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IntroductionRobust Markowitz problem
Robust efficient frontier
Robust Markowitz portfolio selection underambiguous covariance matrix
Huyen PHAM ∗
∗University Paris Diderot, LPMASorbonne Paris Cite
Based on joint work with
A. Ismail, Natixis
MFOMarch 2, 2017
Huyen PHAM Robust Markowitz portfolio selection
IntroductionRobust Markowitz problem
Robust efficient frontier
Outline
1 Introduction
2 Robust Markowitz problemRobust settingExplicit solutionsMcKean-Vlasov DP approach : elements of proof
3 Robust efficient frontier and Sharpe ratio
Huyen PHAM Robust Markowitz portfolio selection
IntroductionRobust Markowitz problem
Robust efficient frontier
Classical Markowitz formulation in continuous time
I Xα = (Xαt )t wealth process with α = (αt) : amount invested in risky
assets at any time t ∈ [0,T ], T < ∞ investment horizon
• Markowitz criterion : on (Ω,F ,P)
maximize over α : E[XαT ] subject to Var(Xα
T ) 6 ϑ
→ U0(ϑ) : maximal expected return given risk ϑ > 0
→ Graph of U0 : Efficient frontier
• Lagrangian mean-variance criterion :
V0(λ) ← minimize over α : λVar(XαT )− E[Xα
T ],
I Duality relation :V0(λ) = infϑ>0
[λϑ− U0(ϑ)
], λ > 0,
U0(ϑ) = infλ>0
[λϑ− V0(λ)
], ϑ > 0.
Huyen PHAM Robust Markowitz portfolio selection
IntroductionRobust Markowitz problem
Robust efficient frontier
Optimal MV portfolio in BS model
• Multidimensional BS model : risk-free asset ≡ 1, d stocks with
b ∈ Rd : vector of assets returnΣ ∈ Sd>+ : covariance matrix of assets
I Optimal amount invested in the d stocks :
α∗t =
(E[X ∗
t ]− X ∗t +
1
2λeR(T−t)
)Σ−1b
=(x0 +
1
2λeRT − X ∗
t
)Σ−1b, 0 6 t 6 T .
R := bᵀΣ−1b ∈ R : (square) of risk premium of the d stocksI
ϑ ↔ λ = λ(ϑ) =
√eRT − 1
4ϑ.
→ Efficient frontier : straight line in mean/standard deviation diagram
U0(ϑ) = x0 +√
eRT − 1√ϑ, ϑ > 0.
• Ref : Zhou, Li (00), Andersson-Djehiche (11), Fisher-Livieri (16), P., Wei
(16).
Huyen PHAM Robust Markowitz portfolio selection
IntroductionRobust Markowitz problem
Robust efficient frontier
Robust portfolio optimization
• Inacurracy in parameters estimation :
Drift : well-known
Correlation : asynchronous data and lead-lag effect
I Portfolio optimization with Knightian uncertainty (ambiguity) onmodel ↔ set of prior subjective probability measures :
ambiguity on return/drift : Hansen, Sargent (01), Gundel (05),Schied (11), Tevzadze et al. (12), etc
ambiguity on volatility matrix : Denis, Kervarec (07), Matoussi,Possamai, Zhou (12), Fouque, Sun, Wong (15), Riedel, Lin (16), etc
• Our main contributions :
Markowitz criterion
Ambiguity on covariance matrix
Explicit solutions, robust efficient frontier, and lower bound forrobust Sharpe ratio
Huyen PHAM Robust Markowitz portfolio selection
IntroductionRobust Markowitz problem
Robust efficient frontier
Robust settingExplicit solutionsMcKean-Vlasov DP approach : elements of proof
Robust framework
• Canonical space Ω = C ([0,T ],Rd) : continuous paths of d stocks
→ B = (Bt)t canonical process, P0 : Wiener measure, F = (Ft)tcanonical filtration
• Drift b ∈ Rd of the assets is assumed to be known (well-estimated orstrong belief) but uncertainty on the covariance matrix, possibly random(even rough !)
Huyen PHAM Robust Markowitz portfolio selection
IntroductionRobust Markowitz problem
Robust efficient frontier
Robust settingExplicit solutionsMcKean-Vlasov DP approach : elements of proof
Ambiguous covariance matrix : Epstein-Ji (11)
• Γ compact set of Sd>+ : prior realizations of covariance matrix
• Γ = Γ(Θ) parametrized by convex set Θ of Rq : there exists somemeasurable function γ : Rq → Sd>+ s.t.
Any Σ in Γ is in the form : Σ = γ(θ) for some θ ∈ Θ.
• Concavity assumption (IC) :
γ(1
2(θ1 + θ2)
) 1
2
(γ(θ1) + γ(θ2)
)(1)
Remark : in examples, we have = in (1).
• Notation : for Σ ∈ Γ, we set,
σ = Σ12 , the volatility matrix.
Huyen PHAM Robust Markowitz portfolio selection
IntroductionRobust Markowitz problem
Robust efficient frontier
Robust settingExplicit solutionsMcKean-Vlasov DP approach : elements of proof
Examples
• Uncertain volatilities for multivariate uncorrelated assets :
Θ =d∏
i=1
[σ2i , σ
2i ], 0 6 σi 6 σi <∞,
γ(θ) =
σ21 . . . 0...
. . ....
0 . . . σ2d
, for θ = (σ21 , . . . , σ
2d).
• Ambiguous correlation in the two-assets case :
γ(θ) =
(σ2
1 σ1σ2θσ1σ2θ σ2
2
), for θ ∈ Θ = [%, %] ⊂ (−1, 1),
for some known constants σ1, σ2 > 0.
Huyen PHAM Robust Markowitz portfolio selection
IntroductionRobust Markowitz problem
Robust efficient frontier
Robust settingExplicit solutionsMcKean-Vlasov DP approach : elements of proof
Prior (singular) probability measures
VΘ : set of F-adapted processes Σ = (Σt)t valued in Γ = Γ(Θ)
PΘ =Pσ : Σ ∈ VΘ
,
with
Pσ := P0 (Bσ)−1, σt = Σ12t , B
σt :=
∫ t
0
σsdBs , P0 a.s.
In other words :
d < B >t = Σtdt under Pσ.
Remark : connection with the theory of G -expectation (Peng), andquasi-sure analysis (Denis/Martini, Soner/Touzi/Zhang, Nutz).We say PΘ − q.s. : Pσ − a.s. for all Σ ∈ VΘ.
Huyen PHAM Robust Markowitz portfolio selection
IntroductionRobust Markowitz problem
Robust efficient frontier
Robust settingExplicit solutionsMcKean-Vlasov DP approach : elements of proof
Assets price and wealth dynamics under covariance matrixuncertainty
• Price process S of d stocks :
dSt = diag(St)(bdt + dBt), 0 6 t 6 T , PΘ − q.s.
• Set A of portfolio strategies : F-adapted processes α valued in Rd s.t.
supPσ∈PΘ Eσ[∫ T
0αᵀt Σtαtdt] < ∞
→ Wealth process Xα :
dXαt = αᵀ
t diag(St)−1dSt
= αᵀt (bdt + dBt), 0 6 t 6 T , Xα
0 = x0, PΘ − q.s.
Huyen PHAM Robust Markowitz portfolio selection
IntroductionRobust Markowitz problem
Robust efficient frontier
Robust settingExplicit solutionsMcKean-Vlasov DP approach : elements of proof
Robust Markowitz mean-variance formulation
• Robust Markowitz problem :
(Mϑ)
maximize over α ∈ A, E(α) := infPσ∈PΘ Eσ[Xα
T ]subject to R(α) := supPσ∈PΘ Varσ(Xα
T ) 6 ϑ.
→ U0(ϑ), ϑ > 0 : robust efficient frontier
• “Lagrangian” robust mean-variance problem : given λ > 0,
(Pλ) V0(λ) = infα∈A
supPσ∈PΘ
(λVarσ(Xα
T )− Eσ[XαT ])
Not clear a priori that U0 and V0 are conjugates of each other !
supPσ∈PΘ
(λVarσ(Xα
T )− Eσ[XαT ])6= λR(α)− E(α)
Huyen PHAM Robust Markowitz portfolio selection
IntroductionRobust Markowitz problem
Robust efficient frontier
Robust settingExplicit solutionsMcKean-Vlasov DP approach : elements of proof
Solution to (Pλ)
• Worst case scenario ↔ constant covariance matrix Σ∗ = γ(θ∗)minimizing the risk premium :
θ∗ ∈ argminθ∈Θ
R(θ), R(θ) := bᵀγ(θ)−1b.
• Optimal robust MV strategy = optimal MV strategy in the BS modelwith covariance matrix Σ∗ ↔ R∗ = bᵀ(Σ∗)−1b :
α∗t =(x0 +
1
2λeR
∗T − X ∗t)(Σ∗)−1b, 0 6 t 6 T .
Key remark :
Eσ[X ∗T ] = x0 +1
2λ
[eR
∗T − 1]
does not depend on Pσ
Huyen PHAM Robust Markowitz portfolio selection
IntroductionRobust Markowitz problem
Robust efficient frontier
Robust settingExplicit solutionsMcKean-Vlasov DP approach : elements of proof
Example : ambiguous correlation in the two-assets case
• Known marginal volatilities σi , and drift bi , i = 1, 2, but unknowncorrelation lying in Θ = [%, %] ⊂ (−1, 1).
→ Instantaneous Sharpe ratio of each asset :
βi =biσi> 0, i = 1, 2,
I We set :
%0 :=min(β1, β2)
max(β1, β2)∈ (0, 1]
as a measure of “proximity” between the two stocks.
Huyen PHAM Robust Markowitz portfolio selection
IntroductionRobust Markowitz problem
Robust efficient frontier
Robust settingExplicit solutionsMcKean-Vlasov DP approach : elements of proof
Case 1 : % < %0
• Worst case scenario : Σ∗ = Σ := γ(%) ↔ highest correlation
α∗t =(x0 +
1
2λeb
ᵀΣ−1bT − X ∗t
)Σ−1b, 0 6 t 6 T , PΘ − q.s.
Moreover, the two components of Σ−1b have the same sign : directionaltrading with worst-case scenario corresponding to highest correlation θ∗
= %, i.e. diversification effect is minimal.
Huyen PHAM Robust Markowitz portfolio selection
IntroductionRobust Markowitz problem
Robust efficient frontier
Robust settingExplicit solutionsMcKean-Vlasov DP approach : elements of proof
Case 2 : % > %0
• Worst case scenario : Σ∗ = Σ = γ(%) ↔ lowest correlation
α∗t =(x0 +
1
2λeb
ᵀΣ−1bT − X ∗t
)Σ−1b, 0 6 t 6 T , PΘ − q.s.
Moreover, the two components of Σ−1b have opposite sign : spreadtrading with worst-case scenario corresponding to lowest correlation θ∗ =%, i.e. spread effect is minimal.
Huyen PHAM Robust Markowitz portfolio selection
IntroductionRobust Markowitz problem
Robust efficient frontier
Robust settingExplicit solutionsMcKean-Vlasov DP approach : elements of proof
Case 3 : % 6 %0 6 %
• Worst-case correlation scenario : θ∗ = %0 (not extreme !)
α∗t =
( [x0 + 1
2λexp
(β2
1T)− X ∗
t
]b1
σ21
0
), 0 6 t 6 T , PΘq.s., if β2
1 > β22 ,
(0[
x0 + 12λ
exp(β2
2T)− X ∗
t
]b2
σ22
), 0 6 t 6 T , PΘq.s., if β2
2 > β21 .
Huyen PHAM Robust Markowitz portfolio selection
IntroductionRobust Markowitz problem
Robust efficient frontier
Robust settingExplicit solutionsMcKean-Vlasov DP approach : elements of proof
Reformulation of robust Markowitz mean-variance problem
• Nonstandard zero-sum stochastic differential game :
infα∈A
supΣ∈VΘ
J(α, σ), with J(α, σ) = λVarσ(XαT )− Eσ[Xα
T ]
I Introduce
ρα,σt := Lσ(Xαt ) law of Xα
t under Pσ valued in P2(R),
P2(R) : Wasserstein space of square-integrable measures→
J(α, σ) = λVar(ρα,σT )− ρα,σT
where for µ ∈ P2(R) :
µ :=
∫Rxµ(dx), Var(µ) :=
∫R
(x − µ)2µ(dx).
→ Standard deterministic differential game in P2(R).Huyen PHAM Robust Markowitz portfolio selection
IntroductionRobust Markowitz problem
Robust efficient frontier
Robust settingExplicit solutionsMcKean-Vlasov DP approach : elements of proof
Method and tools of resolution
• Optimality principle from dynamic programming for the deterministicdifferential game : look for v : [0,T ]× P2 (R) → R s.t.
(i) v(T , µ) = λVar(µ) − µ
(ii) t 7→ v(t, ρα,σ∗
t ) is for all α ∈ A and some Σ∗ ∈ VΘ
(iii) t 7→ v(t, ρα∗,σ
t ) is for some α∗ ∈ A and all Σ ∈ VΘ
• Derivative in P2(R)
• Chain rule along flow of probability measure
Huyen PHAM Robust Markowitz portfolio selection
IntroductionRobust Markowitz problem
Robust efficient frontier
Robust settingExplicit solutionsMcKean-Vlasov DP approach : elements of proof
Derivative in the Wasserstein space in a nutshell
Differentiation on P2 (R)
• Consider u : P2 (R) → RI Lifted version of u ; U : L2(F0;R) → R defined by
U(ξ) = u(L(ξ)),
u is differentiable if U is Frechet differentiable (Lions definition)
• Differential of u :
Frechet derivative of U on on the Hilbert space L2(F0;R) :
DU(ξ) = ∂µu(L(ξ))(ξ), for some function ∂µu(L(ξ)) : R→ R
∂µu(L(ξ)) is called derivative of u at µ = L(ξ), and ∂µu(µ) ∈L2µ(R).
For fixed µ, if x ∈ R → ∂µu(µ)(x) ∈ R is continuouslydifferentiable, its gradient is denoted by ∂x∂µu(µ) ∈ L∞µ (R).
Huyen PHAM Robust Markowitz portfolio selection
IntroductionRobust Markowitz problem
Robust efficient frontier
Robust settingExplicit solutionsMcKean-Vlasov DP approach : elements of proof
Examples of derivative
(1) u(µ) = < ψ, µ > :=∫R ψ(x)µ(dx) → Lifted version : U(ξ) =
E[ψ(ξ)]
U(ξ + ζ) = U(ξ) + E[∂xψ(ξ)ζ] + o(‖ζ‖L2 )
I DU(ξ) = ∂xψ(ξ) → ∂µu(µ) = ∂xψ → ∂x∂µu(µ) = ∂2xψ
(2) u(µ) = Var(µ) :=∫R(x − µ)2µ(dx), with µ :=
∫R xµ(dx), → Lifted
version : U(ξ) = Var(ξ)I
∂µu(µ)(x) = 2(x − µ),
and then ∂x∂µu(µ) = 2.
Huyen PHAM Robust Markowitz portfolio selection
IntroductionRobust Markowitz problem
Robust efficient frontier
Robust settingExplicit solutionsMcKean-Vlasov DP approach : elements of proof
Chain rule for flow of probability measures
Buckdahn, Li, Peng and Rainer (15), Chassagneux, Crisan and Delarue(15) :
• Consider Ito process :
dXt = btdt + σtdWt , X0 ∈ L2(F0;R).
I Let u ∈ C2b(P2 (R)). Then, for all t,
du(L(Xt)) = E[∂µu(L(Xt))(Xt)bt +
1
2∂x∂µu(L(Xt))(Xt)σ
2t
]dt.
Huyen PHAM Robust Markowitz portfolio selection
IntroductionRobust Markowitz problem
Robust efficient frontier
Robust settingExplicit solutionsMcKean-Vlasov DP approach : elements of proof
Hamiltonian function for robust portfolio optimization problem
• Hamiltonian : for (p,M) ∈ R× R∗+, (a,Σ) ∈ Rd × Γ,
H(p,M, a,Σ) = paᵀb +1
2MaᵀΣa
I
H∗(p,M) := infa∈Rd
supΣ∈Γ
H(p,M, a,Σ)
(min-max property under (IC)) = supΣ∈Γ
infa∈Rd
H(p,M, a,Σ)
(saddle point) = H(p,M, a∗(p,M),Σ∗)
where
Σ∗ = argminΣ∈Γ
bᵀΣ−1b, a∗(p,M) = − p
M(Σ∗)−1b.
Huyen PHAM Robust Markowitz portfolio selection
IntroductionRobust Markowitz problem
Robust efficient frontier
Robust settingExplicit solutionsMcKean-Vlasov DP approach : elements of proof
Bellman-Isaacs equation in Wasserstein space
Verification theorem :
Suppose that one can find a smooth function v on [0,T ]× P2(R) with∂x∂µv(t, µ)(x) > 0 for all (t, x , µ) ∈ [0,T )× R× P2 (R), solution to theBellman-Isaacs PDE :
∂tv(t, µ) +
∫RH∗(
∂µv(t, µ)(x), ∂x∂µv(t, µ)(x))µ(dx) = 0, (t, µ) ∈ [0,T )× P
2(R)
v(T , µ) = λVar(µ)− µ, µ ∈ P2
(R).
Moreover, suppose that we can aggregate the family of processes
a∗(∂µv(t,Pσ
XPσt
)(X Pσt ), ∂x∂µv(t,Pσ
XPσt
)(X Pσt )), 0 6 t 6 T , Pσ − p.s.,∀Σ ∈ VΘ
into a PΘ-q.s process α∗, where X Pσ is the solution to the McKean-VlasovSDE under Pσ :
dXt = a∗(∂µv(t,Pσ
Xt)(Xt), ∂x∂µv(t,Pσ
Xt)(Xt)
)[bdt + dBt ],
then α∗ is an optimal portfolio strategy, Σ∗ is the worst-case scenario and
V0(λ) = v(0, δx0 ) = J(α∗, σ∗) = infα
supΣ
J(α, σ) = supΣ
infα
J(α, σ).
Huyen PHAM Robust Markowitz portfolio selection
IntroductionRobust Markowitz problem
Robust efficient frontier
Robust settingExplicit solutionsMcKean-Vlasov DP approach : elements of proof
Explicit resolution
• From the linear-quadratic structure of the problem, the solution to theBellman-Isaacs PDE is
v(t, µ) = K (t)Var(µ)− µ+ χ(t)
for some explicit deterministic functions K and χ.
• Key observation : the MKV SDE under Pσ is linear in X and Eσ[Xt ] →Eσ[Xt ] does not depend on Pσ → we can aggregate X into a PΘ-q.s.solution→ α∗ optimal strategy, and Σ∗ worst-case scenario.
Huyen PHAM Robust Markowitz portfolio selection
IntroductionRobust Markowitz problem
Robust efficient frontier
Duality relation
• Since the solution X ∗ = Xα∗,λto the “Lagrangian” mean-variance
problem has expectation Eσ[X ∗T ] that does not depend on the priorprobability measure Pσ →
supPσ
[λVarσ(X ∗
T )− Eσ[X ∗T ]]
= λ supPσ
Varσ(X ∗T )− inf
PσEσ[X ∗
T ]
→ Robust Markowitz value function U0(ϑ) and mean-variance valuefunction V0(λ) are conjugate :
V0(λ) = infϑ>0
[λϑ− U0(ϑ)
], λ > 0,
U0(ϑ) = infλ>0
[λϑ− V0(λ)
], ϑ > 0.
and solution αϑ to U0(ϑ) is equal to solution α∗,λ to V0(λ) with
λ = λ(ϑ) =
√eR(θ∗)T − 1
4ϑ,
where R(θ∗) = bᵀγ(θ∗)−1b : (square) of minimal risk premium.Huyen PHAM Robust Markowitz portfolio selection
IntroductionRobust Markowitz problem
Robust efficient frontier
Robust lower bound for Sharpe ratio
• Robust efficient frontier :
U0(ϑ) = x0 +√eR(θ∗)T − 1
√ϑ, ϑ > 0.
• Sharpe ratio : for a portfolio strategy α
S(α) :=E[Xα
T ]− x0√Var(Xα
T )computed under the true probability measure .
→ By following a robust Markowitz optimal portfolio αϑ :
S(αϑ) >E(αϑ)− x0√R(αϑ)
=U0(ϑ)− x0√
ϑ
=√eR(θ∗)T − 1 =: S.
Huyen PHAM Robust Markowitz portfolio selection
IntroductionRobust Markowitz problem
Robust efficient frontier
Conclusion
• Explicit solution to robust Markowitz problem under ambiguouscovariance matrix and robust lower bound for Sharpe ratio
• McKean-Vlasov dynamic programming approach
applicable beyond MV criterion to risk measure involving nonlinearfunctionals of the law of the state process
• Open problem : case of drift uncertainty
Aggregation issue for MKV SDE (main difference with expectedutility criterion)
Duality relation does not hold
Huyen PHAM Robust Markowitz portfolio selection