Introduction Robust Markowitz problem Robust ecient frontier Robust Markowitz portfolio selection under ambiguous covariance matrix Huy^ en PHAM University Paris Diderot, LPMA Sorbonne Paris Cit e Based on joint work with A. Ismail, Natixis MFO March 2, 2017 Huy^ en PHAM Robust Markowitz portfolio selection
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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 :
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
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 :
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)