Local Risk-Minimization and BSDEs Under Partial Information
Alessandra Cretarola
based on a joint work with C. Ceci and F. Russo
Frontiers in Financial Mathematics - Dublin, Ireland - June 6th, 2013
Financial motivation
GOAL
I
To provide EXISTENCE and UNIQUENESS results for the solution of BSDEs driven by a general
square-integrable martingale under partial information.
I
I
To prove, via BSDEs, a suitable version of the Föllmer-Schweizer decomposition of a square integrable random
variable working under restricted information.
I
To use this achievement to investigate the local risk-minimization (LRM) approach for a semimartingale
financial market model where agents have access to a partial information.
Examples:
1. incomplete information credit risk models where investors may have a delayed observation of the process driving the default risk;
2. financial markets where the underlying price dynamics depends on an unobservable stochastic factor.
I
Link to the existing literature
I
The scenario characterized by the PARTIAL INFORMATION FRAMEWORK represents an interesting issue
arising in many financial problems;
The valuation and hedging of derivatives in incomplete financial markets is a frequently studied problem in
Mathematical Finance.
LRM under partial information (problem formulation)
The study of the LRM approach under partial information in full generality is still an interesting topic to
discuss . Some contributions in this direction:
Föllmer & Schweizer (1991): they complete the information starting from the reference filtration and recover the optimal strategy
by means of predictable projections with respect to the enlarged filtration.
I Schweizer (1994), Ceci, Cretarola & Russo (2013): they investigate the case where the underlying price process is a (local)
martingale under the real-world probability measure.
I
Fix a probability space (Ω, F , P) endowed with a filtration F := (Ft )0≤t≤T , T ∈ (0, ∞) and F = FT .
Financial market
I one riskless asset with (discounted) price 1;
I one risky asset whose (discounted) price is described by an R-valued square-integrable (càdlàg)
F-semimartingale S = (St )0≤t≤T satisfying the so-called structure condition (SC), that is
Z t
S t = S 0 + Mt +
αs dhMis , 0 ≤ t ≤ T ,
(3)
0
I
For BSDEs driven by a general cádlág martingale beyond the Brownian setting, there exist very few results in
the literature, as far as we are aware :
where M is an R-valued square-integrable (càdlàg) F-martingale with M0 = 0 and
hR F-predictable quadratic
i
variation process denoted by hMi and α is an F-predictable process such that E 0T |αt |2dhMit < ∞.
a contingent claim ξ ∈ L2(FT , P);
I investors acting in the market can access ONLY to the information flow H := (Ht )0≤t≤T with Ht ⊆ Ft , for
each t ∈ [0, T ], about trading in the risky asset while a complete information about trading in the riskless asset.
I
Buckdahn (1993) - Backward stochastic differential equations driven by a martingale;
I El Karoui & Huang (1997) - A general result of existence and uniqueness of backward stochastic differential equations;
I Briand, Delyon & Mémin (2002) - On the robustness of backward stochastic differential equations;
I Carbone, Ferrario & Santacroce (2008) - Backward stochastic differential equations driven by cádlág martingales.;
I Ceci, Cretarola, Russo (2013) - GKW representation theorem under restricted information. An application to risk-minimization.
I
Problem: Hedging in incomplete markets
Local risk-minimization approach: looking for a hedging strategy (in general not self-financing) with minimal
cost, which keeps the replication constraint.
BSDEs under partial information (problem formulation)
Fix a probability space (Ω, F , P) endowed with a filtration F := (Ft )0≤t≤T , T ∈ (0, ∞) and F = FT ;
Ft = full information at time t.
The Föllmer-Schweizer decomposition under partial information
Assumption (C): α boundedness.
There exists a constant K̄ ≥ 0 such that the process α in (3) satisfies:
Partial information framework. We consider an additional filtration H := (Ht )0≤t≤T such that
Ht ⊆ Ft , ∀t ∈ [0, T ].
|αt (ω)| ≤ K̄ ,
(P ⊗ hMi) − a.e. on Ω × [0, T ].
Ht = available information level at time t.
Data of the problem.
I An R-valued square-integrable (cádlág) F-martingale M = (Mt )0≤t≤T with F-predictable quadratic variation
process denoted by hMi = (hM, Mi)0≤t≤T ;
I
Proposition. Let Assumptions (B) and (C) hold. Then, every ξ ∈ L2(FT , P) can be written as
Z T
ξ = U0 +
βtHdSt + AT , P − a.s.,
0
where U0 ∈ L2(F0, P), β H = (βtH)0≤t≤T ∈ M2H(0, T ) and A ∈ L2F (0, T ) is weakly orthogonal to M.
a terminal condition ξ ∈ L2(FT , P);
Definition (Optimal strategy under partial info)
a coefficient f : Ω × [0, T ] × R × R −→ R representing the driver of the equation, such that, for each
(y , z) ∈ R × R, the process f (·, ·, y , z) = (f (·, t, y , z))0≤t≤T , is F-predictable.
? A strategy Ψ = (θ, η) is (H, F)-admissible if θ ∈ M2H(0, T ) and η is an R-valued F-adapted process s.t.
the value process V (Ψ) := θS + η is right-continuous and satisfies Vt (Ψ) ∈ L2(Ft , P) for each t ∈ [0, T ].
Assumption (A): conditions on the driver f .
1. f is uniformly Lipschitz with respect to (y , z): there exists a constant K ≥ 0 such that for every
(y , z), (y 0, z 0) ∈ R × R,
0
0
0
0
|f (ω, t, y , z) − f (ω, t, y , z )| ≤ K |y − y | + |z − z | , (P ⊗ hMi) − a.e. on Ω × [0, T ];
? An (H, F)-admissible strategy Ψ = (θ, η) is called mean-self-financing if its cost process C (Ψ), that is given
by
Z
I
t
Ct (Ψ) = Vt (Ψ) −
is an F-martingale.
? Let ξ ∈ L2(FT , P) be a contingent claim. An (H, F)-admissible strategy Ψ = (θ, η) with VT (ψ) = ξ
P-a.s. is called (H, F)-optimal for ξ if Ψ is mean-self-financing and the F-martingale C (Ψ) is weakly
orthogonal to M.
0
Notations.
h
i
2
2 := E sup
2 < ∞};
I S (0, T ) := {φ = (φt )0≤t≤T càdlàg F − adapted s.t. kφk
|φ
|
t
0≤t≤T
F
S2
hR
i
T
2
2
2dhMi < ∞};
I M (0, T ) := {ϕ = (ϕt )0≤t≤T H − predictable s.t. kϕk
:=
E
|ϕ
|
s
s
2
0
H
Characterization in terms of the FS decomposition under partial info
Proposition. A contingent claim ξ ∈ L2(FT , P) admits an (H, F)-optimal strategy Ψ = (θ, η) with
VT (Ψ) = ξ P-a.s. if and only if ξ can be written as
Z T
ξ = U0 +
βtHdSt + AT , P − a.s.,
M
2
2
I M (0, T ) := {ϕ̃ = (ϕ̃t )0≤t≤T F − predictable s.t. kϕ̃k
< ∞};
F
M2
2
2 := E [hψi ] =< ∞}.
I L (0, T ) := {ψ = (ψt )0≤t≤T F − martingales with ψ0 = 0 s.t. kψk
T
F
L2
A solution of the BSDE
t
Z T
f (s, Ys−, Zs )dhMis −
t
Zs dMs − (OT − Ot ),
0 ≤ t ≤ T,
(1)
with data (ξ, f , H) under partial information, is a triplet (Y , Z , O) = (Yt , Zt , Ot )0≤t≤T of processes with
values in R × R × R satisfying (1), such that
2 (0, T ) × M2 (0, T ) × L2 (0, T ),
(Y , Z , O) ∈ SF
H
F
where O satisfies the orthogonality condition
"
#
Z T
ϕt dMt = 0,
E OT
(4)
0
Definition (Solution of BSDEs under partial info)
Yt = ξ +
∀t ∈ [0, T ],
0
2. the following integrability condition is satisfied:
"Z
#
T
E
|f (t, 0, 0)|2dhMit < ∞.
Z T
θs dSs ,
with U0 ∈ L2(F0, P), β H ∈ M2H(0, T ) and A ∈ L2F (0, T ) weakly orthogonal to M. The strategy Ψ is
then given by
θt = βtH, 0 ≤ t ≤ T
with minimal cost
Ct (Ψ) = U0 + At , 0 ≤ t ≤ T .
If (4) holds, the optimal portfolio value is
Z t
Z t
Vt (Ψ) = Ct (Ψ) +
θs dSs = U0 +
βsHdSs + At , 0 ≤ t ≤ T
0
and
(2)
0
ηt = Vt (Ψ) − βtHSt ,
0 ≤ t ≤ T.
Characterization in terms of the solution to a suitable BSDE under partial info
0
Under Assumptions (B) and (C),
for all processes ϕ ∈ M2H(0, T ).
the (H, F)-optimal strategy Ψ = (θ, η)
I the optimal portfolio value V (Ψ)
I the corresponding minimal cost C (Ψ)
I
? We will say that a square-integrable F-martingale O is weakly orthogonal to M if condition (2) holds for all
processes ϕ ∈ M2H(0, T ).
BSDEs under partial information (Existence and Uniqueness of solution)
can be characterized in terms of the unique solution (Y , Z , O) to the BSDE (1) with the particular choice of the
driver f (t, y , z) = −αt z ; more precisely,
Assumption (B): condition on the behavior of hMi.
There exists a deterministic function ρ : R+ → R+ with ρ(0+) = 0 such that, P-a.s.,
hMit − hMis ≤ ρ(t − s),
∀0 ≤ s ≤ t ≤ T .
Theorem
Let Assumptions (A) and (B) hold. Given data (ξ, f , H), there exists a unique triplet (Y , Z , O) which solves
the BSDE (1) under partial information.
Idea
To use existence and uniqueness of solution to the BSDE (1) under
partial information to prove a suitable version of the
Föllmer-Schweizer decomposition.
=⇒ Revisiting the LRM approach under partial information.
University of Perugia, Department of Mathematics and Computer Science - Via Vanvitelli 1, 06123 Perugia - Italy
V (Ψ) = Y ,
θ = Z,
C (Ψ) = O + Y0
Characterization in terms of the solution of a problem under full info
2 (0, T ) × M2 (0, T ) × L2 (0, T ) be a solution to
Let Assumptions (B) and (C) hold. Let (Ỹ , Z̃ , Õ) ∈ SF
F
F
the problem under complete information
Z T
Z T
Ỹt = ξ −
Ẑs αs dhMis −
Z̃s dMs − (ÕT − Õt ), 0 ≤ t ≤ T ,
t
t
where Õ is strongly orthogonal to M and
Z t
H
dLt
Ẑt :=
, Lt :=
Z̃s dhMis , 0 ≤ t ≤ T .
H
dhMit
0
Then the (H, F)-optimal strategy Ψ = (β H, η), the optimal portfolio value and the minimal cost are given by
βtH = Ẑt ,
Z t
Vt (Ψ) = Ỹt ,
Ct (Ψ) = Ỹ0 + Õt +
(Z̃s − Ẑs )dMs
∀t ∈ [0, T ]
0
respectively.
Contact: [email protected]