Time-Consistent Mean-Variance Portfolio Selection in Discrete and Continuous Time

Transcription

1 ime-consistent Mean-Variance Portfolio Selection in Discrete and Continuous ime Christoph Czichowsky Department of Mathematics EH Zurich BFS 21 oronto, 26th June 21 Christoph Czichowsky (EH Zurich) Mean-variance portfolio selection oronto, 26th June 21 1 / 17

2 Mean-variance portfolio selection in one period Harry Markowitz (Portfolio selection, 1952): maximise return and minimise risk return=expectation risk=variance Mean-variance portfolio selection with risk aversion γ > in one period: U(ϑ) = E[x + ϑ S] γ 2 Var[x + ϑ S] = max ϑ! Solution is the so-called mean-variance efficient strategy, i.e. ϑ := 1 γ Cov[ S F ] 1 E[ S F ] =: ϑ. Question: How does this extend to multi-period or continuous time? Christoph Czichowsky (EH Zurich) Mean-variance portfolio selection oronto, 26th June 21 2 / 17

3 Basic problem Markowitz problem: [ U(ϑ) = E x + ] ϑ uds u γ [ 2 Var x + ] ϑ uds u = max! (ϑ s) s Static: criterion at time determines optimal ϑ via g = ϑds. Question: more explicit dynamic description of ϑ on [, ] from g? Christoph Czichowsky (EH Zurich) Mean-variance portfolio selection oronto, 26th June 21 3 / 17

4 Basic problem Markowitz problem: [ U(ϑ) = E x + ] ϑ uds u γ [ 2 Var x + ] ϑ uds u = max! (ϑ s) s Static: criterion at time determines optimal ϑ via g = ϑds. Question: more explicit dynamic description of ϑ on [, ] from g? Dynamic: Use ϑ on (, t] and determine optimal strategy on (t, ] via [ U t (ϑ) = E x + ] ϑ Ft uds u γ [ 2 Var x + ] ϑ Ft uds u = max! (ϑ s) t s Christoph Czichowsky (EH Zurich) Mean-variance portfolio selection oronto, 26th June 21 3 / 17

5 Basic problem Markowitz problem: [ U(ϑ) = E x + ] ϑ uds u γ [ 2 Var x + ] ϑ uds u = max! (ϑ s) s Static: criterion at time determines optimal ϑ via g = ϑds. Question: more explicit dynamic description of ϑ on [, ] from g? Dynamic: Use ϑ on (, t] and determine optimal strategy on (t, ] via [ U t (ϑ) = E x + ] ϑ Ft uds u γ [ 2 Var x + ] ϑ Ft uds u = max! (ϑ s) t s ime inconsistent: this optimal strategy is different from ϑ on (t, ]! Christoph Czichowsky (EH Zurich) Mean-variance portfolio selection oronto, 26th June 21 3 / 17

6 Basic problem Markowitz problem: [ U(ϑ) = E x + ] ϑ uds u γ [ 2 Var x + ] ϑ uds u = max! (ϑ s) s Static: criterion at time determines optimal ϑ via g = ϑds. Question: more explicit dynamic description of ϑ on [, ] from g? Dynamic: Use ϑ on (, t] and determine optimal strategy on (t, ] via [ U t (ϑ) = E x + ] ϑ Ft uds u γ [ 2 Var x + ] ϑ Ft uds u = max! (ϑ s) t s ime inconsistent: this optimal strategy is different from ϑ on (t, ]! ime-consistent mean-variance portfolio selection: Find a strategy ϑ, which is optimal for U t (ϑ) and time-consistent. Christoph Czichowsky (EH Zurich) Mean-variance portfolio selection oronto, 26th June 21 3 / 17

7 Previous literature Strotz (1956): choose the best plan among those that [you] will actually follow. Recursive approach to time inconsistency for a different problem. In Markovian models: Deterministic functions, HJB PDEs and verification thm. Ekeland et al. (26): game theoretic formulation for different problems. Basak and Chabakauri (27): results for mean-variance portfolio selection. Björk and Murgoci (28): General theory of Markovian time inconsistent stochastic optimal control problems (for various forms of time inconsistency.) Christoph Czichowsky (EH Zurich) Mean-variance portfolio selection oronto, 26th June 21 4 / 17

8 Previous literature Strotz (1956): choose the best plan among those that [you] will actually follow. Recursive approach to time inconsistency for a different problem. In Markovian models: Deterministic functions, HJB PDEs and verification thm. Ekeland et al. (26): game theoretic formulation for different problems. Basak and Chabakauri (27): results for mean-variance portfolio selection. Björk and Murgoci (28): General theory of Markovian time inconsistent stochastic optimal control problems (for various forms of time inconsistency.) 1) How to formulate and to obtain the solution in a more general model? 2) Rigorous justification of the continuous-time formulation? Christoph Czichowsky (EH Zurich) Mean-variance portfolio selection oronto, 26th June 21 4 / 17

9 Previous literature Strotz (1956): choose the best plan among those that [you] will actually follow. Recursive approach to time inconsistency for a different problem. In Markovian models: Deterministic functions, HJB PDEs and verification thm. Ekeland et al. (26): game theoretic formulation for different problems. Basak and Chabakauri (27): results for mean-variance portfolio selection. Björk and Murgoci (28): General theory of Markovian time inconsistent stochastic optimal control problems (for various forms of time inconsistency.) 1) How to formulate and to obtain the solution in a more general model? 2) Rigorous justification of the continuous-time formulation? Financial market: R d -valued semimartingale S wlog. S = S + M + A S 2 (P). Θ = Θ S := {ϑ L(S) ϑds S 2 (P)} = L 2 (M) L 2 (A). Christoph Czichowsky (EH Zurich) Mean-variance portfolio selection oronto, 26th June 21 4 / 17

10 Outline 1 Discrete time 2 Continuous time 3 Convergence of solutions Christoph Czichowsky (EH Zurich) Mean-variance portfolio selection oronto, 26th June 21 5 / 17

11 Local mean-variance efficiency in discrete time Use x + ϑ S := x + ϑ uds u = x + ϑ i S i and suppose d = 1. Definition A strategy ϑ Θ is locally mean-variance efficient (LMVE) if U k 1 ( ϑ) U k 1 ( ϑ + δ½ {k} ) P-a.s. for all k = 1,..., and any δ = (ϑ ϑ) Θ. Recursive optimisation (Källblad 28): ϑ Θ is LMVE if and only if [ ϑ k = 1 E[ S k F k 1 ] Cov S k, ϑ ] Fk 1 γ Var [ S k F k 1 ] i=k+1 i S i = 1 Var [ S k F k 1 ] γ λ k ξ k ( ϑ) for k = 1,...,. Christoph Czichowsky (EH Zurich) Mean-variance portfolio selection oronto, 26th June 21 6 / 17

12 Local mean-variance efficiency in discrete time Use x + ϑ S := x + ϑ uds u = x + ϑ i S i and suppose d = 1. Definition A strategy ϑ Θ is locally mean-variance efficient (LMVE) if U k 1 ( ϑ) U k 1 ( ϑ + δ½ {k} ) P-a.s. for all k = 1,..., and any δ = (ϑ ϑ) Θ. Recursive optimisation (Källblad 28): ϑ Θ is LMVE if and only if [ ϑ k = 1 E[ S k F k 1 ] Cov S k, ϑ ] Fk 1 γ Var [ S k F k 1 ] i=k+1 i S i Var [ S k F k 1 ] = 1 γ A E k E [( M k ) 2 F k 1 ] for k = 1,...,. [ M k E [ i=k+1 ϑ i S i Fk ] Fk 1 ] E [( M k ) 2 F k 1 ] = 1 γ λ k ξ k ( ϑ) Christoph Czichowsky (EH Zurich) Mean-variance portfolio selection oronto, 26th June 21 6 / 17

13 Structure condition and mean-variance tradeoff process S satisfies the structure condition (SC), i.e. there exists a predictable process λ such that A k = k λ i E [ ( M i ) 2 ] k F i 1 = λ i M i for k =,..., and the mean-variance tradeoff process (MV) K k := ( k E[ Si F i 1 ] ) 2 Var [ S i F i 1 ] = k λ 2 i M i = k λ i A i for k =,..., is finite-valued, i.e. λ L 2 loc (M). If the LMVE strategy ϑ exists, then λ L 2 (M), i.e. K L 1 (P). Comments: 1) SC and MV also appear naturally in other quadratic optimisation problems in mathematical finance; see Schweizer (21). 2) No arbitrage condition: A M. Christoph Czichowsky (EH Zurich) Mean-variance portfolio selection oronto, 26th June 21 7 / 17

14 Expected future gains For each ϑ Θ, define the expected future gains Z(ϑ) and the square integrable martingale Y (ϑ) by [ ] [ ] k Z k (ϑ) : = E ϑ i S i F k = E ϑ i A i F k ϑ i A i i=k+1 k =: Y k (ϑ) ϑ i A i k k = Y (ϑ) + ξ i (ϑ) M i + L k (ϑ) ϑ i A i for k =, 1,..., inserting the GKW decomposition of Y (ϑ). Lemma he LMVE strategy ϑ exists if and only if 1) S satisfies (SC) with λ L 2 (M), i.e. K L 1 (P), and 2) ϑ = 1 γ λ ξ( ϑ). Christoph Czichowsky (EH Zurich) Mean-variance portfolio selection oronto, 26th June 21 8 / 17

15 Global description of ξ( ϑ) via FS decomposition Combining both representations we obtain ϑ i A i = 1 γ K = 1 γ = Y ( ϑ) + ( ) 1 γ λ i ξ i ( ϑ) A i ξ i ( ϑ) M i + L ( ϑ) λ i A i = Y ( ϑ) + ξ i ( ϑ) S i + L ( ϑ) (1) (1) is almost the Föllmer Schweizer (FS) decomposition of 1 γ K. he integrand ξ( ϑ) =: 1 γ ξ in the FS decomposition yields the locally risk-minimising strategy for the contingent claim 1 γ K. Global description: ϑ Θ exists iff (1) and ϑ = 1 γ (λ ξ). Christoph Czichowsky (EH Zurich) Mean-variance portfolio selection oronto, 26th June 21 9 / 17

16 Continuous time setting Increasing, integrable, predictable process B called operational time such that: A = a B, M, M = c M B and a = c M λ + η with η Ker( c M ). S satisfies the structure condition (SC), if η =, i.e. A = d M λ, and the mean-variance tradeoff process (MV) K t := t λ u d M uλ u = t λ u da u < +. Expected future gains Z(ϑ) and GKW decomposition of Y(ϑ) [ ] [ ] t Z t (ϑ) : = E ϑ u ds u F t = E ϑ u da u F t ϑ u da u t =: Y t (ϑ) = Y (ϑ) + t t ϑ u da u t ξ u (ϑ)dm u + L t (ϑ) ϑ u da u Christoph Czichowsky (EH Zurich) Mean-variance portfolio selection oronto, 26th June 21 1 / 17

17 Local mean-variance efficiency in continuous time Idea: Combine recursive optimisation with a limiting argument. Definition A strategy ϑ Θ is locally mean-variance efficient (in continuous time) if U ti( ϑ) Uti( ϑ + δ½(ti,ti+1]) lim u Πn [ ϑ, δ] := lim ½ (ti,t n n E[B ti+1 B ti F ti ] i+1] P B-a.e. t i,t i+1 Π n for any increasing sequence (Π n ) of partitions such that Π n and any δ Θ. Inspired by the concept of local risk-minimisation (LRM); Schweizer (88, 8). ( lim [ ϑ, δ] = γ ( ξ( ϑ) + ϑ ) λ + γ ) n uπn 2 δ c M δ δ η P B-a.e. Remarks: 1) Convergence without any additional assumptions, i.e. boundedness assumptions on δ and continuity of A. 2) Generalises also results for LRM. Christoph Czichowsky (EH Zurich) Mean-variance portfolio selection oronto, 26th June / 17

18 he LMVE strategy ϑ in continuous time heorem 1) he LMVE strategy ϑ Θ exists if and only if i) S satisfies (SC) with λ L 2 (M), i.e. K L 1 (P). ii) ϑ = 1 γ λ ξ( ϑ), i.e. Ĵ( ϑ) = ϑ, where Ĵ(ψ) := 1 γλ ξ(ψ) for ψ Θ and ξ(ψ) is the integrand in the GKW decomposition of ψ uda u. 2) If K is bounded and continuous, Ĵ( ) is a contraction on (Θ,. β, ) where ( 1 ) 1 ϑ β, := ϑ 2 u d M u ϑ u L E( βk) ϑ L 2 (M) + ϑ L2 (A). u 2 (P) In particular, the LMVE strategy ϑ is given as the limit ϑ = lim n ϑ n in (Θ,. β, ), where ϑ n+1 = Ĵ(ϑn ) for n 1, for any ϑ = ϑ Θ. Christoph Czichowsky (EH Zurich) Mean-variance portfolio selection oronto, 26th June / 17

19 Global description of ξ( ϑ) via FS decomposition heorem he LMVE strategy ϑ Θ exists if and only if S satisfies (SC) and the MV process K L 1 (P) and can be written as K = K + ξds + L (2) with K L 2 (F ), ξ L 2 (M) such that ξ λ L 2 (A) and L M 2 (P) strongly orthogonal to M. In that case, ϑ = γ( 1 λ ξ), ξ( ϑ) = 1 ξ γ and U( ϑ) =... (2). If the minimal martingale measure exists, i.e. d P b dp := E( λ M) L 2 (P) and strictly positive, and K L 2 (P), then Z t ( ϑ) = 1 ( t ) K + ξds + γ L t K t = 1 γ Ê[K K t F t ], and ξ is related to the GKW of K under P; see Choulli et al. (21). Application in concrete models: 1) λ, 2) K, 3) E( λ M) and 4) ξ... Christoph Czichowsky (EH Zurich) Mean-variance portfolio selection oronto, 26th June / 17

20 Discretisation of the financial market Let (Π n ) n N increasing such that Π n and S = S + M + A. Discretisation of processes S n t := S ti, M n t := M ti and A n t := A ti for t [t i, t i+1 ) and all t i Π n. Discretisation of filtration F n t i := F ti for t [t i, t i+1 ) and all t i Π n and F n := (F n t ) t. Canonical decomposition of S n = S + M n + Ān S 2 (P, F n ) Ā n t := i k=1 E[ An t k F tk 1 ] = A n t MA,n t M n t := M n t + M A,n t for t [t i, t i+1 ) where the discretisation error is given by the F n -martingale M A,n t := i ( A n t k E[ A n t k F tk 1 ]) for t [t i, t i+1 ). k=1 Christoph Czichowsky (EH Zurich) Mean-variance portfolio selection oronto, 26th June / 17

21 Convergence of solutions ϑ n Due to time inconsistency usual abstract arguments don t work. Work with global description directly to show ϑ n = 1 γ ( λ n ξ n) L 2 (M) ϑ = 1 γ ( λ ξ), as Π n. Discrete- and continuous-time FS decomposition K n = K n + ξn ti St n i + L n and K = K + ξ u ds u + L. t i Π n For this we establish 1) λ n = 2) K n = Ān t i+1 E[( t M ½ t n i,t i+1 Π n i+1 ) 2 F ti ] (t i,t i+1] t i,t i+1 Π n λ n t i+1 Ā n t i+1 3) 2), Π n implies ξ n L2 (M) ξ. L 2 (M) λ L 2 (P) K = λ uda u Problem to control the discretisation error M A,n. Simple sufficient condition: K = dk dk dt dt and dt uniformly bounded. Christoph Czichowsky (EH Zurich) Mean-variance portfolio selection oronto, 26th June / 17

22 Some references Basak and Chabakauri. Dynamic Mean-Variance Asset Allocation. (27). Forthcoming in Review of Financial Studies. Björk and Murgoci. A General heory of Markovian ime Inconsistent Stochastic Control Problems. Working paper, Stockholm School of Economics, (28). Ekeland and Lazrak. Being serious about non-commitment: subgame perfect equilibrium in continuous time, (26). Preprint, Univ. of British Columbia. Choulli, Vandaele and Vanmaele. he Föllmer-Schweizer decomposition: Comparison and description. Stoch. Pro. and their Appl., (21), Schweizer. Hedging of Options in a General Semimartingale Model. Diss. EHZ no. 8615, EH Zürich (1988). Schweizer. A Guided our through Quadratic Hedging Approaches. In Option Pricing, Inerest Rates and Risk Management, Cambridge Univ. Press (21). Schweizer. Local risk-minimization for multidimensional assets and payment streams. In Advances in Mathematics of Finance, Banach Center Publ. (28). Strotz. Myopia and inconsistency in dynamic utility maximization. Review of Financial Studies (1956), Christoph Czichowsky (EH Zurich) Mean-variance portfolio selection oronto, 26th June / 17

23 hank you for your attention! czichowc Christoph Czichowsky (EH Zurich) Mean-variance portfolio selection oronto, 26th June / 17