On portfolio delegation with moral hazard under translation ... - HIM

On portfolio delegation with moral hazard under
translation invariance
Julio Daniel Backhoff Veraguas 1
Humboldt-Universität zu Berlin
Department of Mathematics
May 27, 2013
1 Work is part of thesis-project under supervision of Ulrich Horst

Overview
• Portfolio delegation as a Principal-Agent game
• Reduction to a single recursion/BS∆E
• Existence of optimal contract and strategies
• Markovian setting
• A related problem and continuous-time analog
• Further research

Context
• There is an investor (P) and a manager (A)
• (P) delegates management of her investments to (A)
• (A) has to be rewarded for his effort, which is costly
• (P) designs a contract specifying payment structure of deal
• (A) acts in his own self-interest; has an opportunity cost R
• (P) wants to maximize her own well-being
• (P) should not monitor/observe everything (A) does

Setup
• Time is discrete: t ∈ T := {0, .., .T }
• Uncertainty is described by (Ω, F , {F t }, P)
• The N-dim. price process {P t } is F t −adapted
• At time t ∈ T, (A) and (P) have access to F t
• (A) has fixed initial cash endowment W 0

The financial market, wealth and cost
• (A) trades in the financial market comprising the N assets
with price at time t equal P t
Denote ∆˜P t+1 := diag(P t ) −1 ∆P t+1
• Choosing a strategy {A t }, the (A) generates wealth for (P):
∆W t+1 = A t · ∆˜P t+1
• Associated with each strategy {A t } is a cost (of effort)
C := ∑ t
c t (A t )∆t

Payments and preferences
• The payment from (P) to (A) (contract) is assumed linear:
T∑
−1
S = ε T + β t · ∆W t+1
where ε T is F T -measurable (not just F 0 -measurable; key!)
• Terminal wealth of (P), respectively (A), is
0
W T − S respectively S − ∑ t∈T
c t (A t ) · ∆t
• At each time t ∈ T, (A) and (P) maximize conditionally
concave, translation invariant, time-consistent, ... preference
functionals 2 U a t , U p t : L 0 (F T ) → L 0 (F t )
2 See work of Cheridito, Horst or Kupper

Comments
• The Agent’s optimization problems is:
Find an optimal strategy taking into account the
contractual agreements.
• The Principal’s problem is
Find an optimal contract taking into account
agent’s reaction function and outside option.
• For an optimal contract it will be enough to depend on final
wealth rather than its whole path 3
• Main idea: make utility of (A) a control variable for (P) 4
3 As obtained by Ou-Yang
4 As done by Sannikov

The Agent’s problem
• (A)’s problem is a standard conditional optimization problem
• Let H a t
utility enjoyed by/promised to (A) from t ∈ T on:
H a T = ε T
H a t = ess sup
A∈L 0 (Ft ) N {U a t (H a t+1 +βtA·∆˜P t+1)−c t(A)∆t}
• Formally if H a t+1 = E ( H a t+1 |F t)
+ x
a
t+1 , we get a BS∆E:
∆Ht+1 a =xa t+1 − ess sup {U a
A∈L 0 (Ft ) N t (xt+1 a +βtA·∆˜P t+1)−c t(A)∆t}
• Controlling ε T steers H a (hence x a ) and makes H a 0 = R.
Therefore, reinterpret BS∆E as an S∆E starting from R

Attainability
Theorem (Existence of optimal strategies)
Assume c t (·) is convex and lsc, Ut
a is seq-usc, and suppose
Ht+1 a < ∞ has been found. Then, Ha t is attained by some
A ∗ ∈ L 0 (F t ), under any of these conditions:
• U a t (·) is sensitive to large losses and c t (·) ≥ κ t + λ t | · |
• U a t (·) ≤ K t + E (·|F t ) and
c
lim t(A)
|A|→±∞
|A|
= ∞
Remark: Idea is to use No-Arbitrage arguments plus randomized
Bolzano-Weierstrass

The Principal’s problem
• (P)’s problem is a constrained conditional optimization one
• (P) choses strategies and contract subject to two conditions:
• incentive compatibility (IC)
• individual rationality (IR)
• (P)’s problem consists in solving for all time s ∈ T:
u p s =
ess sup Us p (W T − S(A))
ε,(β k ) T −1
k=0
A k ∈arg max H
k
a
H
0 a ≥R
The first constraint is the (IC), the second the (IR) constraint.

The Principal’s problem
• If A is optimal for (A), then the value of the contract is
S = H a 0 +∑ {c t(A t)∆t+H a t+1 −Ua t (H a t+1 +βtAt·∆˜P t+1)
+β tA t·∆˜P t+1 }
• (P)’s utility from t ∈ T on, under a given contract is thus:
Ut
p ( ∑
)
s≥t Ua s (...)−c s(A s)∆t−(β s−1)A s·∆˜P s+1 −Hs+1
a
• (P) can steer H a by controlling ε T

The principal’s problem
Theorem
Let h p s denote (P)’s optimal wealth from time s onwards. Then:
h p T
= 0
h p t = ess sup U p t (h p t+1 −x−(β−1)A·∆˜P t+1)−c(A)∆t
x∈F t+1 (A,β)
+U a t (x+βA·∆˜P t+1)
and u p 0 = W 0 − R + h p 0

The principal’s problem
• (P) controls (A)’s wealth through x(= Ht+1 a − E ( )
t H
a
t+1 )
• x ∈ F t+1 (A, β) means that when faced with contract (β) and
a future wealth prospect x, (A) chooses A
• Problem can be reformulated in terms of (β, A) and
Γ := X + βA · ∆˜P t+1 ∈ L 0 (F t+1 ) alone:
ht p }
= ess sup
{U p t (h p t+1 +A·∆˜P t+1 −Γ)−c(A)∆t+Ut a(Γ) (A,β,Γ)
(A,Γ)∈Ct (β)

Attainability of Principal’s problem
• It is sufficient to consider an unconstrained problem:
V (A,Γ):=U p t (h p t+1 +A·∆˜P t+1 −Γ)−c(A)∆t+U a t (Γ)
• Recall standard convolution problems arising in moles of
optimal risk sharing are of the form:
U p t (Y − Γ) + U a t (Γ)
and this quantity is to be maximized over Γ ∈ L 0 (F t+1 )
• We have complex dependencies/constraints, because
h p t = ess sup {(A,Γ)∈Ct(β)}V (A, Γ)

Attainability of Principal’s problem
Theorem
• Suppose the unrestricted problem ess sup (A,Γ)
V (A, Γ) is finite
and attained at (A ∗ , Γ ∗ ). Then
(
1, Γ ∗ , A ∗)
forms an optimal contract and effort tuple.
• The unconstrained problem has a solution if V (·, ·) ∈ L 0 (F t ),
Ut
a,p (·) = ess inf Z∈L 1
t (F T ) {αa,p t (Z) + E[·Z|F t ]} plus technical
conditions
Remark: Idea is to use Komlos theorem for the Γs and random
Bolzano-Weierstrass for the As

Attainability of Principal’s problem
Theorem
Suppose Ut a,p (·) = 1
γ
U a,p t (γ a,p·) and that U satisfies conditions for
attainability of Agent’s problem.
Then unconstrained problem has a solution, the optimal A ∗ attains
ess sup
{−c(A)∆t + 1 (
])
A
γ U γ
[h }
p t+1 + A∆˜P t+1
where γ = γa γ p
γ a +γ p , and Γ ∗ =
[
]
γp
γ a +γ
h p p t+1 + A∗ ∆˜P t+1 is optimal 5
5 Inspired by a similar result by Barrieu and El-Karoui

Predictable representation property
Definition
The model has the Predictable representation property (PRP) if
there is an adapted m-dimensional (uncorrelated) process M such
that:
L 0 (F t+1 )={z+Z 1·∆P t+1 +Z 2·∆M t+1 :z∈L 0 (F t),Z 1 ∈[L 0 (F t)] d ,Z 2 ∈[L 0 (F t)] M }
and R = (P, M) has the No-Arbitrage property.
Identify Γ ∈ L 0 (F t+1 ), where E[Γ|F t ] = 0, with γ ∈ [L 0 (F t )] d+M
Theorem
Suppose the PRP holds as well as the usual mild conditions on Ut a ,
Ut
p and c. If the Us are sensitive to large losses then unconstrained
problem is attained.

Markovian models
Suppose ∆P t+1 = P t [µ∆t + σ∆B t+1 ] 6
Definition (Markovian model)
If the function g(z) := U t (z · ∆B t+1 ) is deterministic, and PRP
holds for B, we say that the problem is Markovian.
• Under suitable conditions we have the following FOCs:
0 = [βµ − ∇c(A)] ∆t + βσ∇g a (γ)
0 = [µ − ∇c(A)]∆t + σ∇g p (σ ′ A − γ)
0 = ∇g a (γ) − ∇g p (σ ′ A − γ)
• Problem reduces to an “unconstrained” one as before and
necessarily (β − 1)[µ∆t + σ∇g a (γ)] = 0
6 This is more related to Ou-Yang’s work

Markovian models
Remark: Optimal ε is of the form k + ∑ γ t ∆B t+1 − ˜W T with k
and γs constant and ˜W T the optimal portfolio wealth
Remark: If further utilities have same structure,
ε = k −
γa ˜W γ a +γ p T and hence optimal contract has the structure 7
A ↦→ S(A) = k +
Example (Entropic utility function)
γp
γ a + γ p W T A +
γa
γ a + γ p [W T A − ˜W T ]
If uncertainty is spanned by Bernoulli walks and
U t (X ) = − 1 γ log (E t [exp (−γX )]) , then
g(z) = − 1 ( (cosh
γ log −γz √ ))
∆t
7 As in Ou-Yang

A related problem
Suppose contracts had had the form
with ε ∈ L 0 (F 0 ) only.
S = ε + ∑ [β t ∆W t+1 + ν t ∆˜P t+1 ]
• (A)’s utility not so readily controllable
• Translation Invariance still useful: we get two BS∆Es
• Conditional optimization problems easier than before: problem
at time t involves only F t −measurable rvs. even without PRP
• Still β t ≡ 1 is optimal if a related unconstrained problem is
attainable

Continuous-time analog
• Define utilities through BSDEs with suitable generators:
∫ T
∫ T
U s (X ) = X + g t (Z t ) dt − Z t · dB t
s
s
• By assumption, PRP and Markovianity hold (simpler problem)
• H a and h p satisfy analogous BSDEs, by comparison principle
• Generalizes HJB-approach typical of these type of problems
• Non translation-invariant preferences lead to FBSDEs 8
8 See work of Cvitanic et al.

Comments
• Essentially, translation invariance makes BSDE-approach
tractable
• With conditional analysis, we can go well beyond classical
utility functions
• The price-driving process can be much general than the usual
ones
• Makes no use of Maximum Principle, which in general is
troublesome

To be better understood
• Explore conditional law invariance and relate to risk sharing
and the problems above
• Study the asymmetric information case directly (through
having different filtrations for (A) and (P))
• Move on to more general continuous-time models; extend
conditional analysis to continuous time
• ...

Thanks