under ambiguous covariance matrix University Paris Diderot, LPMA Sorbonne Paris Cité Based on joint work with A. Ismail, Natixis MFO March 2, 2017
Outline Introduction 1 Introduction 2 3 and Sharpe ratio
Classical Markowitz formulation in continuous time X α = (X α t ) t wealth process with α = (α t ) : amount invested in risky assets at any time t [0, T ], T < investment horizon Markowitz criterion : on (Ω, F, P) maximize over α : E[X α T ] subject to Var(X α T ) ϑ U 0 (ϑ) : maximal expected return given risk ϑ > 0 Lagrangian mean-variance criterion : Graph of U 0 : Efficient frontier V 0 (λ) minimize over α : λvar(x α T ) E[X α T ], Duality relation : { V0 (λ) = inf ϑ>0 [ λϑ U0 (ϑ) ], λ > 0, U 0 (ϑ) = inf λ>0 [ λϑ V0 (λ) ], ϑ > 0.
Optimal MV portfolio in BS model Multidimensional BS model : risk-free asset 1, d stocks with b R d : vector of assets return Σ S d >+ : covariance matrix of assets Optimal amount invested in the d stocks : ( αt = E[Xt ] Xt + 1 t)) 2λ er(t Σ 1 b ( = x 0 + 1 ) 2λ ert Xt Σ 1 b, 0 t T. R := b Σ 1 b R : (square) of risk premium of the d stocks e RT 1 ϑ λ = λ(ϑ) =. 4ϑ Efficient frontier : straight line in mean/standard deviation diagram U 0(ϑ) = x 0 + e RT 1 ϑ, ϑ > 0. Ref : Zhou, Li (00), Andersson-Djehiche (11), Fisher-Livieri (16), P., Wei (16).
Robust portfolio optimization Inacurracy in parameters estimation : Drift : well-known Correlation : asynchronous data and lead-lag effect Portfolio optimization with Knightian uncertainty (ambiguity) on model 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, robust efficient frontier, and lower bound for robust Sharpe ratio
Robust framework Introduction Canonical space Ω = C([0, T ], R d ) : continuous paths of d stocks B = (B t ) t canonical process, P 0 : Wiener measure, F = (F t ) t canonical filtration Drift b R d of the assets is assumed to be known (well-estimated or strong belief) but uncertainty on the covariance matrix, possibly random (even rough!)
Ambiguous covariance matrix : Epstein-Ji (11) Γ compact set of S d >+ : prior realizations of covariance matrix Γ = Γ(Θ) parametrized by convex set Θ of R q : there exists some measurable function γ : R q S d >+ s.t. Any Σ in Γ is in the form : Σ = γ(θ) for some θ Θ. Concavity assumption (IC) : γ ( 1 2 (θ 1 + θ 2 ) ) 1 ( γ(θ1 ) + γ(θ 2 ) ) (1) 2 Remark : in examples, we have = in (1). Notation : for Σ Γ, we set, σ = Σ 1 2, the volatility matrix.
Examples Introduction Uncertain volatilities for multivariate uncorrelated assets : d Θ = [σ 2 i, σ i 2 ], 0 σ i σ i <, i=1 γ(θ) = σ1 2... 0....., 0... σd 2 Ambiguous correlation in the two-assets case : for θ = (σ 2 1,..., σ 2 d). γ(θ) = ( σ 2 1 σ 1 σ 2 θ σ 1 σ 2 θ σ 2 2 ), for θ Θ = [ϱ, ϱ] ( 1, 1), for some known constants σ 1, σ 2 > 0.
Prior (singular) probability measures V Θ : set of F-adapted processes Σ = (Σ t ) t valued in Γ = Γ(Θ) P Θ = { P σ } : Σ V Θ, with P σ := P 0 (B σ ) 1, σ t = Σ 1 2 t, Bt σ := In other words : t d < B > t = Σ t dt under P σ. 0 σ s db s, P 0 a.s. Remark : connection with the theory of G-expectation (Peng), and quasi-sure analysis (Denis/Martini, Soner/Touzi/Zhang, Nutz). We say P Θ q.s. : P σ a.s. for all Σ V Θ.
Assets price and wealth dynamics under covariance matrix uncertainty Price process S of d stocks : ds t = diag(s t )(bdt + db t ), 0 t T, P Θ q.s. Set A of portfolio strategies : F-adapted processes α valued in R d s.t. sup Pσ P Θ E σ[ T 0 α t Σ t α t dt] < Wealth process X α : dx α t = α t diag(s t ) 1 ds t = α t (bdt + db t ), 0 t T, X α 0 = x 0, P Θ q.s.
Robust Markowitz mean-variance formulation : { maximize over α A, E(α) := infp (M ϑ ) σ P E σ[x α Θ T ] subject to R(α) := sup Pσ P Var σ(x α Θ T ) ϑ. U 0 (ϑ), ϑ > 0 : robust efficient frontier Lagrangian robust mean-variance problem : given λ > 0, ( ) (P λ ) V 0 (λ) = inf sup λvar σ (XT α ) E σ [XT α ] α A P σ P Θ Not clear a priori that U 0 and V 0 are conjugates of each other! ( ) sup P σ P Θ λvar σ (XT α ) E σ [XT α ] λr(α) E(α)
Solution to (P λ ) Introduction Worst case scenario constant covariance matrix Σ = γ(θ ) minimizing the risk premium : θ arg min θ Θ R(θ), R(θ) := b γ(θ) 1 b. Optimal robust MV strategy = optimal MV strategy in the BS model with covariance matrix Σ R = b (Σ ) 1 b : Key remark : αt = ( x 0 + 1 T 2λ er Xt ) (Σ ) 1 b, 0 t T. E σ [XT ] = x 0 + 1 [ e R T 1 ] does not depend on P σ 2λ
Example : ambiguous correlation in the two-assets case Known marginal volatilities σ i, and drift b i, i = 1, 2, but unknown correlation lying in Θ = [ϱ, ϱ] ( 1, 1). Instantaneous Sharpe ratio of each asset : We set : β i = b i σ i > 0, i = 1, 2, ϱ 0 := min(β 1, β 2 ) max(β 1, β 2 ) (0, 1] as a measure of proximity between the two stocks.
Case 1 : ϱ < ϱ 0 Worst case scenario : Σ = Σ := γ( ϱ) highest correlation α t = ( x 0 + 1 2λ eb Σ 1 bt Xt ) Σ 1 b, 0 t T, P Θ q.s. Moreover, the two components of Σ 1 b have the same sign : directional trading with worst-case scenario corresponding to highest correlation θ = ϱ, i.e. diversification effect is minimal.
Case 2 : ϱ > ϱ 0 Worst case scenario : Σ = Σ = γ(ϱ) lowest correlation α t = ( x 0 + 1 ) 2λ eb Σ 1 bt Xt Σ 1 b, 0 t T, P Θ q.s. Moreover, the two components of Σ 1 b have opposite sign : spread trading with worst-case scenario corresponding to lowest correlation θ = ϱ, i.e. spread effect is minimal.
Case 3 : ϱ ϱ 0 Introduction ϱ Worst-case correlation scenario : θ = ϱ 0 (not extreme!) α t = ( [ x 0 + 1 exp ( β 2 2λ 1T ) ] Xt b1 σ1 2 0 ( [ 0 x 0 + 1 exp ( β 2 2λ 2T ) Xt ] b2 σ 2 2 ), 0 t T, P Θ q.s., if β1 2 > β2, 2 ), 0 t T, P Θ q.s., if β2 2 > β1. 2
Reformulation of robust Markowitz mean-variance problem Nonstandard zero-sum stochastic differential game : inf sup J(α, σ), with J(α, σ) = λvar σ (XT α ) E σ [XT α ] α A Σ V Θ Introduce ρ α,σ t := L σ (X α t ) law of X α t under P σ valued in P 2 (R), P 2 (R) : Wasserstein space of square-integrable measures R J(α, σ) = λvar(ρ α,σ T ) ρα,σ T where for µ P 2 (R) : µ := xµ(dx), Var(µ) := (x µ) 2 µ(dx). Standard deterministic differential game in P 2 (R). R
Method and tools of resolution Optimality principle from dynamic programming for the deterministic differential game : look for v : [0, T ] P 2 (R) R s.t. (i) v(t, µ) = λvar(µ) µ (ii) t v(t, ρ α,σ t ) is for all α A and some Σ V Θ (iii) t v(t, ρ α,σ t ) is for some α A and all Σ V Θ Derivative in P 2 (R) Chain rule along flow of probability measure
Derivative in the Wasserstein space in a nutshell Differentiation on P 2 (R) Consider u : P 2 (R) R Lifted version of u U : L 2 (F 0 ; R) R defined by U(ξ) = u(l(ξ)), u is differentiable if U is Fréchet differentiable (Lions definition) Differential of u : Fréchet derivative of U on on the Hilbert space L 2 (F 0 ; R) : DU(ξ) = µ u(l(ξ))(ξ), for some function µ u(l(ξ)) : R R µ u(l(ξ)) is called derivative of u at µ = L(ξ), and µ u(µ) L 2 µ(r). For fixed µ, if x R µ u(µ)(x) R is continuously differentiable, its gradient is denoted by x µ u(µ) L µ (R).
Examples of derivative (1) u(µ) = < ψ, µ > := ψ(x)µ(dx) Lifted version : U(ξ) = R E[ψ(ξ)] U(ξ + ζ) = U(ξ) + E[ x ψ(ξ)ζ] + o( ζ L 2 ) DU(ξ) = x ψ(ξ) µ u(µ) = x ψ x µ u(µ) = 2 x ψ (2) u(µ) = Var(µ) := R (x µ)2 µ(dx), with µ := xµ(dx), Lifted R version : U(ξ) = Var(ξ) and then x µ u(µ) = 2. µ u(µ)(x) = 2(x µ),
Chain rule for flow of probability measures Buckdahn, Li, Peng and Rainer (15), Chassagneux, Crisan and Delarue (15) : Consider Itô process : dx t = b t dt + σ t dw t, X 0 L 2 (F 0 ; R). Let u Cb 2(P (R)). Then, for all t, 2 du(l(x t)) = E [ µu(l(x t))(x t)b t + 1 2 x µu(l(xt))(xt)σ2 t ] dt.
Hamiltonian function for robust portfolio optimization problem Hamiltonian : for (p, M) R R +, (a, Σ) R d Γ, H(p, M, a, Σ) = pa b + 1 2 Ma Σa H (p, M) := inf sup a R d Σ Γ H(p, M, a, Σ) (min-max property under (IC)) = sup inf H(p, M, a, Σ) a R d Σ Γ (saddle point) = H(p, M, a (p, M), Σ ) where Σ = arg min Σ Γ b Σ 1 b, a (p, M) = p M (Σ ) 1 b.
Bellman-Isaacs equation in Wasserstein space Verification theorem : Suppose that one can find a smooth function v on [0, T ] P 2 (R) with x µ v(t, µ)(x) > 0 for all (t, x, µ) [0, T ) R P 2 (R), solution to the Bellman-Isaacs PDE : tv(t, µ) + H ( µv(t, µ)(x), x µv(t, µ)(x) ) µ(dx) = 0, (t, µ) [0, T ) P 2 (R) R v(t, µ) = λvar(µ) µ, µ P 2 (R). Moreover, suppose that we can aggregate the family of processes a ( µv(t, P σ X Pσ t )(X Pσ t ), x µv(t, P σ Xt Pσ )(Xt Pσ ) ), 0 t T, P σ p.s., Σ V Θ into a P Θ -q.s process α, where X Pσ is the solution to the McKean-Vlasov SDE under P σ : dx t = a ( µv(t, P σ X t )(X t), x µv(t, P σ X t )(X t) ) [bdt + db t], then α is an optimal portfolio strategy, Σ is the worst-case scenario and V 0(λ) = v(0, δ x0 ) = J(α, σ ) = inf sup α Σ J(α, σ) = sup Σ inf J(α, σ). α
Explicit resolution Introduction From the linear-quadratic structure of the problem, the solution to the Bellman-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 P σ is linear in X and E σ [X t ] E σ [X t ] does not depend on P σ we can aggregate X into a P Θ -q.s. solution α optimal strategy, and Σ worst-case scenario.
Duality relation Introduction Since the solution X = X α,λ to the Lagrangian mean-variance problem has expectation E σ [XT ] that does not depend on the prior probability measure P σ [ ] sup λvar σ(xt ) E σ[xt ] = λ sup Var σ(xt ) inf Eσ[X P σ P σ P σ T ] Robust Markowitz value function U 0 (ϑ) and mean-variance value function V 0 (λ) are conjugate : { V0 (λ) = inf ϑ>0 [ λϑ U0 (ϑ) ], λ > 0, U 0 (ϑ) = inf λ>0 [ λϑ V0 (λ) ], ϑ > 0. and solution ˆα ϑ to U 0 (ϑ) is equal to solution α,λ to V 0 (λ) with e R(θ )T 1 λ = λ(ϑ) =, 4ϑ where R(θ ) = b γ(θ ) 1 b : (square) of minimal risk premium.
Robust lower bound for Sharpe ratio : U 0 (ϑ) = x 0 + Sharpe ratio : for a portfolio strategy α e R(θ )T 1 ϑ, ϑ > 0. S(α) := E[X α T ] x 0 Var(X α T ) computed under the true probability measure. By following a robust Markowitz optimal portfolio ˆα ϑ : S(ˆα ϑ ) E(ˆαϑ ) x 0 R(ˆαϑ ) = U 0(ϑ) x 0 ϑ = e R(θ )T 1 =: S.
Conclusion Introduction Explicit solution to robust Markowitz problem under ambiguous covariance matrix and robust lower bound for Sharpe ratio McKean-Vlasov dynamic programming approach applicable beyond MV criterion to risk measure involving nonlinear functionals of the law of the state process Open problem : case of drift uncertainty Aggregation issue for MKV SDE (main difference with expected utility criterion) Duality relation does not hold