Robust Markowitz Portfolio Selection in a Stochastic Factor Model

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1 Robust Markowitz Portfolio Selection in a Stochastic Factor Model Dariusz Zawisza Institute of Mathematics, Jagiellonian University in Krakow, Poland 5 May, 2010

2 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)

3 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

4 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]).

5 Robust Investor The worst case scenario criterion: maximize inf Q Q EQ x,y U(XT π ) over π A.

6 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 ).

7 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 ),

8 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 ),

9 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.

10 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.

11 Differential operator L π,η V (x, y, t) := V t a2 (y)v yy 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

12 Differential operator L π,η V (x, y, t) := V t a2 (y)v yy 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.

13 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.

14 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.

15 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.

16 HJBI equation After substitution V t 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.

17 HJBI equation After substitution V t 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).

18 Hopf-Cole Transformation F t 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

19 Hopf-Cole Transformation F t 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 F (y, t) = ( α(y, t)) δ, δ = 1 1 2ρ 2.

20 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.

21 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 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

22 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 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.

23 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)

24 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)

25 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 η

26 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

27 References I Goldfarb, D.; Iyengar, G. Robust portfolio selection problems. Math. Oper. Res. 28 (2003), Mataramvura S., Øksendal B., Risk minimizing portfolios and HJBI equations for stochastic differential games, Stochastics 80 (2008), Tütüncü, R. H. Koenig, M. Robust asset allocation. Ann. Oper. Res. 132 (2004), Schied A.Robust optimal control for a consumption-investment problem. Math. Meth. Opera. Res. 67. (2008), Zariphopoulou, T. A solution approach to valuation with unhedgeable risks, Finance Stoch. 5 (2001),

28 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),

29 Thank you for your attention.