Introduction
What has been done?
What is being done here?
State-dependent utility maximization in
Lévy markets
José Figueroa-López1
1 University
Jin Ma2
of California at Santa Barbara
University
2 Purdue
KENT-PURDUE MINI SYMPOSIUM ON FINANCIAL MATHEMATICS, 2007
Conclusions
Introduction
What has been done?
What is being done here?
Outline
1
Introduction
The portfolio optimization problem
Relation to the replication of contingent claims
2
What has been done?
Solution using convex duality
3
What is being done here?
An “explicit" solution for Lévy-driven markets
4
Conclusions
Conclusions
Introduction
What has been done?
What is being done here?
Conclusions
The portfolio optimization problem
Formulation
1
Market:
A risky asset with price process St .
A bond with value process Bt .
Frictionless with continuous trading.
2
3
Goal:
Allocate an initial wealth w so that to maximize the agent’s
expected final utility during a finite time horizon [0, T ].
State-dependent utility: U(w, ω) : R+ × Ω → R s.t.
Increasing and concave with the wealth w, for each state of
nature ω ∈ Ω.
Constant for wealths w above certain threshold H(ω) > 0.
4
Problem:
Maximize E {U(VT , ω)} s.t. the agent’s wealth process
{Vt }0≤t≤T satisfies V0 ≤ w and V· ≥ 0.
Utility function graph
Introduction
What has been done?
What is being done here?
Conclusions
Relation to the replication of contingent claims
Super-replication
1
Setting
The discounted asset price {Bt−1 St } is a (locally bounded)
semimartingale.
The class M of Equivalent Martingale Measures (EMM) is
non-empty.
2
The hedging problem
An agent needs to deliver the payoff H of a claim at time T .
He wishes to hedge away the risk by investing in the market
3
Super-replication: Any liability H can be hedged away
completely, when starting with large enough wealth.
The Fundamental Theorem
(Kramkov 97)
−1 For any w ≥ w̄ := supQ∈M EQ BT H , there exists an
admissible {Vt }t≤T such that V0 = w and VT ≥ H
Introduction
What has been done?
What is being done here?
Conclusions
Relation to the replication of contingent claims
Super-replication
1
Setting
The discounted asset price {Bt−1 St } is a (locally bounded)
semimartingale.
The class M of Equivalent Martingale Measures (EMM) is
non-empty.
2
The hedging problem
An agent needs to deliver the payoff H of a claim at time T .
He wishes to hedge away the risk by investing in the market
3
Super-replication: Any liability H can be hedged away
completely, when starting with large enough wealth.
The Fundamental Theorem
(Kramkov 97)
For any w ≥ w̄ := supQ∈M EQ BT−1 H , there exists an
admissible {Vt }t≤T such that V0 = w and VT ≥ H
Introduction
What has been done?
What is being done here?
Conclusions
Relation to the replication of contingent claims
Super-replication
1
Setting
The discounted asset price {Bt−1 St } is a (locally bounded)
semimartingale.
The class M of Equivalent Martingale Measures (EMM) is
non-empty.
2
The hedging problem
An agent needs to deliver the payoff H of a claim at time T .
He wishes to hedge away the risk by investing in the market
3
Super-replication: Any liability H can be hedged away
completely, when starting with large enough wealth.
The Fundamental Theorem
(Kramkov 97)
For any w ≥ w̄ := supQ∈M EQ BT−1 H , there exists an
admissible {Vt }t≤T such that V0 = w and VT ≥ H
Introduction
What has been done?
What is being done here?
Conclusions
Relation to the replication of contingent claims
Super-replication
1
Setting
The discounted asset price {Bt−1 St } is a (locally bounded)
semimartingale.
The class M of Equivalent Martingale Measures (EMM) is
non-empty.
2
The hedging problem
An agent needs to deliver the payoff H of a claim at time T .
He wishes to hedge away the risk by investing in the market
3
Super-replication: Any liability H can be hedged away
completely, when starting with large enough wealth.
The Fundamental Theorem
(Kramkov 97)
For any w ≥ w̄ := supQ∈M EQ BT−1 H , there exists an
admissible {Vt }t≤T such that V0 = w and VT ≥ H
Introduction
What has been done?
What is being done here?
Conclusions
Relation to the replication of contingent claims
Super-replication
1
Setting
The discounted asset price {Bt−1 St } is a (locally bounded)
semimartingale.
The class M of Equivalent Martingale Measures (EMM) is
non-empty.
2
The hedging problem
An agent needs to deliver the payoff H of a claim at time T .
He wishes to hedge away the risk by investing in the market
3
Super-replication: Any liability H can be hedged away
completely, when starting with large enough wealth.
The Fundamental Theorem
(Kramkov 97)
For any w ≥ w̄ := supQ∈M EQ BT−1 H , there exists an
admissible {Vt }t≤T such that V0 = w and VT ≥ H
Introduction
What has been done?
What is being done here?
Conclusions
Relation to the replication of contingent claims
Super-replication
1
Setting
The discounted asset price {Bt−1 St } is a (locally bounded)
semimartingale.
The class M of Equivalent Martingale Measures (EMM) is
non-empty.
2
The hedging problem
An agent needs to deliver the payoff H of a claim at time T .
He wishes to hedge away the risk by investing in the market
3
Super-replication: Any liability H can be hedged away
completely, when starting with large enough wealth.
The Fundamental Theorem
(Kramkov 97)
For any w ≥ w̄ := supQ∈M EQ BT−1 H , there exists an
admissible {Vt }t≤T such that V0 = w and VT ≥ H
Introduction
What has been done?
What is being done here?
Conclusions
Relation to the replication of contingent claims
Optimal partial replication
1
Motivation:
w̄ is typically “too high".
Initial wealth < w̄ =⇒ Shortfall risk (sometimes, VT < H).
2
The shortfall minimization problem:
Minimize E L (H − VT )+
s.t. V0 ≤ w
and Vt ≥ 0,
where L is increasing, convex, and null at 0.
3
Equivalent formulation: (Föllmer & Leukert, 2000)
Maximize E {U (VT , ω)}
s.t. V0 ≤ w
and
Vt ≥ 0,
with U(w; ω) := L(H(ω)) − L((H(ω) − w)+ ).
Utility function graph
Introduction
What has been done?
What is being done here?
Solution using convex duality
The dual problem
1
Primal problem:
p∗ (w) := sup E {U(VT , ω)}.
V :V0 ≤w
2
The dual domain e
Γ:
Nonnegative supermartingales {ξt }t≥0 ;
0
n ≤ ξ0 ≤ 1;
o
ξt Bt−1 Vt
is a supermartingale for all {Vt }t≤T .
t≤T
3
The dual problem:
n
o
e
d ∗ (λ) := inf E U(λξ
BT−1 , ω) ,
T
ξ∈e
Γ
e is the convex dual function of U.
where U
Conclusions
Introduction
What has been done?
What is being done here?
Solution using convex duality
Relationship between the dual and primal problems
Theorem ([KrSch 99], [FllmLkrt, 2000])
1
Weak duality: p∗ (w) ≤ d ∗ (λ) + λw, for all λ > 0.
2
Strong duality: p∗ (w) = d ∗ (λ∗ ) + λ∗ w, for some λ∗ > 0.
3
The dual problem is attainable at some ξ ∗ .
4
The primal problem is attainable at some admissible V ∗ .
5
Dual characterization of the optimal final wealth:
VT∗ = I λ∗ ξT∗ BT−1 ,
where I(z, ω) := (U 0 )−1 (z) ∧ H.
Conclusions
Introduction
What has been done?
What is being done here?
Conclusions
Solution using convex duality
Refinements for concrete models
A natural problem
For a specific market model and a given utility function,
Can one narrow down the dual domain Γ ⊂ e
Γ where to
search ξ ∗ ?
1
Itô incomplete market: (Karatzas et. al. 91)
dSt = St n{bt dt + σt dWt }.
o
Rt
Rt
R·
ξt = exp 0 G(s)dWs − 12 0 kG(s)k2 ds = E( 0 G(s)dWs ).
2
Lévy market: (Kunita 03)
n
o
R
e
dSt = St − bt dt + σt dWt + Rd h(t, z)N(dt,
dz)
R
R·R
·
e
ξt = E 0 G(s)dWs + 0 Rd F (t, z)N(dt,
dz) .
Introduction
What has been done?
What is being done here?
Conclusions
Solution using convex duality
Refinements for concrete models
A natural problem
For a specific market model and a given utility function,
Can one narrow down the dual domain Γ ⊂ e
Γ where to
search ξ ∗ ?
1
Itô incomplete market: (Karatzas et. al. 91)
dSt = St n{bt dt + σt dWt }.
o
Rt
Rt
R·
ξt = exp 0 G(s)dWs − 12 0 kG(s)k2 ds = E( 0 G(s)dWs ).
2
Lévy market: (Kunita 03)
n
o
R
e
dSt = St − bt dt + σt dWt + Rd h(t, z)N(dt,
dz)
R
R·R
·
e
ξt = E 0 G(s)dWs + 0 Rd F (t, z)N(dt,
dz) .
Introduction
What has been done?
What is being done here?
Conclusions
Solution using convex duality
Refinements for concrete models
A natural problem
For a specific market model and a given utility function,
Can one narrow down the dual domain Γ ⊂ e
Γ where to
search ξ ∗ ?
1
Itô incomplete market: (Karatzas et. al. 91)
dSt = St n{bt dt + σt dWt }.
o
Rt
Rt
R·
ξt = exp 0 G(s)dWs − 12 0 kG(s)k2 ds = E( 0 G(s)dWs ).
2
Lévy market: (Kunita 03)
n
o
R
e
dSt = St − bt dt + σt dWt + Rd h(t, z)N(dt,
dz)
R
R·R
·
e
ξt = E 0 G(s)dWs + 0 Rd F (t, z)N(dt,
dz) .
Introduction
What has been done?
What is being done here?
Conclusions
Solution using convex duality
Refinements for concrete models
A natural problem
For a specific market model and a given utility function,
Can one narrow down the dual domain Γ ⊂ e
Γ where to
search ξ ∗ ?
1
Itô incomplete market: (Karatzas et. al. 91)
dSt = St n{bt dt + σt dWt }.
o
Rt
Rt
R·
ξt = exp 0 G(s)dWs − 12 0 kG(s)k2 ds = E( 0 G(s)dWs ).
2
Lévy market: (Kunita 03)
n
o
R
e
dSt = St − bt dt + σt dWt + Rd h(t, z)N(dt,
dz)
R
R·R
·
e
ξt = E 0 G(s)dWs + 0 Rd F (t, z)N(dt,
dz) .
Introduction
What has been done?
What is being done here?
Conclusions
Solution using convex duality
What is being used and assumed?
Key result
Kunita-Watanabe (1967)
ξ is a positive local martingale iff ξt = ξ0 E(X ) with
Z
Xt :=
t
Z tZ
G(s)dWs +
0
e
F (s, z)N(ds,
dz),
F > −1.
0
ξ is a positive supermartingale iff ξt = ξ0 E(X − A) where X
is as above and A is increasing predictable s.t. ∆A < 1.
Assumptions on the utility function
Strictly increasing and concave.
Inada conditions: U 0 (0+ ) = ∞ and U 0 (∞) = 0.
U is unbounded.
Introduction
What has been done?
What is being done here?
Conclusions
An “explicit" solution for Lévy-driven markets
Set-up
1
The model:
dSt = St − (b dt + d Zt ), where Z is a Lévy process.
Bt ≡ 1.
2
Primal problem:
p∗ (w) := sup E {U(VT , ω)}.
V :V0 ≤w
where U is a bounded state-dependent utility function.
3
Dual problem:
n
o
e
dΓ∗ (λ) := inf E U(λξ
, ω) ,
T
ξ∈Γ
where Γ is a suitable subclass of e
Γ to be chosen so that the
“dual theorem" holds.
Introduction
What has been done?
What is being done here?
An “explicit" solution for Lévy-driven markets
The Dual Theorem
Let Γ be a convex subclass of e
Γ such that
(i) w̄Γ := supξ∈Γ E {ξT H} < ∞
(ii) Γ is closed under “Fatou convergence”.
Then, for each 0 < w < w̄Γ , there exist λ∗ > 0 and ξ ∗ ∈ Γ s.t.
o
n
e ∗ ξ , ω) is attainable at ξ ∗
1
dΓ∗ (λ∗ ) := infξ∈Γ E U(λ
T
∗
2
E ξT I λ∗ ξT∗ = w, where I(z) := (U 0 )−1 (z) ∧ H
3
p∗ (w) ≤ E U I λ∗ ξT∗
Furthermore, if
i
h
for any EMM Q ∈ M
|F
(iii) Γ contains ξt := E dQ
t
dP
then I λ∗ ξT∗ is super-replicable by an admissible V ∗ s.t.
V0∗ = w. Hence, V ∗ solves the primal problem.
Conclusions
Introduction
What has been done?
What is being done here?
Conclusions
An “explicit" solution for Lévy-driven markets
Construction of the dual class Γ
1
Let S be a family of loc. bounded local martingales s.t.
S is predictable convex
S is closed under Émery distance
Then, the class
b
Γ := {ξ := ξ0 E (X − A) : X ∈ S, A increasing, and ξ ≥ 0},
2
is convex and closed under “Fatou convergence".
The class
Z t
Z tZ
e
S := {Xt :=
G(s)dWs +
F (s, z)N(ds,
dz) : F ≥ −1},
0
3
0
meets the above conditions.
The class Γ := b
Γ∩e
Γ fulfills the conditions necessary for the
Dual Theorem.
Introduction
What has been done?
What is being done here?
Conclusions
Conclusions
The method here is more explicit in the sense that the dual
domain enjoys an explicit parametrization.
Such a parametrization could lead to certain discrete time
approximations.
It can accommodate more general jump-diffusion models
driven by Lévy processes such as
R
P
e
dS i (t) = S i (t − ){bi dt + d σ ij dW j + d h(t, z)N(dt,
dz)}.
t
j=1
t
t
R
What about optimal portfolio problems with consumption,
(
)
Z T
sup
E U1 (VT ) +
U2 (t, c(t))dt ,
V :V0 ≤w & V· ≥0
0
where dVt := rVt dt + βt dSt − ct dt?
Graphs
Bibliography
Utility function and its convex dual function
e
U(z,
ω) := sup {U(w, ω) − z w} = U(I(z)) − z I(z),
0≤w≤H
e 0 (z).
I(z) := (U 0 )−1 (z) ∧ H = −U
Return 1
Return 2
Return 3
Graphs
Bibliography
Bibliography
Figueroa-Lopez and Ma. State-dependent utility maximization in Lévy markets, Preprint. Available at
www.pstat.ucsb.edu/faculty/figueroa.
Kramkov and Schachermayer. The asymptotic elasticity of utility functions and optimal investment in
incomplete markets. Finance and Stochastics, 1999.
Föllmer and Leukert. Efficient hedging: Cost versus shortfall risk. Finance and Stochastics, 2000.
Karatzas, Lehoczky, Shreve, and Xu. Martingale and duality methods for utility maximization in an
incomplete market. SIAM J. Control and Optimization, 1991.
Kunita. Variational equality and portfolio optimization for price processes with jumps. In Stoch. Proc. and
Appl. to Mathem. Fin., 2003.
Kramkov. Optional decomposition of supermartingales and pricing of contigent claims in incomplete security
markets. Prob. Th. and Rel. fields, 1996.