Semimartingale Setting
Value process and BSDE
Diffusion Model with correlation
Sufficiency of filtrations
Exponential utility maximization under partial
information and sufficiency of information
Marina Santacroce
Politecnico di Torino
Joint work with M. Mania
WORKSHOP “FINANCE and INSURANCE”
March 16-20, Jena 2009
Semimartingale Setting
Value process and BSDE
Diffusion Model with correlation
Outline
Expected utility maximization under partial information:
semimartingale setting
Semimartingale model
Expected utility and partial information
Equivalent problem and solution
Value process and BSDE
Assumptions
Equivalent problem
Value process and BSDE
FS ⊆ G
An example with explicit solution.
Diffusion model with correlation
Sufficiency of filtrations
Sufficiency of filtrations
Sufficiency of filtrations
Semimartingale Setting
Value process and BSDE
Diffusion Model with correlation
Sufficiency of filtrations
The model
• Let S = (St , t ∈ [0, T ]) be a continuous semimartingale which
represents the price process of the traded asset.
• (Ω, A, A = (At , t ∈ [0, T ]), P), where A = AT and T < ∞ is a fixed
time horizon.
• Assume the interest rate equal to zero.
The price process S admits the decomposition
Z t
St = S0 + Nt +
λu dhNiu , hλ · NiT < ∞ a.s.,
0
where N is a continuous A -local martingale and λ is a A -predictable
process (Structure condition).
Semimartingale Setting
Value process and BSDE
Diffusion Model with correlation
Sufficiency of filtrations
Expected utility maximization and partial information
Denote by G = (Gt , t ∈ [0, T ]) a filtration smaller than A
Gt ⊆ A t ,
for every t ∈ [0, T ].
G represents the information available to the hedger.
We consider the exponential utility maximization problem with random payoff
H at time T when G is the available information,
to maximize
E[−e−α(x+
RT
0
πu dSu −H)
]
over all
π ∈ Π(G ).
• x represents the initial endowment (w.l.g. we take x = 0)
• Π(G ) is a certain class of self-financing strategies (G -predictable and
S-integrable processes).
R
• ( 0t πu dSu , t ∈ [0, T ]) represents the wealth process related to the
self-financing strategy π.
This problem is equivalent to
to minimize
E[e−α(
RT
0
πu dSu −H)
]
over all
π ∈ Π(G ).
Semimartingale Setting
Value process and BSDE
Diffusion Model with correlation
Sufficiency of filtrations
In most papers, under various setups, (see, e.g., Lakner (1998), Pham and
Quenez (2001), Zohar (2001)) expected utility maximization problems have
been considered for market models where only stock prices are observed,
while the drift can not be directly observed.
=⇒ under the hypothesis F S ⊆ G .
We consider the case when G does not necessarily contain all information on
the prices of the traded asset i.e.
S is not a G -semimartingale in general!
=⇒ In this case, we solve the problem in 2 steps:
• Step 1: Prove that the expected exponential maximization problem is
equivalent to another maximization problem of the filtered terminal net
wealth (reduced problem)
• Step 2: Apply the dynamic programming method to the reduced
problem.
(In Mania et al. (2008) a similar approach is used in the context of mean
variance hedging).
Semimartingale Setting
Value process and BSDE
Diffusion Model with correlation
Sufficiency of filtrations
Filtration F and decomposition of S w.r.t. F
• Let F = (Ft , t ∈ [0, T ]) be the augmented filtration generated by F S
and G .
• S is a F -semimartingale:
t
Z
St = S0 +
)
b(F
λ
dhMiu + Mt ,
u
0
(Decomposition of S with respect to F )
Z t
)
b(F
Mt = Nt +
[λu − λ
]dhNiu is F -local martingale
u
0
b(F ) the F -predictable projection of λ.
where we denote by λ
• Note that hMi = hNi are F S -predictable.
Semimartingale Setting
Value process and BSDE
Diffusion Model with correlation
Sufficiency of filtrations
Assumptions
In the sequel we will make the following assumptions:
bF = λ
bG , hence for each t
A) hMi is G -predictable and dhMit dP a.e. λ
S
E(λt |Ft−
∨ Gt ) = E(λt |Gt ), P − a.s.
B) any G -martingale is a F -local martingale,
C) the filtration G is continuous,
D) for any G -local martingale m(g) hM, m(g)i is G -predictable,
E) H is an AT -measurable bounded random variable, such that P- a.s.
E[eαH |FT ] = E[eαH |GT ],
Semimartingale Setting
Value process and BSDE
Diffusion Model with correlation
Sufficiency of filtrations
Remarks
Note that
• If F S ⊆ G , then hMi is G -predictable. Besides, in this case G = F and
conditions A), B), D) and the equality in E) are automatically satisfied.
• Condition B) is satisfied if and only if the σ-algebras FtS ∨ Gt and GT are
conditionally independent given Gt for all t ∈ [0, T ] (Jacod 1978).
• The continuity of filtration G is weaker than the assumption that the
filtration F is continuous.
Filtration F continuous + condition B) ⇒ Filtration G is continuous
The converse is not true in general.
• The filtrations F S and F can be discontinuous.
Semimartingale Setting
Value process and BSDE
Diffusion Model with correlation
Sufficiency of filtrations
G -projection processes
bt = E(St |Gt ) be the G -optional projection of St .
Let S
bF = λ
bG = λ,
b we have:
Note that since λ
Z t
bt = E(St |Gt ) = S0 +
bt
bu dhMiu + M
λ
S
0
b t is the G -local martingale E(Mt |Gt ).
where M
• Under conditions A-D,
Z
bt =
M
0
t
dhM, m(g)iu
dmu (g) + Lt (g),
dhm(g)iu
where m(g) is any G -local martingale and L(g) is a G -local martingale
orthogonal to m(g).
b m(g)it ⇒ hM, Mi
b t = hMi
b t.
• Hence, hM, m(g)it = hM,
Using this decomposition and the Doléans equation, it is possible to show
L EMMA Under conditions A)-D),
b
E\
t (M) = E(Et (M)|Gt ) = Et (M).
Semimartingale Setting
Value process and BSDE
Diffusion Model with correlation
Sufficiency of filtrations
Equivalent problem
We recall that our aim is
E[e−α(
to minimize
RT
0
πu dSu −H)
]
over all
π ∈ Π(G ).
(1)
where the class of strategies is defined as
Π(G ) = {π : G − predictable, π · M ∈ BMO(F )}
P ROPOSITION Let conditions A)-E) be satisfied. Then the optimization
problem (1) is equivalent to
to minimize E[e−α(
RT
0
α2
bu −H)+
e
πu d S
2
RT
0
πu2 (1−κ2u )dhMiu
e = 1 ln E[eαH |GT ],
H
α
κ2t =
], over all π ∈ Π(G )
b t
dhMi
.
dhMit
Moreover, for any π ∈ Π(G )
E[e−α(
RT
t
πu dSu −H)
|Gt ] = E[e−α(
RT
t
2
α
bu −H)+
e
πu d S
2
RT
t
πu2 (1−κ2u )dhMiu
|Gt ].
(2)
Semimartingale Setting
Value process and BSDE
Diffusion Model with correlation
Sufficiency of filtrations
Sketch of the proof
Let t = 0. Taking the conditional expectation with respect to FT and using
condition E[eαH |FT ] = E[eαH |GT ], we have that
RT
RT
RT
e
E[e−α( 0 πu dSu −H) ] = E e−α 0 πu dSu E[eαH |FT ] = E e−α( 0 πu dSu −H) .
(3)
Resorting to
• the F -decomposition of S,
• the previous lemma,
b
• the decomposition of S,
we have that (3) is equal to
E[ET (−απ · M)e
α2
2
RT
0
R
e
b u dhMiu +αH
πu2 dhMiu −α 0T πu λ
= E[E(ET (−απ · M)|GT )e
b
= E[ET (−απ · M)e
= E[e−α
=
α2
2
α2
b
0 πu d M u − 2
RT
RT
0
RT
0
]
RT
e
b u dhMiu +αH
πu2 dhMiu −α 0 πu λ
]
R
e
b u dhMiu +αH
πu2 dhMiu −α 0T πu λ
2
α2
b
0 πu dhMiu + 2
RT
α2
b
e
E[e−α( 0 πu d Su −H)+ 2
RT
α2
2
RT
RT
2
0 πu dhMiu −α 0
RT
2
2
0 πu (1−κu )dhMiu
].
]
e
b u dhMiu +αH
πu λ
]
Semimartingale Setting
Value process and BSDE
Diffusion Model with correlation
Sufficiency of filtrations
Remarks
• The previous proposition says that the optimization problems (1) and (2)
are equivalent.
• It is sufficient to solve problem (2), which is formulated in terms of
G -adapted processes.
• We can see (2) as an exponential hedging problem under complete
information with a (multiplicative) correction term and we can solve it
using methods for complete information.
Let
Vt = ess inf E[e−α(
RT
t
2
α
bu −H)+
e
πu d S
2
RT
t
πu2 (1−κ2u )dhMiu
π∈Π(G )
be the value process related to the equivalent problem.
|Gt ],
Semimartingale Setting
Value process and BSDE
Diffusion Model with correlation
Sufficiency of filtrations
From now on let assume
RT 2
bt dhMit ≤ C, P − a.s..
F) 0 λ
Note that:
• c ≤ Vt ≤ C for two positive constants:
• H is bounded ⇒ Vt ≤ C.
• condition F) ⇒ Vt ≥ c for a positive c (by duality issues).
RT
Indeed, if Vt (F ) = ess infπ∈Π(F ) E[e−α( t πu dSu −H) |Ft ], then it is
well known that under condition F), Vt (F ) ≥ c, (see e.g. Delbaen
et al. (2002)). Therefore
i
h RT
Vt = ess inf E E e−α( t πu dSu −H) |Ft |Gt ≥ E(Vt (F )|Gt ) ≥ c.
π∈Π(G )
e = H and problem (1) is equivalent
• If H is GT -measurable, then H
to minimize
E[e−α(
RT
0
2
bu −H)+ α
πu d S
2
RT
0
πu2 (1−κ2u )dhMiu
].
Semimartingale Setting
Value process and BSDE
Diffusion Model with correlation
Sufficiency of filtrations
Main result
T HEOREM Under assumptions A)-F). Then, the value process V related to
the equivalent problem (2) is the unique bounded strictly positive solution of
the following BSDE
Yt = Y0 +
1
2
t
Z
0
b u Y u )2
(ψu κ2u + λ
dhMiu +
Yu
t
Z
b u + Lt
ψu d M
(4)
0
YT = E[eαH |GT ]
Moreover the optimal strategy exists in the class Π(G ) and is equal to
πt∗ =
1 b
ψt κ2t
(λ t +
).
α
Yt
(5)
• The existence of a solution to the BSDE (4) is derived by using results in
Tevzadze (2008) (see also Morlais (2008)).
• Unicity is proved by directly showing that the unique bounded solution of
the BSDE is the value process of the problem.
Semimartingale Setting
Value process and BSDE
Diffusion Model with correlation
FS ⊆ G.
What happens if FtS ⊆ Gt ?
• Gt = Ft ≡ FtS ∨ Gt
The “F decomposition” of S is the “G decomposition of S”:
Z t
bu dhMiu + Mt , M ∈ Mloc (G )
λ
St = S0 +
0
Note
b t = Mt and κ2t = 1 for all t
• M
• A), B), D) and E[eαH |FT ] = E[eαH |GT ] are satisfied
=⇒ If G is continuous, the initial problem (1) is equivalent to
to minimize
E(e−α(
RT
0
e
πu dSu −H)
)
over all
π ∈ Π(G ).
Sufficiency of filtrations
Semimartingale Setting
Value process and BSDE
Diffusion Model with correlation
Sufficiency of filtrations
C OROLLARY Let F S ⊆ G ⊆ A .
Assume G continuous, H to be a bounded AT -measurable random variable,
RT 2
bt dhMit ≤ C, P − a.s.. Then, the value process V is the unique
λ
0
bounded positive solution of the BSDE
Yt = Y0 +
1
2
t
Z
0
bu Yu )2
(ψu + λ
dhMiu +
Yu
Z
t
ψu dMu +Lt ,
YT = E(eαH |GT ). (6)
0
Moreover, the optimal strategy is equal to
πt∗ =
1 b
ψt
(λt + ).
α
Yt
R EMARK: In the case of full information Gt = At , we also have
b t = Mt = Nt ,
M
b t = λt ,
λ
YT = eαH
and Equation (6) takes on the form
Z
Z t
1 t (ψu + λu Yu )2
dhNiu +
ψu dNu + Lt ,
Yt = Y0 +
2 0
Yu
0
YT = eαH .
Semimartingale Setting
Value process and BSDE
Diffusion Model with correlation
Sufficiency of filtrations
Diffusion model with correlation
As an example we deal with a diffusion market model consisting of two
correlated risky assets one of which has no liquid market.
• First we consider an agent who is finding the best strategy to hedge a
contingent claim H, trading with the liquid asset but using just the
information on the non tradable one (partial information).
• We solve the problem for full information when H is function only of the
nontraded asset
• Then, we compare the results obtained in the cases of partial/full
information.
Semimartingale Setting
Value process and BSDE
Diffusion Model with correlation
Sufficiency of filtrations
Basis risk model
The price of the two risky assets follow the dynamics
et =S
et (µ(t, η)dt + σ(t, η)dWt1 ),
dS
dηt =b(t, η)dt + a(t, η)dWt ,
(7)
subjected to initial conditions.
• W 1 and W are standard Brownian motions with constant correlation ρ,
i.e. EdWt1 dWt = ρdt, ρ ∈ (−1, 1).
Wt = ρWt1 +
q
1 − ρ2 Wt0 ,
where W 0 and W 1 are independent Brownian motions.
• η represents the price of a nontraded asset (e.g. an index);
• Besides, we denote
dSt = µ(t, η)dt + σ(t, η)dWt1
dSt =
e
dS
t
e
S
t
⇒ S represents the process of returns of the tradable asset.
Semimartingale Setting
Value process and BSDE
Diffusion Model with correlation
Sufficiency of filtrations
Assumptions
Assume that the coefficients µ, σ, a and b are non anticipative functionals
such that:
RT 2
1) 0 σµ2 (t,η)
dt is bounded,
(t,η)
2) σ 2 > 0, a2 > 0
3) Eq. (7) admits a unique strong solution (η),
4) H is bounded and FTη ∨ σ(ξ)-measurable, where ξ is a random variable
independent of S and η.
Under conditions 2), 3) we have F S,η = F W
1
,W
and F η = F W .
Semimartingale Setting
Value process and BSDE
Diffusion Model with correlation
Sufficiency of filtrations
Partial information
Problem: An agent is hedging a contingent claim H trading with the liquid
asset S using only observations coming from η.
to minimize
E[e−α(
RT
0
πu dSu −H)
]
over all
π ∈ Π(F η ),
where π represents the dollar amount the agent invests in the stock which
depends only on η.
Ft = FtS,η ⊆ At and Gt = Ftη .
Under conditions 1)–4) (which imply that our assumptions A)–F) are
satisfied),
=⇒ we give an explicit expression of the optimal amount of money that
should be invested in the liquid asset.
Semimartingale Setting
Value process and BSDE
In this model,
Z t
Mt =
σ(u, η)dWu1 ,
0
Diffusion Model with correlation
t
Z
hMit =
σ 2 (u, η)du,
λt =
0
t
Z
bt = ρ
M
Sufficiency of filtrations
µ(t, η)
.
σ 2 (t, η)
b t = ρ2 hMit and κ2t = ρ2 .
σ(u, η)dWu ⇒ hMi
0
Conditions 1)–4) imply that assumptions A)–F) hold, indeed
• µ, σ, b, a are F η -measurable ⇒ conditions A) and D);
1
• F W ,W (resp. F W ) is the (augmented) filtration generated by W 1 and W
(resp. W ) ⇒ G = F η is continuous (condition C));
• W is a Brownian motion (also) with respect to F W
• since FT =
FTS,η ,
assumption 4) ⇒ condition E);
• condition F) is here represented by 1).
1
,W
⇒ condition B);
Semimartingale Setting
Value process and BSDE
Diffusion Model with correlation
Sufficiency of filtrations
We find that the value process related to the optimization problem is equal to
e
12
R
2
e 1 T 2
Vt = E Q [e(1−ρ )(αH− 2 t θu du) |Ftη ] 1−ρ .
(8)
The optimal strategy π ∗ is identified by
πt∗
1
=
ασ(t, η)
ρht
θt +
Rt
fu )
(1 − ρ2 )(c + 0 hu d W
• θ stands for the market price of risk θt =
!
,
µ
σ
• ht is the integrand of the integral representation
Z T
R
2
e 1 T 2
ft .
e(1−ρ )(αH− 2 0 θt dt) = c +
ht d W
0
e defined by d Qe = ET (−ρ θ · W ) is a new measure and
• Q,
R dP
e
ft = Wt + ρ t θu du is a Q-Brownian
W
motion.
0
(9)
Semimartingale Setting
Value process and BSDE
Diffusion Model with correlation
Sufficiency of filtrations
Full information
e in order to solve the
Problem: An agent is trading with a portfolio of stocks S
optimization problem
RT
e
e
to minimize E e−α( 0 πt d St −H )
over all π ∈ Π(F S,η ),
(10)
where πt represents the number of stocks held at time t and is adapted to the
e
filtration FtS,η and H = f (η) is written on the non traded asset.
• the agent builds his strategy using all market information, and
e
Gt = FtS,η = At ;
This problem was earlier studied in, e.g., Musiela and Zariphopoulou (2004),
Henderson and Hobson (2005), Davis (2006), for the basis risk model with
constant µ and σ. Assuming the Markov structure of b and a, Musiela and
Zariphopoulou gave an explicit expression for the related value function.
• Under suitable conditions on the coefficients of the model [i.e. 1)–3)]
and assuming H = f (η) and bounded, we show that the value process
related to (10) coincides with (8), and that the optimal amount of money
depends only on the observation coming from the non traded asset.
Semimartingale Setting
Value process and BSDE
Diffusion Model with correlation
Sufficiency of filtrations
Comparison
Therefore, under conditions 1)–3) and assuming H = f (η) and bounded
• we recover the result of Musiela and Zariphopoulou in a non Markovian
setting
• we find that the optimal amount of money depends only on the
observation coming from the non traded asset.
Indeed, the two optimization problems (partial/full information) are equivalent:
the corresponding value processes coincide and the optimal strategies of
these problems are related by the equality
∗
∗
et .
πt partial = π
et full S
=⇒ This means that, if H = f (η), the optimal dollar amount invested in the
assets is the same in both problems and is based only on the information
coming from the non traded asset η.
=⇒ In other words, the information coming from η is sufficient for the full
information problem...
Semimartingale Setting
Value process and BSDE
Diffusion Model with correlation
Sufficiency of filtrations
Sufficiency of filtrations
• Let Vt (A ) and Vt (G ) be the value processes of the problems
respectively in the cases of full and partial information, i.e.
Vt (A ) = ess inf E[e−α(
RT
t
πu dSu −H
RT
πu dSu −H)
π∈Π(A )
Vt (G ) = ess inf E[e−α(
t
π∈Π(G )
)|At ],
|Gt ].
• We express the optimization problem as the set
OΠ = {(Ω, A, A , P), S, H}
and we give the following definition of sufficient filtration.
D EFINITION The filtration G is said to be sufficient for the optimization
problem OΠ if V0 (G ) = V0 (A ), i.e., if
inf
π∈Π(A )
E[e−α(
RT
0
πu dSu −H)
]=
inf
π∈Π(G )
E[e−α(
⇒ Our aim is to give sufficient conditions for (11).
RT
0
πu dSu −H)
].
(11)
Semimartingale Setting
Value process and BSDE
Diffusion Model with correlation
Sufficiency of filtrations
Assumptions and G projections
We recall the A -decomposition of S
t
Z
St = S0 + Nt +
λu dhNiu .
0
We make the following assumptions, which are similar to the previous ones,
but are written with respect to A instead of F (so they are stronger):
0
A ) hNi is G -predictable,
0
B ) any G -martingale is an A -local martingale,
0
C ) the filtration G is continuous,
0
D ) λ and hN, m(g)i (for any G -local martingale m(g)) are G -predictable,
0
E ) H is a GT -measurable bounded random variable,
0 RT
F ) 0 λ2t dhNit ≤ C, P − a.s..
0
0
0
Similarly to previously, we can prove that under conditions A ), B ), C )
R
bt = E(St |Gt ) = S0 + t λ
bt ,
b dhNiu + N
• S
0 u
R
\u
b t = t ( dhN,m(g)i
• N
)dmu (g) + Lt (g), for any G local martingale m(g).
dhm(g)iu
0
b m(g)i for any G local martingale m(g), where AG
• hN, m(g)iG = hN,
denotes the dual G -predictable projection of A.
Semimartingale Setting
Value process and BSDE
Diffusion Model with correlation
Sufficiency of filtrations
Sufficient conditions
Now, we give necessary and sufficient conditions for G to be sufficient.
0
0
0
0
P ROPOSITION Let conditions A ), B ), C ) and F ) be satisfied. Then the two
value processes Vt (A ) and Vt (G ) coincide if and only if H is GT -measurable
and bounded and the process Vt (A ) satisfies the BSDE
Z
Z t
b u Y u )2
1 t (ψu ku2 + λ
bu + N
e t , YT = eαH , (12)
ψu d N
Yt = Y0 +
dhNiu +
2 0
Yu
0
e is a G -local martingale orthogonal to N
b and kt2 = dhNi
b t /dhNit .
where N
b which
• The “necessity part” relies essentially on the decomposition of N
enables to prove that, if Vt (A ) = Vt (G ), the BSDE satisfied by Vt (A )
can be written as Eq. (12).
Remark Note that we do not require condition D 0 , which is needed to find “the
equivalent problem” in the partial information case.
Semimartingale Setting
Value process and BSDE
Diffusion Model with correlation
Sufficiency of filtrations
• If F S ⊆ G , we can reduce the necessary and sufficient conditions as
follows.
C OROLLARY If F S ⊆ G and C 0 and F 0 be satisfied. Then
a) Vt (A ) = Vt (G ) if and only if H is GT -measurable and bounded and λ
is G -predictable.
0
b) If in addition condition B ) is satisfied and H is bounded and
GT -measurable, then V0 (A ) = V0 (G ) if and only if λ is G -predictable.
• Relying on the previous results, we have
0
0
T HEOREM Let conditions A )–F ) be satisfied. Then Vt (A ) = Vt (G ) and
the filtration G is sufficient.
R EMARK Note that under conditions A)–F) the filtration G is sufficient for
the auxiliary filtration F = FtS ∨ G , i.e. for the optimization problem
OΠ = {(Ω, F, F , P), S, H}.
Semimartingale Setting
Thank you.
Value process and BSDE
Diffusion Model with correlation
Sufficiency of filtrations
Appendix
References
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mathematical finance, Springer, Berlin, 2006, 169–187.
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Exponential Hedging and Entropic Penalties. Mathematical Finance 12, N.2, 2002,
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Journal of Economic Dynamics and Control 29, N.7, 2005, 1237–1266
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