robust markowitz portfolio selection under ambiguous ...ajacquie/oberwolfach2017/pham.pdf · march...

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

Outline

1 Introduction

2 Robust Markowitz problemRobust settingExplicit solutionsMcKean-Vlasov DP approach : elements of proof

3 Robust efficient frontier and Sharpe ratio

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

I Duality relation :V0(λ) = infϑ>0

[λϑ− U0(ϑ)

], λ > 0,

U0(ϑ) = infλ>0

[λϑ− V0(λ)

], ϑ > 0.

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 +

2λeR(T−t)

)Σ−1b

=(x0 +

2λeRT − X ∗

)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

→ 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

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

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 !)

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) :

2(θ1 + θ2)

(γ(θ1) + γ(θ2)

Remark : in examples, we have = in (1).

• Notation : for Σ ∈ Γ, we set,

σ = Σ12 , the volatility matrix.

Examples

• Uncertain volatilities for multivariate uncorrelated assets :

Θ =d∏

[σ2i , σ

2i ], 0 6 σi 6 σi <∞,

γ(θ) =

σ21 . . . 0...

. . ....

0 . . . σ2d

, for θ = (σ21 , . . . , σ

• Ambiguous correlation in the two-assets case :

γ(θ) =

1 σ1σ2θσ1σ2θ σ2

), for θ ∈ Θ = [%, %] ⊂ (−1, 1),

for some known constants σ1, σ2 > 0.

Prior (singular) probability measures

VΘ : set of F-adapted processes Σ = (Σt)t valued in Γ = Γ(Θ)

PΘ =Pσ : Σ ∈ VΘ

Pσ := P0 (Bσ)−1, σt = Σ12t , B

σt :=

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.

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.

Robust Markowitz mean-variance formulation

• Robust Markowitz problem :

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)

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 +

∗T − X ∗t)(Σ∗)−1b, 0 6 t 6 T .

Key remark :

Eσ[X ∗T ] = x0 +1

∗T − 1]

does not depend on Pσ

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.

Case 1 : % < %0

• Worst case scenario : Σ∗ = Σ := γ(%) ↔ highest correlation

α∗t =(x0 +

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.

Case 2 : % > %0

• Worst case scenario : Σ∗ = Σ = γ(%) ↔ lowest correlation

α∗t =(x0 +

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.

Case 3 : % 6 %0 6 %

• Worst-case correlation scenario : θ∗ = %0 (not extreme !)

α∗t =

( [x0 + 1

2λexp

1T)− X ∗

), 0 6 t 6 T , PΘq.s., if β2

1 > β22 ,

x0 + 12λ

exp(β2

2T)− X ∗

), 0 6 t 6 T , PΘq.s., if β2

2 > β21 .

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

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) :

∫Rxµ(dx), Var(µ) :=

(x − µ)2µ(dx).

→ Standard deterministic differential game in P2(R).Huyen PHAM Robust Markowitz portfolio selection

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

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).

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.

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 +

2∂x∂µu(L(Xt))(Xt)σ

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

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∗)

Σ∗ = argminΣ∈Γ

bᵀΣ−1b, a∗(p,M) = − p

M(Σ∗)−1b.

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

v(T , µ) = λVar(µ)− µ, µ ∈ P2

Moreover, suppose that we can aggregate the family of processes

a∗(∂µv(t,Pσ

)(X Pσt ), ∂x∂µv(t,Pσ

)(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

J(α, σ) = supΣ

J(α, σ).

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.

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 ∗

→ 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

where R(θ∗) = bᵀγ(θ∗)−1b : (square) of minimal risk premium.Huyen PHAM Robust Markowitz portfolio selection

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√

=√eR(θ∗)T − 1 =: S.

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

robust markowitz portfolio selection under ambiguous ...ajacquie/oberwolfach2017/pham.pdf · march...

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