’s formula yields (2) with γ1 (t) = ζt St σ(t, θt , St ) and γ2 (t) = ϕ2 (t). The
rest follows the same reasoning as in the proof of Theorem 3.1 of Kohlmann, Xiong and
Ye(2006).
We now consider the solution of the BSDE (2) under measure Q0
1 (t) + δ2 (t)dNt
dmt = δ1 (t)dW
0,G
mT = ZT .
(3)
The solution of (3) is a triplet of G-predictable processes (m, δ1 , δ2 ) satisfying (3) and that
m is a uniformly integrable positive (Q0 , G)-martingale. By Theorem 2.8, it is easy to see
that (2) has a unique solution.
Theorem 3.3 For a G-predictable N -integrable process δ2 such that
δ2 (u)dNu
nt = 1 +
]0,t]
is a positive (Q0 , G)-martingale and EQ0 [ZT0,G (nT )2 ] < ∞. Let (m, δ1 , δ2 ) be the solution of
(3), and let (Y (δ2 ,δ2 ) , θ) be the solution of the following BSDE
⎧
(δ2 ,δ2 )
t
(δ2 ,δ2 )
⎪
=
θ
1
d
W
(t)
+
δ
(u)
δ
(u)λ
du
−
Y
dNt , t ≤ T,
dY
⎨
t t≤τ
1
2
u
t
t−
0 2
τ ∧T
(4)
⎪
⎩ YT(δ2 ,δ2 )
=
δ2 (u)δ2 (u)λu du ,
0
Let Vt = m0 +
(δ ,δ )
Y0 2 2
+
0
t
1 (u). If there exists a δ2 such that
{nu− δ1 (u) + θu 1u≤τ }dW
mt− δ2 (t) + nt− δ2 (t) + δ2 (t)δ2 (t) +
t
0
(δ ,δ )
δ2 (u)δ2 (u)λu du − Yt−2 2 = 0,
for all t ≤ τ,
then V = (Vt )t∈[0,T ] is a positive (Q0 , G)-martingale then the solution of (2) is given by
⎧
nt− δ1 (t) + θt 1t≤τ
⎪
⎪
,
⎨ γ1 (t) =
Vt
(5)
δ2 (t)
⎪
⎪
⎩ γ2 (t) = −
.
nt−
Proof. It is easy to see that n can be rewritten as
γ2 (u)dNu )t .
nt = E (−
]0,·]
8
From (4), we have
τ ∧T
(δ2 ,δ2 )
δ2 (u)δ2 (u)λu du = Y0
+
0
0
So ZT0,G nT ∈ L2 (P ) and
ZT0,G nT
= mT n T
T
1 (t)+
θt 1t≤τ dW
T
]0,T ]
t
0
(δ2 ,δ2 )
δ2 (u)δ2 (u)λu du − Yt−
dNt .
T
nu− dmu + [m, n]T − [m, n]0
T
1 (u)
= m0 +
mu− δ2 (u)dNu +
nu− δ1 (u)dW
0
]0,T
]
+
nu− δ2 (u)dNu +
δ2 (u)δ2 (u)dHu
]0,T ]
]0,T ]
T
= m0 +
nu− δ1 (u)dW1 (u) +
mu− δ2 (u)dNu
0
]0,T ]
τ ∧T
+
nu− δ2 (u)dNu +
δ2 (u)δ2 (u)dNu +
δ2 (u)δ2 (u)λu du
0
]0,T ]
]0,T ]
T
(δ2 ,δ2 )
1 (t)
+
{nt− δ1 (t) + θt 1t≤τ }dW
= m0 + Y0
0
t
(δ ,δ ) δ2 (u)δ2 (u)λu du − Yt−2 2 dNt
mt− δ2 (t) + nt− δ2 (t) + δ2 (t)δ2 (t) +
+
]0,T ]
0
T
(δ2 ,δ2 )
1 (t)
= m0 + Y0
+
{nt− δ1 (t) + θt 1t≤τ }dW
0
·
(δ ,δ )
1 (u))T .
= VT = (m0 + Y0 2 2 )E ( γ1 (u)dW
= m0 +
mu− dnu +
0
0
0
4
The solution of mean variance hedging
We now consider the mean-variance hedging problem for a defaultable claim. We begin
with the following lemma:
Lemma 4.1 There exists a measure Q ∈ M2e (G) satisfying the reverse Hölder inequality
R2 (P, G), i.e., there exists a constant C ∈ (0, ∞) such that the density process Z Q satisfies
E[(ZTQ )2 |Gt ] ≤ C(ZtQ )2 ,
P -a.s.
1 is also
1 is a (Q0 , G)-Brownian motion, it is a (P, G)-semimartingale. W
Proof. Since W
a continuous (P, F)-semimartingale with the following canonical decomposition
t
l1 (u)du + W1 (t).
W1 (t) =
0
Since l1 is a bounded process,
·
0
l1 (u)du is an F-predictable process with integrable varia-
tion. According to Proposition (9.24) of Jacod(1979)[11] , we see that the canonical decom1 with respect to (P, G) is given by
position of W
t
(p,G)
1 (t),
l1 (u)du + W
W1 (t) =
0
9
where
(p,G)
1 (t) := W
1 (t)−
l1 is the G-predictable projection of l1 under P and W
t
0
(p,G)
l1 (u)du
1 is a (P, G)-Brownian motion. Fur1 t = t, hence W
is a local (P, G)-martingale with W
thermore, the canonical decomposition of S with respect to (P, G) is given by
t
t
(p,G)
1 (u).
Su σ(u, θu , Su )
l1 (u)du +
Su σ(u, θu , Su )dW
St = S0 +
0
0
Thus we define a new measure Q by
dQ :=
E
−
dP GT
·
(p,G)
0
1 (u) ,
l1 (u)dW
T
and it is easy to see that S is a continuous local (Q, G)-martingale and Q ∈ M2e (G). As l1
(p,G)
is a bounded process,
l1 is also a bounded process and thus Q ∈ R2 (P, G). Corollary 4.2 There exists a positive constant c such that
1
c
Zt0,G
≤ at ≤
1
Zt0,G
,
Q0 ( or P )-a.s.
for all t ∈ [0, T ].
Proof. From Theorem 4.1 of Delbaen et al.(1997) and Lemma 4.1, we have that P satisfies
the reverse Hölder inequality R2 (P, G). Thus there exists a constant C ∈ (0, ∞) such that
the density process Z P satisfies
(ZtP )2 ≤ E[(ZTP )2 |Gt ] ≤ C(ZtP )2 ,
From
0
P -a.s.
(ZtP )2
Z P ,Q
ZP
,
=
= 0,G t at = t
Zt
Zt EP (ZTP |Gt )
Zt0,G E (ZTP )2 Gt
we get that
1
1 1
≤ at ≤ 0,G ,
0,G
C Zt
Zt
Q0 ( or P )-a.s.
Given H ∈ L2 (GT , P ), we now consider the following backward stochastic differential
equation (BSDE)
1 (t) + φt dNt − λt γ2 (t)1t≤τ dt ,
dht = ψt dW
(6)
hT = H ,
where γ2 is part of the solution (a, γ1 , γ2 ) of the BSDE (2). A solution of (6) is a 4-tuple
(h, ψ, φ) satisfying (6) and
EQ0
T
0
1
0,G
Zu−
(ψu )2 du} < ∞,
10
EQ0
1
ZT0,G
(φτ )2 1τ ≤T
< ∞.
(7)
Lemma 4.3 For all H ∈ L2 (GT , P ), (6) has a unique solution. If (h, ψ, φ) is the solution
of (6), then h is a uniformly (P, G)-martingale, i.e., ht = EP (H|Gt ).
t := Nt −
Proof. First, from Girsanov’s theorem it is seen that N
0
t
λu γ2 (u)1u≤τ du is a
(P, G)-martingale. As EP (|H|) < ∞, let ht := EP (H|Gt ), by Theorem 2.8 there exists a
pair (ψ, φ) such that
t
u .
ψu dW1 (u) +
φu dN
ht = h0 +
0
]0,t]
Since h is a uniformly integrable (P, G)-martingale, and P satisfies the reverse Hölder
inequality R2 (P, G), by Proposition 1 of Rheinländer M. and Schweizer M.(1997)[23] , there
exists a constant C such that
E[( sup |ht |)2 ] ≤ CE[h2T ] = CE[H 2 ] < ∞,
0≤t≤T
and
E([h, h]T ) = E[
T
(ψu )2 du + (φτ )2 1τ ≤T ] ≤ CE[( sup |ht |)2 ] < ∞ .
0≤t≤T
0
(8)
(9)
Thus (ψ, φ) satisfies (7), which implies that (h, ψ, φ) is the solution of BSDE (6).
On the other hand, if (h, ψ, φ) is the solution of (6), using the reverse Hölder inequality
and Proposition 1 of [23] once again yields
t
t
1 (u)|
1 (u)|
ψu dW
≤ ZTP 2 sup |
ψu dW
EP sup |
2
L (P ) 0≤t≤T
0≤t≤T
0
0
L (P,G)
1
T
2
≤ C ZTP 2
(ψu )2 du]
<∞
E[
L (P )
0
1 (u) is a uniformly integrable (P, G)-martingale. In the same way, one
ψu dW
u is also a uniformly integrable (P, G)-martingale. shows that ]0,·] φu dN
Let
π:
π is a G-predictable process with
T
.
Adm(x) =
E[ 0 (πu )2 du] < ∞
so that
For π ∈ Adm(x), we introduce
Xtπ
: =x+
=x+
t
0 t
0
πu
1
dSu
Su
1 (u),
πu σ(u, θu , Su )dW
which is called as the wealth process with respect to π with initial wealth x. By the
boundedness of σ(t, θt , St ), we know that E[(supt∈[0,T ] |Xtπ |)2 ] < ∞.
11
Lemma 4.4 Let
πu∗ =
where
∗
Xtπ
=x+
t
0
∗
ψu − (hu− − Xuπ )γ1 (u)
,
σ(u, θu , Su )
(10)
πu∗
dSu , then π ∗ ∈ Adm(x).
Su
Proof. Obviously
∗
∗
1 (t) + φt dNt − λt γ2 (t)1t≤τ dt
d(ht − Xtπ ) = γ1 (t)(ht− − Xtπ )dW
∗
Let Lt := at (ht − Xtπ ), then by Itô’s formula, L is the solution of following stochastic
differential equation (SDE)
dLt = Lt− γ2 (t)dNt + at− φt (1 + γ2 (t))dNt
L0 = a0 (h0 − x),
whose solution can be represented as
⎞
⎛
0
1
1
a φ dNu −
a φ λ γ (u)1u≤τ du⎠
Lt = ZtP ,Q ⎝L0 +
,Q0 u− u
,Q0 u− u u 2
P
P
]0,t] Z
]0,t] Z
u−
u−
where
0
ZtP ,Q
:= LZ 0,G
L
= E(
0
γ2 dN )t . Recall that Z P = Z 0,G Z P ,Q and a =
]0,·]
t
t := Nt −
and N
λu γ2 (u)1u≤τ du, then
0
!
t = (at Z 0,Gt )Zt
L
Since 0 < c ≤
aZ 0,G
≤ 1, and nt :=
]0,t]
L0 +
]0,t]
1
u
φu dN
Zu−
0
Z P ,Q
, if we let
Z
"
.
u is a uniformly integrable (P, G)-martingale
φu dN
1 ] = 0, so by Lemma 7 of [23], we have
null at 0 and [n, W
1
u ∈ L2 (P ).
φu dN
sup Zt
u−
0≤t≤T ]0,t] Z
L
∗
2
π∗ = h −
, thus sup0≤t≤T |Xtπ | ∈ L2 (P ).
Therefore, sup0≤t≤T L
t ∈ L (P ). As X
t
0,G
aZ
Thus by Proposition 1 of [23], there exists a constant C such that
E{
T
0
∗
∗
(πu∗ )2 du} = E{[X π ]T } ≤ CE{ sup |Xtπ |} < ∞ .
0≤t≤T
12
Theorem 4.5 Given H ∈ L2 (GT , P ), let (h, ψ, φ) be a solution of BSDE (6), then
T ∧τ
EQ0
au− λu (1 + γ2 (u))(φu )2 du < ∞ ,
(11)
0
and the optimal solution of the following problem
∗
J (H) = min E[(x +
π∈Adm(x)
T
0
πu
dSu − H)2 ]
Su
is given by (15) and the optimal cost is given by
T ∧τ
au− λu (1 + γ2 (u))(φu )2 du .
J ∗ (H) = a0 (h0 − x)2 + EQ0
0
Remark. Theorem 4.5 can be viewed as a special case of Theorem 4.3 in Kohlmann, Xiong
and Ye(2006). Note however that here we can not find two positive constants c1 and c2
such that c1 < Zt0,G < c2 .
Proof. For given π ∈ Adm(x), ht − Xtπ can be rewritten in the following way
t
π
1 (u)
(ψu − σ(u, θu , Su )πu )dW
ht − Xt = (h0 − x) +
0
t
φu dNu −
φu γ2 (u)λu 1u≤τ du .
+
0
]0,t]
Therefore,
(ht − Xtπ )2 = (h0 − x)2
t
2 +
−2(hu− − Xuπ )φu γ2 (u)λu 1u≤τ + (φu )2 λu 1u≤τ + ψu − σ(u, θu , Su )πu
du
0 t
1 (u) +
2(hu− − Xuπ ) ψu − σ(u, θu , Su )πu dW
{2(hu− − Xuπ )φu + (φu )2 }dNu .
+
0
]0,t]
By Itô formula, we have
t∧τ
= a0 (h0 −
+
au− λu (φu )2 (1 + γ2 (u))du
at (ht −
0
t
2
au− σ(u, θu , Su )πu − ψu − (hu− − Xuπ )γ1 (u)
du
+
0 t
π
π
1 (u)
au− (hu− − Xu ) 2 ψu − σ(u, θu , Su )πu − (hu− − Xu )γ1 (u) dW
+
0
π
2
π 2
au− 2(1 + γ2 (u))(hu− − Xu )φu + (1 + γ2 (u))(φu ) + (hu− − Xu− ) γ2 (u) dNu .
+
Xtπ )2
x)2
]0,t]
There exists a sequence of G-stopping times (τi ) such that τi increases to T , Q0 -a.s., as
π
); u ∈ [0, T ]} and au∧τi − are two bounded
i → ∞, and for each i, both {(hu∧τi − − Xu∧τ
i
G-predictable processes. Thus, for any t ∈ [0, T ],
t∧τ ∧τi
π )2 ] = a (h − x)2 + E 0
au− λu (φu )2 (1 + γ2 (u))du
EQ0 [at∧τi (ht∧τi − Xt∧τ
0 0
Q
i
0
t∧τi
(12)
2
au− σ(u, θu , Su )πu − ψu − (hu− − Xuπ )γ1 (u)
du ,
+EQ0
0
13
which implies
τ ∧τi
EQ0
au− λu (φu )2 (1 + γ2 (u))du ≤ EQ0 [aτi (hτi − Xτπi )2 ] ≤ E[( sup |ht − Xtπ |)2 ]
0≤t≤T
0
As τi ↑ T , (11) follows from the monotone convergence theorem.
On the other hand, since {ht − Xtπ } is left continuous at T , by (12) and the monotone
convergence theorem as well as Lebesgue’s dominated convergence theorem, we obtain
E[(H − XTπ )2 ] = EQ0 [
1
ZT0,G
(hT − XTπ )2 ] = EQ0 [aT (hT − XTπ )2 ]
= limτi ↑T EQ0 [aτi (hτi − Xτπi )2 ]
T ∧τ
2
au− λu (φu )2 (1 + γ2 (u))du
= a0 (h0 − x) + EQ0
0
T
2
au− σ(u, θu , Su )πu − ψu − (hu− − Xuπ )γ1 (u)
du
+EQ0
0
T ∧τ
au− λu (φu )2 (1 + γ2 (u))du
≥ a0 (h0 − x)2 + EQ0
∗
= E[(hG (T ) − XTπ )2 ],
where πu∗ =
5
0
ψu − (hu− − Xuπ )γ1 (u)
, which is the same as the strategy given by (15).
σ(u, θu , Su )
The mean-variance hedging of ZT0,G
As ZT0,G ∈ L2 (GT , P ), we can view ZT0,G as a contingent claim, and so we now consider the
mean-variance hedging of ZT0,G . Let (h0 , ψ0 , φ0 ) be the unique solution of BSDE (6) with
terminal value ZT0,G , i.e.,
1 (t) + φ0 (t) dNt − λt γ2 (t)1t≤τ dt
dh0 (t) = ψ0 (t)dW
(13)
h0 (T ) = ZT0,G ,
then the optimal strategy with initial wealth x̂ = h0 (0) is given by
ψ0 (t) − (h0 (t−) − X x̂,0 (t))γ1 (t)
,
σ(t, θt , St )
t x̂,0
t
πu
x̂,0
1 (u) is the optimal wealth
dSu = x̂ +
πux̂,0 σ(u, θu , Su )dW
where X (t) := x̂ +
0 Su
0
process with respect to π 0,x . Let Rx̂,0 (t) := h0 (t) − X x̂,0 (t), it is easy to see that X x̂,0 is
the solution of the following SDE
t
x̂,0
1 (u).
{ψ0 (u) − Rx̂,0 (u−)γ1 (u)}dW
X (t) = x̂ +
πtx̂,0 =
0
Rx̂,0
itself satisfies the following BSDE
t
x̂,0
x̂,0
R (u−)γ1 (u)dW1 (u) +
R (t) =
On the other hand,
0
]0,t]
14
u ,
φ0 (u)dN
whose solution can be represented as
R
x̂,0
t
(t) = Z
]0,t]
1
u ,
φ0 (u)dN
Zu
(14)
P
1 (u))t by the proof of Lemma 3.2.
where Zt := EP (ZT |Gt ) = Z0 E ( γ1 (u)dW
Lemma 5.1 E{Rx̂,0 (T )} = 0.
Proof. From Lemma 7 of [23], we see that
⎧!
⎫
t
"2 ⎬
⎨
1
t t
E
sup Z
φ0 (u)dN
⎭ < ∞,
u
⎩ 0≤t≤T
0 Z
thus
EP
sup 0≤t≤T
which implies that
(14),
·
t
0
&
t
&
1
1
t ≤ E sup Z
t < ∞,
t
φ0 (u)dN
φ0 (u)dN
u
0≤t≤T
Zu
0 Z
1
u φ0 (u)dNt
0 Z
is a uniformly integrable (P , G)-martingale. Thus by
ER
x̂,0
(T ) = EP
]0,T ]
1
u = 0 .
φ0 (u)dN
Zu
From the above lemma, we see that ZT0,G can be decomposed into two parts
ZT0,G = X x̂,0 (T ) + Rx̂,0 (T ),
and for all θ ∈ Adm(x) and for all x ∈ R, we have
E{RTx̂,0 x +
T
0
1 (u) } = 0.
θu dW
By making use of [7] or [26], we have the following lemma:
T = Z P = X x̂,0 .
Lemma 5.2 Z
T
T
As to the solution of BSDE (13), we may state the following theorem
Theorem 5.3 Let (a, γ1 , γ2 ) be the
(h0 , ψ0 , φ0 ) is given by (13)
⎧
⎪
h0 (t)
⎪
⎪
⎪
⎪
⎨
ψ0 (t)
⎪
⎪
⎪
⎪
⎪
⎩ φ0 (t)
solution of the SRE (2), then the solution of the BSDE
1
,
at
γ1 (t)
=
,
at−
=
=
1
−1 +
1 + γ2 (t)
15
1
.
at−
t γ1 (t)
Z
and the optimal cost is given by
σ(t, θt , St )
T ∧τ
λu γ2 (u)2
du .
J ∗ (ZT0,G ) = EQ0
au− (1 + γ2 (u))
0
Furthermore, the optimal strategy πtx̂,0 =
1
. On the other
Proof. The first part follows from (2), by applying Itô’s formula to
a
hand, we see from Lemma 5.2 that
T
1 (u) ,
{ψ0 (u) − Rx̂,0 (u−)γ1 (u)}dW
ZT = Z0 E ( γ1 (u)dW1 (u))T = x̂ +
0
from which we can see that
u γ1 (u) = ψ0 (u) − Rx̂,0 (u−)γ1 (u),
Z
thus
πtx̂,0 =
t γ1 (t)
Z
ψ0 (t) − (h0 (t−) − X x̂,0 (t))γ1 (t)
=
.
σ(t, θt , St )
σ(t, θt , St )
Now we can rewrite Theorem 4.5 in the following form
Theorem 5.4 Given H ∈ L2 (GT , P ), let (h, ψ, φ) be a solution of BSDE (6) and (h0 , ψ0 , φ0 )
be a solution of BSDE (13), then the optimal solution to the following problem
T
πu
∗
dSu − H)2 ]
J (H) = min E[(x +
π∈Adm(x)
0 Su
can be rewritten as
ψu
− Rt− (x, H)πtx̂,0 ,
(15)
σ(u, θu , Su )
1
h0 − x
1
∗
π
u , which can be interpreted as
(ht − Xt ) =
+
φu dN
where Rt (x, H) :=
Zt
Z0
]0,t] Zu
the relative ????residual risk???? of H with respect to ZT0,G (??). Further more,
the optimal cost can also be rewritten as
T ∧τ
λu (φu )2
∗
2
du .
J (H) = a0 (h0 − x) + EQ0
h0 (u−) + φ0 (u)
0
πu∗ =
6
The mean-variance hedging of a defaultable claim
We begin with the following lemma
Lemma 6.1 For all given ξ ∈ bFTS , we have
EP (ξ|Gt ) = EQ0 (ξ|Gt ) = EQ0 (ξ|FtS ),
for all t ∈ [0, T ],
which implies that each (P , FS )-martingale is a (P, G)-martingale.
16
Proof. Let nt := EQ0
that
dP Gt , it follows that n is a uniformly (Q0 , G)-martingale and
dQ0
γ2 (u)dNu )t .
nt = E (
]0,·]
Let mt := EQ0 (ξ|Gt ), it follows that m is a bounded (Q0 , G)-martingale. By Lemma 2.2
and Lemma 2.1, we have that mt = EQ0 (ξ|FtS ) which can be represented as the following
form
t
1 (u)
ϑu dW
mt = m0 +
0
with a FS -predictable process ϑ. mn is a uniformly integrable (Q0 , G)-martingale. Therefore,
1
EP (ξ|Gt ) = EQ0 (mT nT |Gt ) = mt . nt
Remark. From Corollary 2.5, we can choose a solution of (2) such that γ2 is a FS t := Nt − t∧τ λu γ2 (u)du =
predictable process. And it follows from Girsanov’s theorem that N
0
t∧τ
Ht − 0 λu (1 + γ2 (u))du is a (P, G)-martingale. Thus the density of the hazard rate of τ
under P with respect to FS is given by λ(1 + γ2 ). Let FtP := EP (1τ ≤t|FtS ), as in Remark of
Assumption 2.3, we have
t
P
λu (1 + γ2 (u))du).
Ft = 1 − exp(−
0
6.1
The case of H = ξ1T <τ
We first consider the mean-variance hedging of a defaultable option claim of the form ξ1T <τ .
Assume ξ is a positive bounded FTS -measurable random variable, it is easy to see that for
all t ∈ [0, T ]
ht = EP (ξ1τ >T |Gt )
E (ξ1τ >T |FtS )
1t<τ
= P
EP (1t<τ |FtS )
t
= exp( λu (1 + γ2 (u))du)EP ξ(1 − FTP )
FtS 1t<τ
0
T
t
S
= exp
λu (1 + γ2 (u))du EP ξ exp −
λu (1 + γ2 (u))du Ft 1t<τ
0 t
0 T
S
= exp
λu (1 + γ2 (u))du EQ0 ξ exp −
λu (1 + γ2 (u))du Ft 1t<τ .
0
Let mξt := EQ0 ξ exp −
0
0
T
S
λu (1 + γ2 (u))du Ft , then mξ is a uniformly integrable
(Q0 , FS )-martingale. By Lemma 2.1, there exists an FS -predictable process ϑ such that
mξt = mξ0 +
t
0
17
1 (u) .
ϑu dW
Thus
t
ht = exp
λu (1 + γ2 (u))du mξt 1t<τ
0
t
s
ξ
1 (s)
= m0 +
exp
λu (1 + γ2 (u))du ϑs dW
0
0
t
s
ξ
ms λs (1 + γ2 (s)) exp
λu (1 + γ2 (u))du ds 1t<τ
+
0
0 t
s
t
ξ
= m0 +
exp
λu (1 + γ2 (u))du ϑs dW1 (s) +
hs λs (1 + γ2 (s))ds 1t<τ
0
t∧τ
0 s
0 t∧τ
1 (s) +
= mξ0 +
exp
λu (1 + γ2 (u))du ϑs dW
hs λs (1 + γ2 (s))ds − hτ 1τ ≤τ
0
0
0
t
s
t
s
1 (s) −
s .
= mξ0 +
exp
λu (1 + γ2 (u))du ϑs 1s≤τ dW
exp
λu (1 + γ2 (u))du mξs dN
0
0
0
0
Therefore, the solution of (6) is then given by
s
ψs = exp 0 λu (1 + γ2 (u))du ϑs 1s≤τ
s
φs = − exp 0 λu (1 + γ2 (u))du mξs .
The case H = ξ 1{τ >T } + fτ ∧T 1{τ ≤T }
6.2
Now we assume that there exists a random recovery rate fτ ∧T , where f is a Borel function
such that 0 ≤ f ≤ 1. We still assume that ξ is a bounded positive FTS -measurable random.
The defaultable contingent claim is then given by H = ξ 1{τ >T } + fτ ∧T 1{τ ≤T } . For such
H, we now give an explicit solution of BSDE (6). For all t ∈ [0, T ], we have
ht = EP ξ 1{τ >T } + fτ ∧T 1{τ ≤T } Gt
= EP ξ 1 + (fτ − 1)1{τ ≤T } Gt
S
EP ξ(fτ − 1)1{t<τ ≤T } FtS
S
1
+
E
(fτ − 1)1τ ≤t
= EQ0 (ξ|Ft ) +
0 ξ Ft
t<τ
Q
1 − FtP
:= mξ1 (t) + ht 1t<τ + mξ1 (t)(fτ − 1)1τ ≤t ,
where
mξ1 (t)
:=
EQ0 (ξ|FtS )
and ht :=
EP ξ(fτ − 1)1{t<τ ≤T } FtS
1 − FtP
S
2.1 that there exists an F -predictable process ϑ1 such that
t
ξ
ξ
1 (u),
ϑ1 (u)dW
m1 (t) = m (0) +
. It follows from Lemma
0
from which we obtain
mξ1 (t)(fτ
− 1)1t≥τ =
mξ1 (fτ
− 1)(τ )1t≥τ +
0
t
1 (u).
ϑ1 (u)(fτ − 1)1u>τ dW
We only need to compute h, and so first we compute EP ξ(fτ − 1)1{t<τ ≤T } FtS . Introduce
At = ξ1τ ≤t .
18
As ξ is a positive bounded random variable, it follows that A is a P-integrable increasing
process. It is stressed that A is not adapted to G or FS . By Lemma 6.1, we have
EP (ξ1τ ≤t |FtS ) = EP (ξ|FtS )EP (1τ ≤t |FtS )
= mξ1 (t)FtP
=
mξ1 (0)
t
FsP dmξ1 (s)
+
0
t
+
0
mξ1 (s)dFsP .
Thus the predictable dual projection of A with respect to (P, FS ) is then given by
,FS ,p)
(P
At
=
mξ1 (0)
t
+
0
mξ1 (s)dFsP .
Therefore,
S
S
(fu − 1)dAu Ft
EP ξ(fτ − 1)1{t<τ ≤T } Ft = EP
]t,T ]
' T
(
(P ,FS ,p) S
(fu − 1)dAu
= EP
Ft
't T
(
ξ
P S
m1 (u)(fu − 1)dFu Ft
= EQ0
' T
(
t t
ξ
ξ
P
P S
m1 (u)(fu − 1)dFu + EQ0
m1 (u)(fu − 1)dFu Ft
=−
0
0 t
mξ1 (u)(fu − 1)dFuP + mξ2 (t)
=−
0
'
where
mξ2 (t)
:= EQ0
0
ϑ2 such that
T
mξ1 (u)(fu
−
1)dFuP FtS
mξ2 (t)
=
mξ2 (0)
+
0
t
(
, there exists an FS -predictable process
1 (u) .
ϑ2 (u)dW
Thus
t
1 (u)
ϑ2 (u)dW
EP ξ(fτ − 1) 1{t<τ ≤T } FtS = mξ2 (0) +
0
s
t
ξ
m1 (s)(fs − 1)λs (1 + γ2 (s)) exp −
λu (1 + γ2 (u))du ds .
−
0
Since ht = exp
we have
0
t
0
λu (1 + γ2 (u))du EP ξ(fτ − 1)1{t<τ ≤T } FtS , then by Itô’s formula,
t
ξ
[mξ1 (s)(fs − 1) − hs ]λs (1 + γ2 (s))ds
ht = m2 (0) −
0
s
t
1 (s) .
ϑ2 (s) exp
λu (1 + γ2 (u))du dW
+
0
0
19
We now can compute ht
ht = mξ1 (t) + ht 1t<τ + mξt (fτ − 1)1t≥τ
= mξ1 (t) + ht∧τ − hτ 1t≥τ + mξt (fτ − 1)1t≥τ
t∧τ
= mξ1 (t) + mξ2 (0) −
[mξ1 (s)(fs − 1) − hs ]λs (1 + γ2 (s))ds
0
t
s
1 (s) − ϑ2 (s) exp
λu (1 + γ2 (u))du 1s≤τ dW
hτ 1t≥τ + mξt (fτ − 1)1t≥τ
+
0
0
t
s
ξ
ξ
1 (s)
= m1 (0) + m2 (0) +
λu (1 + γ2 (u))du 1s≤τ + ϑ1 (s)(fτ − 1)1s>τ dW
ϑ1 (s) + ϑ2 (s) exp
0
0
t∧τ
hτ }1t≥τ −
[mξ1 (s)(fs − 1) − hs ]λs (1 + γ2 (s))ds
+{mξ1 (τ )(fτ − 1) − 0
t
s
ξ
ξ
1 (s)
ϑ1 (s) + ϑ2 (s) exp
λu (1 + γ2 (u))du 1s≤τ + ϑ1 (s)(fτ − 1)1s>τ dW
= m1 (0) + m2 (0) +
0
0
t
s .
hs }dN
+ {mξ1 (s)(fs − 1) − 0
Thus we obtain the following theorem:
Theorem 6.2 For a given bounded FTS -measurable random ξ and a Borel measurable function f such that 0 ≤ f ≤ 1, let H = ξ(1T <τ + fτ 1T ≥τ ). Introduce mξ1 (t) := EQ0 (ξ|FtS )
' T
(
ξ
ξ
P S
m1 (u)(fu − 1)dFu Ft and assume that ϑ1 and ϑ2 are two FS and m2 (t) := EQ0
0
predictable process such that
mξ1 (t)
mξ2 (t)
=
mξ (0)
+
=
mξ2 (0)
+
t
0 t
0
1 (u),
ϑ1 (u)dW
1 (u) .
ϑ2 (u)dW
Furthermore, assume that h is the solution of the following stochastic differential equation
t
ξ
[mξ1 (s)(fs − 1) − hs ]λs (1 + γ2 (s))ds
ht = m2 (0) −
0
s
t
1 (s) .
ϑ2 (s) exp
λu (1 + γ2 (u))du dW
+
0
0
Then the solution of BSDE (6) is given by
s
ψs = ϑ1 (s) + ϑ2 (s) exp 0 λu (1 + γ2 (u))du 1s≤τ + ϑ1 (s)(fτ − 1)1s>τ ,
hs .
φs = mξ1 (s)(fs − 1) − References
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