Time-Consistent Mean-Variance Portfolio Selection in Discrete and ...
Time-Consistent Mean-Variance Portfolio Selection in Discrete and Continuous Time Christoph Czichowsky Department of Mathematics ETH Zurich
BFS 2010 Toronto, 26th June 2010
Christoph Czichowsky (ETH Zurich)
Mean-variance portfolio selection
Toronto, 26th June 2010
1 / 17
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 > 0 in one period: U(ϑ) = E [x + ϑ⊤ ∆S] −
γ Var[x + ϑ⊤ ∆S] = max! ϑ 2
Solution is the so-called mean-variance efficient strategy, i.e. 1 b ϑe := Cov[∆S|F0 ]−1 E [∆S|F0 ] =: ϑ. γ
Question: How does this extend to multi-period or continuous time?
Christoph Czichowsky (ETH Zurich)
Mean-variance portfolio selection
Toronto, 26th June 2010
2 / 17
Basic problem Markowitz problem: i i γ h h RT RT U(ϑ) = E x + 0 ϑu dSu − Var x + 0 ϑu dSu = max ! 2 (ϑs )0≤s≤T Static: criterion at time 0 determines optimal ϑe via ge =
RT 0
e ϑdS.
Question: more explicit dynamic description of ϑe on [0, T ] from ge ?
Christoph Czichowsky (ETH Zurich)
Mean-variance portfolio selection
Toronto, 26th June 2010
3 / 17
Basic problem Markowitz problem: i i γ h h RT RT U(ϑ) = E x + 0 ϑu dSu − Var x + 0 ϑu dSu = max ! 2 (ϑs )0≤s≤T Static: criterion at time 0 determines optimal ϑe via ge =
RT 0
e ϑdS.
Question: more explicit dynamic description of ϑe on [0, T ] from ge ?
Dynamic: Use ϑe on (0, t] and determine optimal strategy on (t, T ] via i i γ h h RT RT Ut (ϑ) = E x + 0 ϑu dSu Ft − Var x + 0 ϑu dSu Ft = max ! (ϑs )t≤s≤T 2
Christoph Czichowsky (ETH Zurich)
Mean-variance portfolio selection
Toronto, 26th June 2010
3 / 17
Basic problem Markowitz problem: i i γ h h RT RT U(ϑ) = E x + 0 ϑu dSu − Var x + 0 ϑu dSu = max ! 2 (ϑs )0≤s≤T Static: criterion at time 0 determines optimal ϑe via ge =
RT 0
e ϑdS.
Question: more explicit dynamic description of ϑe on [0, T ] from ge ?
Dynamic: Use ϑe on (0, t] and determine optimal strategy on (t, T ] via i i γ h h RT RT Ut (ϑ) = E x + 0 ϑu dSu Ft − Var x + 0 ϑu dSu Ft = max ! (ϑs )t≤s≤T 2 Time inconsistent: this optimal strategy is different from ϑe on (t, T ]!
Christoph Czichowsky (ETH Zurich)
Mean-variance portfolio selection
Toronto, 26th June 2010
3 / 17
Basic problem Markowitz problem: i i γ h h RT RT U(ϑ) = E x + 0 ϑu dSu − Var x + 0 ϑu dSu = max ! 2 (ϑs )0≤s≤T Static: criterion at time 0 determines optimal ϑe via ge =
RT 0
e ϑdS.
Question: more explicit dynamic description of ϑe on [0, T ] from ge ?
Dynamic: Use ϑe on (0, t] and determine optimal strategy on (t, T ] via i i γ h h RT RT Ut (ϑ) = E x + 0 ϑu dSu Ft − Var x + 0 ϑu dSu Ft = max ! (ϑs )t≤s≤T 2 Time inconsistent: this optimal strategy is different from ϑe on (t, T ]! Time-consistent mean-variance portfolio selection: b which is “optimal” for Ut (ϑ) and time-consistent. Find a strategy ϑ,
Christoph Czichowsky (ETH Zurich)
Mean-variance portfolio selection
Toronto, 26th June 2010
3 / 17
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. (2006): game theoretic formulation for different problems. Basak and Chabakauri (2007): results for mean-variance portfolio selection. Bj¨ ork and Murgoci (2008): General theory of Markovian time inconsistent stochastic optimal control problems (for various forms of time inconsistency.)
Christoph Czichowsky (ETH Zurich)
Mean-variance portfolio selection
Toronto, 26th June 2010
4 / 17
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. (2006): game theoretic formulation for different problems. Basak and Chabakauri (2007): results for mean-variance portfolio selection. Bj¨ ork and Murgoci (2008): 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 (ETH Zurich)
Mean-variance portfolio selection
Toronto, 26th June 2010
4 / 17
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. (2006): game theoretic formulation for different problems. Basak and Chabakauri (2007): results for mean-variance portfolio selection. Bj¨ ork and Murgoci (2008): 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: Rd -valued semimartingale S wlog. S = S0 + M + A ∈ S 2 (P). R Θ = ΘS := {ϑ ∈ L(S) | ϑdS ∈ S 2 (P)} = L2 (M) ∩ L2 (A). Christoph Czichowsky (ETH Zurich)
Mean-variance portfolio selection
Toronto, 26th June 2010
4 / 17
Outline
1
Discrete time
2
Continuous time
3
Convergence of solutions
Christoph Czichowsky (ETH Zurich)
Mean-variance portfolio selection
Toronto, 26th June 2010
5 / 17
Local mean-variance efficiency in discrete time Use x + ϑ · S := x +
Definition
RT 0
ϑu dSu = x +
PT
i =1
ϑi ∆Si and suppose d = 1.
A strategy ϑb ∈ Θ is locally mean-variance efficient (LMVE) if b − Uk−1 (ϑb + δ 1{k} ) ≥ 0 Uk−1 (ϑ)
P-a.s.
b ∈ Θ. for all k = 1, . . . , T and any δ = (ϑ − ϑ)
Recursive optimisation (K¨allblad 2008): ϑb ∈ Θ is LMVE if and only if i h PT bi ∆Si Fk−1 ϑ Cov ∆S , k i =k+1 1 1 E [∆Sk |Fk−1 ] b − = λk − ξk (ϑ) ϑbk = γ Var [∆Sk |Fk−1 ] Var [∆Sk |Fk−1 ] γ
for k = 1, . . . , T . Christoph Czichowsky (ETH Zurich)
Mean-variance portfolio selection
Toronto, 26th June 2010
6 / 17
Local mean-variance efficiency in discrete time Use x + ϑ · S := x +
Definition
RT 0
ϑu dSu = x +
PT
i =1
ϑi ∆Si and suppose d = 1.
A strategy ϑb ∈ Θ is locally mean-variance efficient (LMVE) if b − Uk−1 (ϑb + δ 1{k} ) ≥ 0 Uk−1 (ϑ)
P-a.s.
b ∈ Θ. for all k = 1, . . . , T and any δ = (ϑ − ϑ)
Recursive optimisation (K¨allblad 2008): ϑb ∈ Θ is LMVE if and only if i h PT bi ∆Si Fk−1 ϑ Cov ∆S , k i =k+1 1 1 E [∆Sk |Fk−1 ] b − = λk − ξk (ϑ) ϑbk = γ Var [∆Sk |Fk−1 ] Var [∆Sk |Fk−1 ] γ i i h hP T bi ∆Si Fk Fk−1 ϑ E ∆M E k i =k+1 ∆Ak 1 − = γ E [(∆Mk )2 |Fk−1 ] E [(∆Mk )2 |Fk−1 ]
for k = 1, . . . , T .
Christoph Czichowsky (ETH Zurich)
Mean-variance portfolio selection
Toronto, 26th June 2010
6 / 17
Structure condition and mean-variance tradeoff process S satisfies the structure condition (SC), i.e. there exists a predictable process λ such that Ak =
k X i =1
k X λi ∆hMii λi E (∆Mi )2 |Fi −1 = i =1
for k = 0, . . . , T and the mean-variance tradeoff process (MVT) Kk :=
2 k k k X X X E [∆Si |Fi −1 ] λ2i ∆hMii = λi ∆Ai = Var [∆Si |Fi −1 ] i =1
i =1
i =1
for k = 0, . . . , T is finite-valued, i.e. λ ∈ L2loc (M). If the LMVE strategy ϑb exists, then λ ∈ L2 (M), i.e. KT ∈ L1 (P).
Comments: 1) SC and MVT also appear naturally in other quadratic optimisation problems in mathematical finance; see Schweizer (2001). 2) No arbitrage condition: A ≪ hMi. Christoph Czichowsky (ETH Zurich)
Mean-variance portfolio selection
Toronto, 26th June 2010
7 / 17
Expected future gains For each ϑ ∈ Θ, define the expected future gains Z (ϑ) and the square integrable martingale Y (ϑ) by " T " T # # X k X X Zk (ϑ) : = E ϑi ∆Ai ϑi ∆Si Fk = E ϑi ∆Ai Fk − i =1
i =1
i =k+1
=: Yk (ϑ) −
k X
ϑi ∆Ai
i =1
= Y0 (ϑ) +
k X
ξi (ϑ)∆Mi + Lk (ϑ) −
i =1
k X
ϑi ∆Ai
i =1
for k = 0, 1, . . . , T inserting the GKW decomposition of Y (ϑ).
Lemma The LMVE strategy ϑb exists if and only if b 1) S satisfies (SC) with λ ∈ L2 (M), i.e. KT ∈ L1 (P), and 2) ϑb = γ1 λ − ξ(ϑ). Christoph Czichowsky (ETH Zurich)
Mean-variance portfolio selection
Toronto, 26th June 2010
8 / 17
b via FS decomposition Global description of ξ(ϑ) Combining both representations we obtain T X i =1
ϑbi ∆Ai =
T X 1
γ
i =1
b λi − ξi (ϑ) ∆Ai
b + = Y0 (ϑ)
T X i =1
b b ξi (ϑ)∆M i + LT (ϑ)
T T X 1 1X b b b KT = λi ∆Ai = Y0 (ϑ) + ξi (ϑ)∆S i + LT (ϑ) γ γ i =1
(1)
i =1
(1) is almost the F¨ ollmer–Schweizer (FS) decomposition of
1 γ KT .
b =: 1 ξb in the FS decomposition yields the locally The integrand ξ(ϑ) γ risk-minimising strategy for the contingent claim γ1 KT . b Global description: ϑb ∈ Θ exists iff (1) and ϑb = γ1 (λ − ξ).
Christoph Czichowsky (ETH Zurich)
Mean-variance portfolio selection
Toronto, 26th June 2010
9 / 17
Continuous time setting Increasing, integrable, predictable process B called “operational time” such that: A = a · B, hM, Mi = e c M · B and a = e c M λ + η with η ∈ Ker(e c M ). S satisfies the structure condition (SC), if η = 0, i.e. Z A = dhMiλ,
and the mean-variance tradeoff process (MVT) Z t Z t Kt := λ⊤ dhMi λ = λu dAu < +∞. u u u 0
0
Expected future gains Z (ϑ) and GKW decomposition of Y (ϑ) "Z "Z # # Z t T T Zt (ϑ) : = E ϑu dSu Ft = E ϑu dAu ϑu dAu Ft − =: Yt (ϑ) − = Y0 (ϑ) +
Z
Z
t
ϑu dAu
0 t
ξu (ϑ)dMu + Lt (ϑ) −
0
Christoph Czichowsky (ETH Zurich)
0
0
t
Mean-variance portfolio selection
Z
t
ϑu dAu
0 Toronto, 26th June 2010
10 / 17
Local mean-variance efficiency in continuous time Idea: Combine recursive optimisation with a limiting argument.
Definition A strategy ϑb ∈ Θ is locally mean-variance efficient (in continuous time) if b δ] := lim lim u Πn [ϑ,
n→∞
n→∞
X
ti ,ti +1 ∈Πn
b − Ut (ϑb + δ 1(t ,t ] ) Uti (ϑ) i i i +1 1(ti ,ti +1 ] ≥ 0 P⊗B-a.e. E [Bti +1 − Bti |Fti ]
for any increasing sequence (Πn ) of partitions such that |Πn | → 0 and any δ ∈ Θ. Inspired by the concept of local risk-minimisation (LRM); Schweizer (88, 08). ⊤ b δ] = γ ξ(ϑ) b + ϑb − λ + γ δ c M δ − δ ⊤ η lim u Πn [ϑ, P ⊗ B-a.e. n→∞ 2
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 (ETH Zurich)
Mean-variance portfolio selection
Toronto, 26th June 2010
11 / 17
The LMVE strategy ϑb in continuous time Theorem
1) The LMVE strategy ϑb ∈ Θ exists if and only if
i) S satisfies (SC) with λ ∈ L2 (M), i.e. KT ∈ L1 (P).
b i.e. b b = ϑ, b where b ii) ϑb = γ1 λ − ξ(ϑ), J(ϑ) J(ψ) := γ1 λ − ξ(ψ) for ψ ∈ Θ and RT ξ(ψ) is the integrand in the GKW decomposition of 0 ψu dAu .
2) If K is bounded and continuous, b J(·) is a contraction on (Θ, k.kβ,∞) where kϑkβ,∞
Z
:=
0
T
12 1
ϑ⊤
2 ∼ kϑkL2 (M) + kϑkL2 (A) . u dhMiu ϑu E(−βK )u L (P)
In particular, the LMVE strategy ϑb is given as the limit ϑb = limn→∞ ϑn in (Θ, k.kβ,∞), where ϑn+1 = b J(ϑn ) for n ≥ 1, for any ϑ0 = ϑ ∈ Θ. Christoph Czichowsky (ETH Zurich)
Mean-variance portfolio selection
Toronto, 26th June 2010
12 / 17
b via FS decomposition Global description of ξ(ϑ) Theorem
The LMVE strategy ϑb ∈ Θ exists if and only if S satisfies (SC) and the MVT process KT ∈ L1 (P) and can be written as b0 + KT = K
Z
0
T
b +b ξdS LT
(2)
b0 ∈ L2 (F0 ), ξb ∈ L2 (M) such that ξb − λ ∈ L2 (A) and b with K L ∈ M20 (P) strongly b = 1 ξb and U(ϑ) b = . . . (2). orthogonal to M. In that case, ϑb = γ1 λ − ξb , ξ(ϑ) γ b
If the minimal martingale measure exists, i.e. ddPP := E(−λ · M)T ∈ L2 (P) and strictly positive, and KT ∈ L2 (P), then Z t 1b 1 b b b b ξdS + Lt − Kt = E Zt (ϑ) = K0 + [KT − Kt |Ft ], γ γ 0 b see Choulli et al. (2010). and ξb is related to the GKW of KT under P; Application in concrete models: 1) λ, 2) K , 3) E(−λ · M) and 4) ξb . . .
Christoph Czichowsky (ETH Zurich)
Mean-variance portfolio selection
Toronto, 26th June 2010
13 / 17
Discretisation of the financial market Let (Πn )n∈N increasing such that |Πn | → 0 and S = S0 + M + A. Discretisation of processes Stn := Sti , Mtn := Mti and Ant := Ati for t ∈ [ti , ti +1 ) and all ti ∈ Πn . Discretisation of filtration Ftni := Fti for t ∈ [ti , ti +1 ) and all ti ∈ Πn and Fn := (Ftn )0≤t≤T . ¯n+A ¯ n ∈ S 2 (P, Fn ) Canonical decomposition of S n = S0 + M ¯ n := Pi E [∆An |Ft ] = An − MA,n A t t tk k−1 t k=1 ¯ tn := Mtn + MtA,n for t ∈ [ti , ti +1 ) M
where the “discretisation error” is given by the Fn -martingale MA,n := t
i X
(∆Antk − E [∆Antk |Ftk−1 ])
for t ∈ [ti , ti +1 ).
k=1
Christoph Czichowsky (ETH Zurich)
Mean-variance portfolio selection
Toronto, 26th June 2010
14 / 17
Convergence of solutions ϑbn
Due to time inconsistency usual abstract arguments don’t work. Work with global description directly to show 1 n bn L2 (M) b 1 λ − ξ −→ ϑ = λ − ξb , ϑbn = γ γ
as |Πn | → 0.
Discrete- and continuous-time FS decomposition Z X n b0 + bn + bn ∆S n + b L and K = K KTn = K ξ T T 0 ti ti
0
ti ∈Πn
For this we establish ¯n P ∆A ti +1 1) λn = ¯ n )2 |F E [(∆M ti ,ti +1 ∈Πn
2) KTn =
P
ti ,ti +1 ∈Πn
ti +1
T
ξbu dSu + b LT .
L2 (M)
ti ]
1(ti ,ti +1 ] −→ λ
2
(P) ¯ nt L−→ KT = λnti +1 ∆A i +1 2
L (M) b 3) 2), |Πn | → 0 implies ξbn −→ ξ.
RT 0
λu dAu
Problem to control the “discretisation error” MA,n . R dK Simple sufficient condition: K = dK dt dt and dt uniformly bounded.
Christoph Czichowsky (ETH Zurich)
Mean-variance portfolio selection
Toronto, 26th June 2010
15 / 17
Some references Basak and Chabakauri. Dynamic Mean-Variance Asset Allocation. (2007). Forthcoming in Review of Financial Studies. Bj¨ ork and Murgoci. A General Theory of Markovian Time Inconsistent Stochastic Control Problems. Working paper, Stockholm School of Economics, (2008). Ekeland and Lazrak. Being serious about non-commitment: subgame perfect equilibrium in continuous time, (2006). Preprint, Univ. of British Columbia. Choulli, Vandaele and Vanmaele. The F¨ ollmer-Schweizer decomposition: Comparison and description. Stoch. Pro. and their Appl., (2010), 853-872. Schweizer. Hedging of Options in a General Semimartingale Model. Diss. ETHZ no. 8615, ETH Z¨ urich (1988). Schweizer. A Guided Tour through Quadratic Hedging Approaches. In Option Pricing, Inerest Rates and Risk Management, Cambridge Univ. Press (2001). Schweizer. Local risk-minimization for multidimensional assets and payment streams. In Advances in Mathematics of Finance, Banach Center Publ. (2008). Strotz. Myopia and inconsistency in dynamic utility maximization. Review of Financial Studies (1956), 165-180. Christoph Czichowsky (ETH Zurich)
Mean-variance portfolio selection
Toronto, 26th June 2010
16 / 17
Thank you for your attention! http://www.math.ethz.ch/∼czichowc
Christoph Czichowsky (ETH Zurich)
Mean-variance portfolio selection
Toronto, 26th June 2010
17 / 17
BFS 2010 Toronto, 26th June 2010
Christoph Czichowsky (ETH Zurich)
Mean-variance portfolio selection
Toronto, 26th June 2010
1 / 17
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 > 0 in one period: U(ϑ) = E [x + ϑ⊤ ∆S] −
γ Var[x + ϑ⊤ ∆S] = max! ϑ 2
Solution is the so-called mean-variance efficient strategy, i.e. 1 b ϑe := Cov[∆S|F0 ]−1 E [∆S|F0 ] =: ϑ. γ
Question: How does this extend to multi-period or continuous time?
Christoph Czichowsky (ETH Zurich)
Mean-variance portfolio selection
Toronto, 26th June 2010
2 / 17
Basic problem Markowitz problem: i i γ h h RT RT U(ϑ) = E x + 0 ϑu dSu − Var x + 0 ϑu dSu = max ! 2 (ϑs )0≤s≤T Static: criterion at time 0 determines optimal ϑe via ge =
RT 0
e ϑdS.
Question: more explicit dynamic description of ϑe on [0, T ] from ge ?
Christoph Czichowsky (ETH Zurich)
Mean-variance portfolio selection
Toronto, 26th June 2010
3 / 17
Basic problem Markowitz problem: i i γ h h RT RT U(ϑ) = E x + 0 ϑu dSu − Var x + 0 ϑu dSu = max ! 2 (ϑs )0≤s≤T Static: criterion at time 0 determines optimal ϑe via ge =
RT 0
e ϑdS.
Question: more explicit dynamic description of ϑe on [0, T ] from ge ?
Dynamic: Use ϑe on (0, t] and determine optimal strategy on (t, T ] via i i γ h h RT RT Ut (ϑ) = E x + 0 ϑu dSu Ft − Var x + 0 ϑu dSu Ft = max ! (ϑs )t≤s≤T 2
Christoph Czichowsky (ETH Zurich)
Mean-variance portfolio selection
Toronto, 26th June 2010
3 / 17
Basic problem Markowitz problem: i i γ h h RT RT U(ϑ) = E x + 0 ϑu dSu − Var x + 0 ϑu dSu = max ! 2 (ϑs )0≤s≤T Static: criterion at time 0 determines optimal ϑe via ge =
RT 0
e ϑdS.
Question: more explicit dynamic description of ϑe on [0, T ] from ge ?
Dynamic: Use ϑe on (0, t] and determine optimal strategy on (t, T ] via i i γ h h RT RT Ut (ϑ) = E x + 0 ϑu dSu Ft − Var x + 0 ϑu dSu Ft = max ! (ϑs )t≤s≤T 2 Time inconsistent: this optimal strategy is different from ϑe on (t, T ]!
Christoph Czichowsky (ETH Zurich)
Mean-variance portfolio selection
Toronto, 26th June 2010
3 / 17
Basic problem Markowitz problem: i i γ h h RT RT U(ϑ) = E x + 0 ϑu dSu − Var x + 0 ϑu dSu = max ! 2 (ϑs )0≤s≤T Static: criterion at time 0 determines optimal ϑe via ge =
RT 0
e ϑdS.
Question: more explicit dynamic description of ϑe on [0, T ] from ge ?
Dynamic: Use ϑe on (0, t] and determine optimal strategy on (t, T ] via i i γ h h RT RT Ut (ϑ) = E x + 0 ϑu dSu Ft − Var x + 0 ϑu dSu Ft = max ! (ϑs )t≤s≤T 2 Time inconsistent: this optimal strategy is different from ϑe on (t, T ]! Time-consistent mean-variance portfolio selection: b which is “optimal” for Ut (ϑ) and time-consistent. Find a strategy ϑ,
Christoph Czichowsky (ETH Zurich)
Mean-variance portfolio selection
Toronto, 26th June 2010
3 / 17
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. (2006): game theoretic formulation for different problems. Basak and Chabakauri (2007): results for mean-variance portfolio selection. Bj¨ ork and Murgoci (2008): General theory of Markovian time inconsistent stochastic optimal control problems (for various forms of time inconsistency.)
Christoph Czichowsky (ETH Zurich)
Mean-variance portfolio selection
Toronto, 26th June 2010
4 / 17
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. (2006): game theoretic formulation for different problems. Basak and Chabakauri (2007): results for mean-variance portfolio selection. Bj¨ ork and Murgoci (2008): 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 (ETH Zurich)
Mean-variance portfolio selection
Toronto, 26th June 2010
4 / 17
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. (2006): game theoretic formulation for different problems. Basak and Chabakauri (2007): results for mean-variance portfolio selection. Bj¨ ork and Murgoci (2008): 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: Rd -valued semimartingale S wlog. S = S0 + M + A ∈ S 2 (P). R Θ = ΘS := {ϑ ∈ L(S) | ϑdS ∈ S 2 (P)} = L2 (M) ∩ L2 (A). Christoph Czichowsky (ETH Zurich)
Mean-variance portfolio selection
Toronto, 26th June 2010
4 / 17
Outline
1
Discrete time
2
Continuous time
3
Convergence of solutions
Christoph Czichowsky (ETH Zurich)
Mean-variance portfolio selection
Toronto, 26th June 2010
5 / 17
Local mean-variance efficiency in discrete time Use x + ϑ · S := x +
Definition
RT 0
ϑu dSu = x +
PT
i =1
ϑi ∆Si and suppose d = 1.
A strategy ϑb ∈ Θ is locally mean-variance efficient (LMVE) if b − Uk−1 (ϑb + δ 1{k} ) ≥ 0 Uk−1 (ϑ)
P-a.s.
b ∈ Θ. for all k = 1, . . . , T and any δ = (ϑ − ϑ)
Recursive optimisation (K¨allblad 2008): ϑb ∈ Θ is LMVE if and only if i h PT bi ∆Si Fk−1 ϑ Cov ∆S , k i =k+1 1 1 E [∆Sk |Fk−1 ] b − = λk − ξk (ϑ) ϑbk = γ Var [∆Sk |Fk−1 ] Var [∆Sk |Fk−1 ] γ
for k = 1, . . . , T . Christoph Czichowsky (ETH Zurich)
Mean-variance portfolio selection
Toronto, 26th June 2010
6 / 17
Local mean-variance efficiency in discrete time Use x + ϑ · S := x +
Definition
RT 0
ϑu dSu = x +
PT
i =1
ϑi ∆Si and suppose d = 1.
A strategy ϑb ∈ Θ is locally mean-variance efficient (LMVE) if b − Uk−1 (ϑb + δ 1{k} ) ≥ 0 Uk−1 (ϑ)
P-a.s.
b ∈ Θ. for all k = 1, . . . , T and any δ = (ϑ − ϑ)
Recursive optimisation (K¨allblad 2008): ϑb ∈ Θ is LMVE if and only if i h PT bi ∆Si Fk−1 ϑ Cov ∆S , k i =k+1 1 1 E [∆Sk |Fk−1 ] b − = λk − ξk (ϑ) ϑbk = γ Var [∆Sk |Fk−1 ] Var [∆Sk |Fk−1 ] γ i i h hP T bi ∆Si Fk Fk−1 ϑ E ∆M E k i =k+1 ∆Ak 1 − = γ E [(∆Mk )2 |Fk−1 ] E [(∆Mk )2 |Fk−1 ]
for k = 1, . . . , T .
Christoph Czichowsky (ETH Zurich)
Mean-variance portfolio selection
Toronto, 26th June 2010
6 / 17
Structure condition and mean-variance tradeoff process S satisfies the structure condition (SC), i.e. there exists a predictable process λ such that Ak =
k X i =1
k X λi ∆hMii λi E (∆Mi )2 |Fi −1 = i =1
for k = 0, . . . , T and the mean-variance tradeoff process (MVT) Kk :=
2 k k k X X X E [∆Si |Fi −1 ] λ2i ∆hMii = λi ∆Ai = Var [∆Si |Fi −1 ] i =1
i =1
i =1
for k = 0, . . . , T is finite-valued, i.e. λ ∈ L2loc (M). If the LMVE strategy ϑb exists, then λ ∈ L2 (M), i.e. KT ∈ L1 (P).
Comments: 1) SC and MVT also appear naturally in other quadratic optimisation problems in mathematical finance; see Schweizer (2001). 2) No arbitrage condition: A ≪ hMi. Christoph Czichowsky (ETH Zurich)
Mean-variance portfolio selection
Toronto, 26th June 2010
7 / 17
Expected future gains For each ϑ ∈ Θ, define the expected future gains Z (ϑ) and the square integrable martingale Y (ϑ) by " T " T # # X k X X Zk (ϑ) : = E ϑi ∆Ai ϑi ∆Si Fk = E ϑi ∆Ai Fk − i =1
i =1
i =k+1
=: Yk (ϑ) −
k X
ϑi ∆Ai
i =1
= Y0 (ϑ) +
k X
ξi (ϑ)∆Mi + Lk (ϑ) −
i =1
k X
ϑi ∆Ai
i =1
for k = 0, 1, . . . , T inserting the GKW decomposition of Y (ϑ).
Lemma The LMVE strategy ϑb exists if and only if b 1) S satisfies (SC) with λ ∈ L2 (M), i.e. KT ∈ L1 (P), and 2) ϑb = γ1 λ − ξ(ϑ). Christoph Czichowsky (ETH Zurich)
Mean-variance portfolio selection
Toronto, 26th June 2010
8 / 17
b via FS decomposition Global description of ξ(ϑ) Combining both representations we obtain T X i =1
ϑbi ∆Ai =
T X 1
γ
i =1
b λi − ξi (ϑ) ∆Ai
b + = Y0 (ϑ)
T X i =1
b b ξi (ϑ)∆M i + LT (ϑ)
T T X 1 1X b b b KT = λi ∆Ai = Y0 (ϑ) + ξi (ϑ)∆S i + LT (ϑ) γ γ i =1
(1)
i =1
(1) is almost the F¨ ollmer–Schweizer (FS) decomposition of
1 γ KT .
b =: 1 ξb in the FS decomposition yields the locally The integrand ξ(ϑ) γ risk-minimising strategy for the contingent claim γ1 KT . b Global description: ϑb ∈ Θ exists iff (1) and ϑb = γ1 (λ − ξ).
Christoph Czichowsky (ETH Zurich)
Mean-variance portfolio selection
Toronto, 26th June 2010
9 / 17
Continuous time setting Increasing, integrable, predictable process B called “operational time” such that: A = a · B, hM, Mi = e c M · B and a = e c M λ + η with η ∈ Ker(e c M ). S satisfies the structure condition (SC), if η = 0, i.e. Z A = dhMiλ,
and the mean-variance tradeoff process (MVT) Z t Z t Kt := λ⊤ dhMi λ = λu dAu < +∞. u u u 0
0
Expected future gains Z (ϑ) and GKW decomposition of Y (ϑ) "Z "Z # # Z t T T Zt (ϑ) : = E ϑu dSu Ft = E ϑu dAu ϑu dAu Ft − =: Yt (ϑ) − = Y0 (ϑ) +
Z
Z
t
ϑu dAu
0 t
ξu (ϑ)dMu + Lt (ϑ) −
0
Christoph Czichowsky (ETH Zurich)
0
0
t
Mean-variance portfolio selection
Z
t
ϑu dAu
0 Toronto, 26th June 2010
10 / 17
Local mean-variance efficiency in continuous time Idea: Combine recursive optimisation with a limiting argument.
Definition A strategy ϑb ∈ Θ is locally mean-variance efficient (in continuous time) if b δ] := lim lim u Πn [ϑ,
n→∞
n→∞
X
ti ,ti +1 ∈Πn
b − Ut (ϑb + δ 1(t ,t ] ) Uti (ϑ) i i i +1 1(ti ,ti +1 ] ≥ 0 P⊗B-a.e. E [Bti +1 − Bti |Fti ]
for any increasing sequence (Πn ) of partitions such that |Πn | → 0 and any δ ∈ Θ. Inspired by the concept of local risk-minimisation (LRM); Schweizer (88, 08). ⊤ b δ] = γ ξ(ϑ) b + ϑb − λ + γ δ c M δ − δ ⊤ η lim u Πn [ϑ, P ⊗ B-a.e. n→∞ 2
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 (ETH Zurich)
Mean-variance portfolio selection
Toronto, 26th June 2010
11 / 17
The LMVE strategy ϑb in continuous time Theorem
1) The LMVE strategy ϑb ∈ Θ exists if and only if
i) S satisfies (SC) with λ ∈ L2 (M), i.e. KT ∈ L1 (P).
b i.e. b b = ϑ, b where b ii) ϑb = γ1 λ − ξ(ϑ), J(ϑ) J(ψ) := γ1 λ − ξ(ψ) for ψ ∈ Θ and RT ξ(ψ) is the integrand in the GKW decomposition of 0 ψu dAu .
2) If K is bounded and continuous, b J(·) is a contraction on (Θ, k.kβ,∞) where kϑkβ,∞
Z
:=
0
T
12 1
ϑ⊤
2 ∼ kϑkL2 (M) + kϑkL2 (A) . u dhMiu ϑu E(−βK )u L (P)
In particular, the LMVE strategy ϑb is given as the limit ϑb = limn→∞ ϑn in (Θ, k.kβ,∞), where ϑn+1 = b J(ϑn ) for n ≥ 1, for any ϑ0 = ϑ ∈ Θ. Christoph Czichowsky (ETH Zurich)
Mean-variance portfolio selection
Toronto, 26th June 2010
12 / 17
b via FS decomposition Global description of ξ(ϑ) Theorem
The LMVE strategy ϑb ∈ Θ exists if and only if S satisfies (SC) and the MVT process KT ∈ L1 (P) and can be written as b0 + KT = K
Z
0
T
b +b ξdS LT
(2)
b0 ∈ L2 (F0 ), ξb ∈ L2 (M) such that ξb − λ ∈ L2 (A) and b with K L ∈ M20 (P) strongly b = 1 ξb and U(ϑ) b = . . . (2). orthogonal to M. In that case, ϑb = γ1 λ − ξb , ξ(ϑ) γ b
If the minimal martingale measure exists, i.e. ddPP := E(−λ · M)T ∈ L2 (P) and strictly positive, and KT ∈ L2 (P), then Z t 1b 1 b b b b ξdS + Lt − Kt = E Zt (ϑ) = K0 + [KT − Kt |Ft ], γ γ 0 b see Choulli et al. (2010). and ξb is related to the GKW of KT under P; Application in concrete models: 1) λ, 2) K , 3) E(−λ · M) and 4) ξb . . .
Christoph Czichowsky (ETH Zurich)
Mean-variance portfolio selection
Toronto, 26th June 2010
13 / 17
Discretisation of the financial market Let (Πn )n∈N increasing such that |Πn | → 0 and S = S0 + M + A. Discretisation of processes Stn := Sti , Mtn := Mti and Ant := Ati for t ∈ [ti , ti +1 ) and all ti ∈ Πn . Discretisation of filtration Ftni := Fti for t ∈ [ti , ti +1 ) and all ti ∈ Πn and Fn := (Ftn )0≤t≤T . ¯n+A ¯ n ∈ S 2 (P, Fn ) Canonical decomposition of S n = S0 + M ¯ n := Pi E [∆An |Ft ] = An − MA,n A t t tk k−1 t k=1 ¯ tn := Mtn + MtA,n for t ∈ [ti , ti +1 ) M
where the “discretisation error” is given by the Fn -martingale MA,n := t
i X
(∆Antk − E [∆Antk |Ftk−1 ])
for t ∈ [ti , ti +1 ).
k=1
Christoph Czichowsky (ETH Zurich)
Mean-variance portfolio selection
Toronto, 26th June 2010
14 / 17
Convergence of solutions ϑbn
Due to time inconsistency usual abstract arguments don’t work. Work with global description directly to show 1 n bn L2 (M) b 1 λ − ξ −→ ϑ = λ − ξb , ϑbn = γ γ
as |Πn | → 0.
Discrete- and continuous-time FS decomposition Z X n b0 + bn + bn ∆S n + b L and K = K KTn = K ξ T T 0 ti ti
0
ti ∈Πn
For this we establish ¯n P ∆A ti +1 1) λn = ¯ n )2 |F E [(∆M ti ,ti +1 ∈Πn
2) KTn =
P
ti ,ti +1 ∈Πn
ti +1
T
ξbu dSu + b LT .
L2 (M)
ti ]
1(ti ,ti +1 ] −→ λ
2
(P) ¯ nt L−→ KT = λnti +1 ∆A i +1 2
L (M) b 3) 2), |Πn | → 0 implies ξbn −→ ξ.
RT 0
λu dAu
Problem to control the “discretisation error” MA,n . R dK Simple sufficient condition: K = dK dt dt and dt uniformly bounded.
Christoph Czichowsky (ETH Zurich)
Mean-variance portfolio selection
Toronto, 26th June 2010
15 / 17
Some references Basak and Chabakauri. Dynamic Mean-Variance Asset Allocation. (2007). Forthcoming in Review of Financial Studies. Bj¨ ork and Murgoci. A General Theory of Markovian Time Inconsistent Stochastic Control Problems. Working paper, Stockholm School of Economics, (2008). Ekeland and Lazrak. Being serious about non-commitment: subgame perfect equilibrium in continuous time, (2006). Preprint, Univ. of British Columbia. Choulli, Vandaele and Vanmaele. The F¨ ollmer-Schweizer decomposition: Comparison and description. Stoch. Pro. and their Appl., (2010), 853-872. Schweizer. Hedging of Options in a General Semimartingale Model. Diss. ETHZ no. 8615, ETH Z¨ urich (1988). Schweizer. A Guided Tour through Quadratic Hedging Approaches. In Option Pricing, Inerest Rates and Risk Management, Cambridge Univ. Press (2001). Schweizer. Local risk-minimization for multidimensional assets and payment streams. In Advances in Mathematics of Finance, Banach Center Publ. (2008). Strotz. Myopia and inconsistency in dynamic utility maximization. Review of Financial Studies (1956), 165-180. Christoph Czichowsky (ETH Zurich)
Mean-variance portfolio selection
Toronto, 26th June 2010
16 / 17
Thank you for your attention! http://www.math.ethz.ch/∼czichowc
Christoph Czichowsky (ETH Zurich)
Mean-variance portfolio selection
Toronto, 26th June 2010
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