Robust Markowitz Portfolio Selection in a Stochastic Factor Model Dariusz Zawisza Institute of Mathematics, Jagiellonian University in Krakow, Poland 5 May, 2010
Financial Market Probability space (Ω, F T, P), T > 0 is an investment horizon. A bank account A risky asset dp t = rp t dt An untradable economic factor ds t = b(y t ) dt + σ(y t ) db 1 t dy t = g(y t ) dt + a(y t ) (ρ db 1 t + 1 ρ 2 db 2 t ) A market price of risk ratio is defined as λ(y) := b(y) r. σ(y)
Portfolio Dynamics π - part of the wealth invested in the risky asset S The portfolio dynamics d X t = r X t dt + (b(y t ) r) π t dt + σ(y t ) π t db 1 t Transformation of X t and π t to the T -forward values X t := e (T t)r X t, π t := e (T t)r π t Then dx t = π t (b(y t ) r)dt + π t σ(y t ) db 1 t Simplifying assumption: r = 0 dx t = π t b(y t )dt + π t σ(y t ) db 1 t
Model Misspecification The investor knows only that the correct measure belongs to a class of possible measures. { Q := Q P dq ( ) } dp = E η 1t dbt 1 +η 2t dbt 2 (η 1, η 2 ) M, T M denotes the set of all progressively measurable processes η = (η 1, η 2 ) taking values in a fixed compact convex set Γ R 2 (Schied [4]).
Robust Investor The worst case scenario criterion: maximize inf Q Q EQ x,y U(XT π ) over π A.
Robust Investor The worst case scenario criterion: maximize inf Q Q EQ x,y U(XT π ) over π A. This can be considered as a zero sum stochastic differential game, where investor is looking for a saddle point (π, Q ) such that E Q x,y U(XT π ) EQ x,y U(XT π ) EQ x,y U(XT π ).
Markowitz Selection Problem Markowitz type investor is looking for a strategy π such that if E x,y (X π T ) = A. Var x,y (X π T ) Var x,y (X π T ),
Markowitz Selection Problem Markowitz type investor is looking for a strategy π such that if E x,y (X π T ) = A. Var x,y (X π T ) Var x,y (X π T ), Robust Markowitz Selection Problem Find a pair of controls (π, Q ) such that if E Q (XT π ) = A, and if E Q (X π T ) = A. Var Q x,y (XT π ) Var Q x,y (X π T ), Varx,y Q (XT π Q ) Varx,y (XT π ),
Auxiliary Problem The auxiliary problem is constructed via Lagrange multipliers: J π,q θ (x, y, t) := E Q x,y,t(x π T A)2 θ(e x,y,t (X π T ) A) = E Q x,y,t(x π T D)2 (A + θ 2 )2 + A 2 + θa, D = A + θ 2, J π,q D (x, y, t) := EQ x,y,t(x π T D)2.
Auxiliary Problem The auxiliary problem is constructed via Lagrange multipliers: J π,q θ (x, y, t) := E Q x,y,t(x π T A)2 θ(e x,y,t (X π T ) A) = E Q x,y,t(x π T D)2 (A + θ 2 )2 + A 2 + θa, D = A + θ 2, J π,q D (x, y, t) := EQ x,y,t(x π T D)2. Lemma Let (π (D), Q (D)) be a family of saddle points for J π,q D. If there exist D such that E Q (D ) x,y (X π (D ) T ) = A, then (π (D ), Q (D )) is a solution to the robust Markowitz selection problem.
Differential operator L π,η V (x, y, t) := V t + 1 2 a2 (y)v yy + 1 2 π2 σ 2 (y)v xx + πσ(y)ρa(y)v xy + a(y)(ρη 1 ρ + ρη 2 )V y + g(y)v y + π(b(y) + σ(y)η 1 )V x
Differential operator L π,η V (x, y, t) := V t + 1 2 a2 (y)v yy + 1 2 π2 σ 2 (y)v xx + πσ(y)ρa(y)v xy + a(y)(ρη 1 ρ + ρη 2 )V y + g(y)v y + π(b(y) + σ(y)η 1 )V x The first step is to find V such that max min π R η Γ Lπ,η V (x, y, t) = min max η Γ π R Lπ,η V (x, y, t) = 0, V (x, y, T ) = (x D) 2 and apply the following verification theorem, which is slight modification of Mataramvura and Øksendal [2] result.
Verification Theorem Suppose there exist a nonnegative function V C 2,2,1 (R 2 [0, T )) C(R 2 [0, T ]) and an admissible Markov control (π (x, y, t), η (x, y, t)) such that L π (x,y,t),η V (x, y, t) 0, L π,η (x,y,t) V (, x, y, t) 0, L π (x,y,t),η (x,y,t) V (x, y, t) = 0, V (x, y, T ) = (x D) 2 for all η Γ, π R, (x, y, t) R 2 [0, T ), and ( ) E Q x,y,t V (Xs π, Y s, s) < + sup t s T for all (x, y, t) R 2 [0, T ], π A, Q Q. Then (π (x, y, t), η (x, y, t), V ) is a solution to the auxiliary problem.
Applying standard minimax results to prove that min max η Γ π R Lπ,η V (x, y, t) = max min π R η Γ Lπ,η V (x, y, t), we can reduce the problem to solving only min max η Γ π R Lπ,η V (x, y, t) = 0, V (x, y, T ) = (x D) 2.
Applying standard minimax results to prove that min max η Γ π R Lπ,η V (x, y, t) = max min π R η Γ Lπ,η V (x, y, t), we can reduce the problem to solving only min max η Γ π R Lπ,η V (x, y, t) = 0, V (x, y, T ) = (x D) 2. The maximum with respect to π is reached at π (x, y, t, η) = ρa(y) V xy b(y) + η 1(y, t)σ(y) σ(y) V xx σ 2 (y) V x V xx.
HJBI equation After substitution V t + 1 2 a2 (y)v yy 1 2 ρa(y)(η 1 + λ(y))v xy V x V xx ρ 2 a 2 (y)v 2 xy V xx ( + g(y)v y + max 1 (η 1 + λ(y)) 2 Vx 2 η Γ 2 V xx + η 1 ρa(y)v y + ) 1 ρ 2 a(y)η 2 V y = 0, V (x, y, T ) = (x D) 2.
HJBI equation After substitution V t + 1 2 a2 (y)v yy 1 2 ρa(y)(η 1 + λ(y))v xy V x V xx ρ 2 a 2 (y)v 2 xy V xx ( + g(y)v y + max 1 (η 1 + λ(y)) 2 Vx 2 η Γ 2 V xx + η 1 ρa(y)v y + ) 1 ρ 2 a(y)η 2 V y = 0, V (x, y, T ) = (x D) 2. The solution should be of the form V (x, y, t) = (x D) 2 F (y, t).
Hopf-Cole Transformation F t + 1 2 a2 (y)f yy ρ2 a 2 (y)fy 2 + (g(y) 2ρa(y)λ(y))F y ( F + max η 1 ρa(y)f y + ) 1 ρ 2 a(y)η 2 F y (η 1 + λ(y)) 2 F = 0 η Γ
Hopf-Cole Transformation F t + 1 2 a2 (y)f yy ρ2 a 2 (y)fy 2 + (g(y) 2ρa(y)λ(y))F y ( F + max η 1 ρa(y)f y + ) 1 ρ 2 a(y)η 2 F y (η 1 + λ(y)) 2 F = 0 η Γ To remove the nonlinear term ρ2 a 2 (y)fy 2 F transformation is used: the Hopf-Cole type Case I: ρ 2 1 2 F (y, t) = ( α(y, t)) δ, δ = 1 1 2ρ 2.
Assuming that ρ 2 < 1 2, we have α t + 1 ( 2 a2 (y)α yy + (g(y) 2ρa(y)λ(y))α y + max η 1 ρa(y)α y η Γ + ) 1 ρ 2 a(y)η 2 α y (1 2ρ 2 )(η 1 + λ(y)) 2 α = 0.
Assuming that ρ 2 < 1 2, we have α t + 1 ( 2 a2 (y)α yy + (g(y) 2ρa(y)λ(y))α y + max η 1 ρa(y)α y η Γ + ) 1 ρ 2 a(y)η 2 α y (1 2ρ 2 )(η 1 + λ(y)) 2 α = 0. Case II: ρ 2 = 1 2 We get F (y, t) = e α(y,t) α t + 1 2 a2 (y)α yy + (g(y) 2ρayλ(y))α y + max ( η 1 ρa(y)α y + ) 1 ρ 2 η 2 a(y)α y (η 1 + λ(y)) 2 = 0. η Γ
Existence of HJBI Solutions Theorem If a is bounded, Lipschitz continuous and uniformly eliptic, λ, g are bounded and Lipschitz continuous, then there exist a solution of the equation F t + 1 2 a2 (y)f yy ρ2 a 2 (y)fy 2 + (g(y) 2ρa y λ(y))f y ( ) ( F + max η 1 ρa(y)f y + ) 1 ρ 2 a(y)η 2 F y (η 1 + λ(y)) 2 F = 0 η Γ bounded together with the derivative F y.
Solution to the Auxiliary Problem Theorem If a is bounded, Lipschitz continuous and uniformly eliptic, λ, g are bounded and Lipschitz continuous, then sup π A t inf Q Q Jπ,Q (x, y, t) = inf sup J π,q (x, y, t) = (x D) 2 F (y, t), Q Q π A t and there exists the optimal pair of controls (η1 (y, t), η 2 (y, t)) such that η = (η1 (y, t), η 2 (y, t)) realizes maximum in equation ( ), and the optimal strategy is given by π ρa(y)(x D) F y (y, x, t) = σ(y) F (λ(y) + η 1 (y, t))(x D). σ(y)
Back to the Robust Markowitz Problem It is sufficient to find D such that E Q x 0,y 0 (X π (D) T ) = A, dx t =(X t D)ζ(Y t, t)(b(y t ) + η 1(Y t, t)σ(y t ))dt + (X t D)ζ(Y t, t)σ(y t )db η1 t, ζ(y, t) := ρa(y) F y σ(y) F (λ(y) + η 1 (y, t)). σ(y)
Back to the Robust Markowitz Problem It is sufficient to find D such that E Q x 0,y 0 (X π (D) T ) = A, dx t =(X t D)ζ(Y t, t)(b(y t ) + η 1(Y t, t)σ(y t ))dt + (X t D)ζ(Y t, t)σ(y t )db η1 t, ζ(y, t) := ρa(y) F y σ(y) F (λ(y) + η 1 (y, t)). σ(y) After simple transformations we get ( T A =D + (x 0 D)E Q y 0 exp ζ(y s )(b(y s ) + η1(y s, s)σ(y s ))ds 0 1 T T ) ζ 2 (Y s, s)σ 2 (Y s )ds + ζ(y s, s)σ(y s )dbs η1. 2 0 0
The Black-Scholes Model ds t = bs t dt + σs t db t, Γ = [ R, R], λ > R. ( ) F t + max (η 1 + λ) 2 F = 0, η 1 Γ η1 = R, π (x) = (λ R)(x D ), D = A x 0e (λ R)2 T σ 1 e (λ R)2 T
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References II Zawisza, D. Robust portfolio selection under exponential preferences, Applicationes Mathematicae to appear. Zhou X., Li D. Continuous time mean variance portfolio selection: A stochastic LQ framework. Applied Mathematics and Optimization 42 (2000), 19 33.
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