Journal of Mathematical Finance, 2017, 7, 751-768 http://www.scirp.org/journal/jmf ISSN Online: 2162-2442 ISSN Print: 2162-2434
Multi-Period Portfolio Selection with No-Shorting Constraints: Duality Analysis Jun Qi, Lan Yi Management School, Jinan University, Guangzhou, China
How to cite this paper: Qi, J. and Yi, L. (2017) Multi-Period Portfolio Selection with No-Shorting Constraints: Duality Analysis. Journal of Mathematical Finance, 7, 751-768. https://doi.org/10.4236/jmf.2017.73040 Received: July 25, 2017 Accepted: August 28, 2017 Published: August 31, 2017 Copyright © 2017 by authors and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY 4.0). http://creativecommons.org/licenses/by/4.0/ Open Access
Abstract This paper considers a multi-period mean-variance portfolio selection problem with no shorting constraint. We assume that the sample space is finite, and the possible securities price vector transitions is equivalent to the number of securities. By making use of the embedding technique of Li and Ng (2000), the original nonseparable problem can be solved by introducing an auxiliary problem. After the risk neutral probability is calculated, the auxiliary problem can be solved by using the martingale method of Pliska (1986). Finally, we derive a closed form of the optimal solution to the original constrained problem.
Keywords Multi-Period Mean-Variance Formulation, Auxiliary Market, Martingale Method, Risk Neutral Probability, Duality, Optimal Trading Strategy
1. Introduction Portfolio theory deals with the question of how to find an optimal distribution of the wealth among various assets. Mean-variance analysis and expected utility formulation are two different tools for dealing with portfolio selections. A fundamental basis for portfolio selection in a single period was provided by Markowitz. Under the assumption that short-selling of stocks is not allowed, analytical expression of the mean-variance efficient frontier in single-period portfolio selection was derived by solving a quadratic programming problem in Markowitz (1952) [1]. Later, an analytical solution to the single-period meanvariance problem with assumption that short-selling is allowed is derived in Merton (1972) [2]. Recently, a multi-period portfolio selection problem has been studied. This problem is more interesting as investors always invest their wealth in multi DOI: 10.4236/jmf.2017.73040 Aug. 31, 2017
J. Qi, L. Yi
periods instead of only one period. Work of Li and Ng (2000) [3] considers the multi-period portfolio selection problem in a mean-variance framework when short-selling of stocks is allowed. Li and Ng have derived the analytical formulation of the frontier of the multi-period portfolio selection by embedding the assets-only multi-period mean-variance problem into a large tractable problem. When short-selling is not allowed, the multi-period portfolio selection problem is much more difficult to deal with. For continuous-time mean-variance portfolio selections, Xun, Xunyu and Andrew (2002) [4] use stochastic optimal linear-quadratic method. For multi-period setting, the portfolio selection problem with no-shorting constraint has been studied in Xu and Shreve (1992) [5] [6]. These papers investigated a utility maximization problem with a no shortselling constraint using a duality analysis. The objective of this paper is to investigate dynamic mean-variance portfolio selection when short-selling is not allowed. Instead of using optimization method, this paper used a martingale approach, which was originally proposed by Pliska (1986) [7]. To our knowledge, no analytical numerical method using martingale measure for finding the optimal portfolio policy with no-short shelling constraint for the multiperiod mean-variance formulation has been reported in the literature. In this sense, this paper extends existing literature by utilizing a martingale approach to solve an optimal portfolio selection problem with no-shorting constraint. This approach also showed that a unique equivalent martingale measure exist in the no-arbitrage complete market model. An effective algorithm is derived for finding the maximum quadratic utility function with no-short selling constraint. To outline of this paper, In Section 2, we build up the security market model. In Section 3, we consider the optimal portfolio selection problem with no shortselling constraint. By transforming the original market to some auxiliary markets, the optimal value of original constrained problem can be derived by the optimal valued of the unconstrained problem in the auxiliary markets. In Section 4, we use martingale approach to solve the unconstrained problem in the auxiliary markets. In Section 5, the optimal terminal wealth was derived by solving a dual problem. In Section 6, the derive the optimal trading strategy based on the optimal terminal wealth. A numerical example is also given in the Section 7. Finally, we conclude the paper.
2. Security Market Model We consider a multi-period security market model with T + 1 trading dates (indexed by 0,1, ,T ), and the time horizon T is finite. There are n risky securities and one bond in the market. Let
( Ω, , P )
be the probability space.
Suppose there are finite states of the world, and let K ( t ) = {1, 2, , N ( t )} be the state space of the economy at time t. The sample space Ω of the economy has a finite number of element ω = {e1 , e2 , , eT } with et ∈ K ( t ) . The filtration= F {= t , t 0,1, , T } where t is generated by {e1 , e2 , , eT } reveals the information on the economy. Specifically, 0 = {∅, Ω} and T = . We
752
J. Qi, L. Yi
claim that the process = K A∈ \ ∅ , P ( A ) > 0 .
{k ( t ) ∈ K ( t ) ;0 ≤ t ≤ T − 1}
is -adapted. For any
The n + 1 securities are traded in the market without transaction cost.
Denote the stochastic process of the security price as = S {St ; 0 ≤ t ≤ T } , where ′ S = S (1) , , S ( n ) is a random vector, and the bond price process as
(
t
t
)
t
where Bt is constant. Let = R { Rt ;1 ≤ t ≤ T } be the risky security return process defined by Rt = ( Rt (1) , , Rt ( n ) )′ and
= B
{Bt ; 0 ≤ t ≤ T } ,
= Rt ( i ) St ( i ) St −1 ( i ) − 1 ≥ −1 for St −1 ( i ) > 0 , and = r
{rt ;1 ≤ t ≤ T }
be the bond
return process defined by= rt Bt Bt −1 − 1 ≥ 0 with rt = r for all t = 1, , T . Assumption 2.1. 1) The state space at time t has N ( t= ) nt + 1 elements; 2)
For any ω = {e1 , e2 , , eT } , if et = j ∈ K ( t ) , then et +1 ∈ K j ( t + 1)= { j , j + 1, , j + n} ⊆ K ( t + 1) ; 3) Denote a matrix D of the securities’ prices
St (1, j ) St ( n, j ) Bt ( j ) Dt = B ( j + n ) S (1, j + n ) S ( n, j + n ) t t t j
( )
rank Dt j = n + 1 for t = 1, 2, , T and j ∈ K ( t )
The above assumption makes the security market a complete one. We can easily verify that the sample space of the market Ω has under assumption 2.1.
( n + 1)
T
elements
We consider an investor in the financial market with initial wealth v. She or he follows a self-financing trading strategies = H
Ht ; t {=
1, 2, , T }
where H t = ( H t (1) , , H t ( n ) )′ and H t ( i ) is the number of units of the ith risky security held between time t − 1 and t . The number of money invested in the bond is ht . Assumption 2.2. The investor invests her or his wealth in the complete market with no short-selling constraint, that is (1)
H t ≥ 0 for t = 1, , T . Let Vt be the value of portfolio at time t , it satisfies
Vt =ht Bt + H t′St =Vt −1 + ht Bt −1rt + H t′St −1 Rt ,
(2)
where
0 St −1 (1) St −1 = . 0 St −1 ( n ) For our convenience, we introduce the discounted price process ′ = S {St* ;0 ≤ t ≤ T } with St* = St* (1) , , St* ( n ) and St* ( i ) = St ( i ) Bt . So the discounted value of portfolio is V *= ht + H t′St* . The change of the discounted prices of risky security is defined as δ ( i ) = δ (1) , , δ ( n ) ′ with *
δ= St* ( i ) − St*−1 ( i ) . t (i )
(
)
t
(
t
t
)
753
J. Qi, L. Yi
We can also defined the self-financing trading strategies as= π {= π t ; t 1, 2, , T } , ′ where π = π (1) , , π ( n ) and π i is the fraction of money invested in t
(
t
t
)
t
()
ith risky security at time t − 1 . Similarly, the no short-selling requires that π t ( i ) is non-negative for any t . Therefore, the value of portfolio can be ren written as V Vt −1 1 + 1 − ∑ i=1 π t ( i ) rt + π t′Rt . It is easy to verify that = t π t = H t′St −1 Vt −1 .
(
)
3. Primal and Auxiliary Problem The multi-period portfolio optimization problem under mean-variance framework in this paper can be formulated as follows:
max s.t.
E (VT ) − ωVar (VT ) (1) & ( 2 ) ,
for ω ≥ 0 . Varying the value of ω yields the set of efficient solutions. As indicated in Li and Ng (2000), above problem is difficult to be solved directly because of the non-separability in the sense of dynamic programming. In Li and Ng (2000), the relation between the multi-period mean-variance portfolio selection problem with a fixed investment horizon and a separable portfolio selection problem with a quadratic utility function is investigated and the analytical solution is derived by using an embedding scheme. Fortunately, Theorems 1 and 2 in Li and Ng (2000) can be also applied in the current subject with an uncertain investment horizon. We now consider the following auxiliary problem:
max s.t.
(
E λVT − ωVT2
)
(1) & ( 2 ) ,
The objective function of the auxiliary problem is equivalent to the quadratic λ 1 utility function U ( x= > 0 . It is concave and twice conβ ) β x − x2 , = 2 2ω tinuously differentiable function. Proposition 3.1. 1) The first derivative of U ( x ) is U ′ ( x = ) β −x. 2) The inverse function of U ′ ( x ) is I ( y )=: β − y .
The optimal portfolio problem is to maximize the expectation of U (VT ) under the no short selling constraint π t ∈ K= {π ∈ ℜn ; π ≥ 0} . So the con-
strained optimal portfolio problem is:
max EU (VT ) s.t
π ∈ , V0 = v
where denote the set of all admissible trading strategies belong in K . Since there is a no short-selling constraint in the optimal portfolio selection problem, it is difficult to be solved directly by dynamic programming. We will try to solve the problem by introducing unconstraint auxiliary problems. Denote the support function σ ( x ) of K by
σ ( x ) ≡ sup ( −π ′x ) . π ∈K
754
J. Qi, L. Yi
In order to eliminate the situation σ ( x ) = ∞ , we defined that the effective = { x ∈ ℜn : σ ( x ) < ∞}= { x ∈ ℜn ; x ≥ 0} , domain of σ is the convex cone K . We introduce the predictable process and σ ( x ) = 0 for x ∈ K for all t ≥ 1 . Let = κ {= κ ; t 0,1,= , T − 1} with κ (κ (1) , , κ ( n ) )′ ∈ K t
t
t
t
denote the set of all such process κ . Define an auxiliary market Mκ for each κ ∈ by modifying the return processes for the bond and the risky securities as:
rt → rt , t ≥ 1 Rt → Rt + Btκ t −1 , t ≥ 1 Specially, the market M 0 with κ = 0 is the original market. We consider the unconstraint optimal portfolio problem in the market Mκ :
( )
max EU VTκ V0 = v
s.t
Let J κ ( v ) denote the corresponding optimal objective value in the market Mκ . Theorem 3.1. Suppose J 0* is the optimal solution of the primal constrained problem, and J * is the optimal solution of the dual problem
min Jκ ( v ) , κ∈
where J κ ( v ) is the optimal objective value in the unconstrained market Mκ , associated with the optimal solution κ * . If the optimal trading strategy π for the unconstrained market Mκ * satisfies. a) π ∈ b) π t′κ t*−1 = 0 , for all t ≥ 1 Then π is the optimal strategy for the original constrained market, and J0 = J * .
Proof. For the market Mκ * and optimal trading strategy π which satisfies
(a) and (b), the value of portfolio at time T is
(
)
T VTκ= (π ) v∏ t =1 1 + rt + π t′ Rt + Btκ t*−1 − rt 1 *
= v∏ t =1 1 + rt + π t′ ( Rt − rt 1) + Btπ t′κ t*−1 T
rt 1) VT0 (π ) = v∏ t =1 1 + rt + π t′ ( Rt − = T
As π is a feasible solution of the original constrained problem, the expected utility of VT0 (π ) is smaller than or equal to the optimal value of the original
( )
( )
constrained problem. So we have J * EU VTκ EU VT0 ≤ J 0* . = = On the overhands, for an arbitrary market Mκ and the optimal trading *
strategy π * of the original constrained portfolio problem, we have,
( )
VTκ = π * v∏ t =1 1 + rt + π t*′ ( Rt + Btκ t −1 − rt 1) T
T = v∏ t =1 1 + rt + π t*′ ( Rt − rt 1) + Btπ t*′κ t −1
( )
T V0 π* ≥ ∏ t =1 1 + rt + π t*′ ( Rt − rt 1) = T
755
J. Qi, L. Yi
(
J ( v ) ≤ J ( v ) for any κ , ( ) ) ≤ EU (V (π ) ) = =EU (V (π ) ) ≤ EU (V (π ) ) ≤ J ( v ) for any κ . Hence, J ≤ J
Since EU VTκ π *
J 0*
0 T
κ*
*
κ*
*
κ
κ*
T
*
* 0
κ
T
therefore,
κ*
J* . (v) =
Putting together the above two inequalities, we have J * = J 0* .
4. Martingale Method Now we try to solve the auxiliary problems
( )
max EU VTκ V0 = v
s.t
Denote the risk neutral probability in the market Mκ as Qκ . Let L (ω ) = Q (ω ) P (ω ) be the state price density. Proposition 4.1. Under the no-arbitrage consideration, The expected discounted terminal wealth based on the risk neutral probability is equal to the initial wealth, i.e.
(
)
EQκ VTκ BTκ = v.
So the problem is equivalent to
( ) (V B ) = v
max EU VTκ s.t
EQκ
κ
T
κ
T
Theorem 4.1. For the above optimal problem with quadratic utility function, the optimal attainable wealth is: κ κ v − β E L BT κ κ VT = β + L BT , 2 2 BTκ E Lκ κ*
( ) ( )
and the optimal objective value is
(
)
2
κ κ 1 2 v − β E L BT EU V= β − . T 2 2 2 BTκ 2 E Lκ
( ) κ*
Proof.
( ) ( )
( ) ( ) EU (V ) − λ E (V L
EU VTκ − λ EQκ VTκ =
κ
κ
T
T
( n+1)
BT
)
T
=
∑ P (ωi ) U (VTκ (ωi ) ) − λVTκ (ωi ) L (ωi ) i =1
BT
The necessary conditions to maximize this expression must be:
(
)
= U ′ VTκ (ω ) λ L (ω ) BT , for all ω ∈ Ω .
This is equivalent to
VTκ (ω )= I ( λ L (ω ) BT )= β − λ L (ω ) BT , for all ω ∈ Ω .
The value of the parameter λ is the one that makes VTκ satisfies EQκ (VTκ * ) = v . Hence EQκ
756
( ( β − λ L BT )
(
)
BT ) = E ( ( β − λ L BT ) L BT ) = β E ( L BT ) − λ E L2 BT2 = v
J. Qi, L. Yi
Therefore, = λ β E ( L BT ) − v E ( L2 BT2 ) . Hence, we have v − β E [L B ] T L BT , all ω ∈ Ω, VTκ *= β + 2 2 E L BT and the optimal objective value
1 2 ( v − β E [ L BT ]) β − . 2 2 E L2 BT2 2
J κ= (v)
5. Optimal Terminal Wealth Now we come to the dual problem:
( v − β E [ L BT ]) . 1 min J κ= (v) β 2 − κ∈ 2 2 E L2 BT2 2
Since E= [ L BT ] E= [ L ] BT 1 BT and E L2 BT2 = E L2 BT2 , the problem is equivalent to min E L2 = κ∈
Definition 5.1. For arbitrary = ωi the price change at time t as
(
i
Dtet −1
δ 1, eti−1 t = δ n, eti−1 t
(
)
( n+1)T
∑ Q 2 (ωi ) . i =1
{e ,, e } ∈ Ω , we defined the matrix of i 1
i T
δ t (1, eti−1 + 1) δ t (1, eti−1 + n )
, i δ t n, et −1 + n
)
δ t ( n, eti−1 + 1)
(
)
where δ t ( j , i ) is the change of discounted price of jth security at time t when the market is at state i ∈ K ( t − 1) at time t − 1 . i Definition 5.2. Denote Dtet −1 ( j ) as an n × n matrix, which comes from i
Dtet −1 by deleting column j .
i Definition 5.3. Denote D tet −1 ( l , j ) as an n × n matrix, which comes from i Dtet −1 ( l ) by replacing the row j with 11×n .
Theorem 5.1. Under the assumption 2.1, the market exists an unique risk neutral probability T
Q (ωi ) = ∏
( −1)
eti
( )
(
)
j
eti−1
(
) (
)
i i D eti−1 ei + n D eti−1 ei , j S ∑ j=1 t t t t −1 j , et −1 κ t −1 j , et −1 t ,
∑ j=1 ( −1) n+1
t =1
Dt
( j)
St ( i, j ) is the price of the ith risky security at time t when the market is at the state j ∈ K ( t ) . T Proof. We can see that Q (ωi ) > 0 for any i 1, 2, , ( n + 1) . First, we try to =
prove that the sum of Q (ωi ) is equal to one. For T = 1 , we have
Q (ωi ) =
( −1)
e1i
( )
(
)
D11 e1i + ∑ n D11 e1i , j S0 ( j ) κ 0 ( j ) j =1 , n +1 j 1 − 1 D j ∑ j=1 ( ) 1 ( ) 757
J. Qi, L. Yi
and ei (1) = i .
∑ Q (ω ) =
n +1 i ∑ ( −1)
=j 1 i
n +1 n i −1) ∑ j 1 D11 ( i, j ) S0 ( j ) κ 0 ( j ) D11 (= i ) + ∑ j 1 (=
∑ j=1 ( −1) n +1
i
j
D11 ( j )
.
Because n +1
n
n +1
n
∑ ( −1) ∑ D11 ( i, j ) S0 ( j ) κ 0 ( j ) = ∑ ∑ ( −1) D11 ( i, j ) S0 ( j ) κ 0 ( j ) i
i
=j 1 = i 1 n +1 1 1 1 = ∑ ( −1) D1 (1, j ) + D1 ( 2, j ) + + ( −1) D1 ( n + 1, j ) S0 ( j ) κ 0 ( j ) j =1 =i 1
=j 1
n
=
n
A1 ( j ) S0 ( j ) κ 0 ( j ) ∑=
0,
j +1
where
1 δ t (1,1 + n ) δ t ( j − 1,1 + n ) , 1 δ t ( j + 1,1 + n ) δ t ( n,1 + n ) n +1 × n +1
1 1 δ (1,1) δ 1, 2) ( t t δ ( j − 1,1) δ ( j − 1, 2 ) t At ( j ) = t 1 1 δ ( j + 1,1) δ ( j + 1, 2 ) t t δ ( n,1) δ t ( n, 2 ) t
(
) (
)
hence
( −1) D11 ( i ) ∑ i =1 = n +1 j ∑ j=1 ( −1) D11 ( j ) n +1
= ∑ Q (ωi ) i
i
1.
(
)
Suppose for T = k , the sum of the Qk = (ωi ) i 1, 2, , ( n + 1) is one.For T= k + 1 , the sample space is Ω k +1 . Under the assumption 2.1, Ω k +1 can be k k n +1 Ω1k +1 Ω 2k +1 Ω(k +1 ) , where Ωlk +1 divided into ( n + 1) subspace Ω k +1 = includes all the ω which has the same state in the first k period (i.e. l 1) ekl , ekl + 1, , ekl + n . Ωik +1 Ω kj +1 = ∅ ( e1l ,, ekl ) = ωl ), and ek +1 ∈ K ek ( k += for any i ≠ j . So for arbitrary subspace Ωlk +1 , k
{
∑
}
Qk +1 (ωi )
ωi ∈Ωlk +1 k +1
∑ ∏
=
( −1)
eti
∑ j =1 ( −1) n +1
ωi ∈Ωlk +1 t =1
k
=∏
( −1)
eti
n +1
( −1)
eki +1
=∏ t =1
758
( −1)
)
Dtet −1 ( j ) i
j
(
)
j
)
(
)
Dtet −1 ( j ) i
) (
(
( )
∑ j =1 ( −1) n +1
eti
(
)
) (
n ei D eki ei + ∑ j =1 D t k eki +1 , j Sk j , eki κ k j , eki k +1 k +1
ωi ∈Ωlk +1
k
(
( )
∑ j =1 ( −1) ∑
)
(
i i D eti−1 ei + n D eti−1 ei , j S ∑ j =1 t t t t −1 j , et −1 κ t −1 j , et −1 t
t =1
×
)
(
( )
i i D eti−1 ei + n D eti−1 ei , j S ∑ j =1 t t t t −1 j , et −1 κ t −1 j , et −1 t
( )
j
Dk k+1 ( j ) ei
)
(
(
)
(
)
i i D eti−1 ei + n D eti−1 ei , j S ∑ j =1 t t t t −1 j , et −1 κ t −1 j , et −1 t
∑ j =1 ( −1) n +1
j
eti−1
Dt
( j)
J. Qi, L. Yi
we can verify that
∑
Qk +1 (ωi ) = Qk (ω l ) .
ωi∈Ωlk +1
Therefore, ( n+1)k +1
( n+1)k
Q (ω ) ∑ ∑=
k +1 i =i 1 =l 1
( n+1) Qk +1 (ωi ) = Qk (ω l ) 1. ∑= ω ∈Ωl = ∑ i k +1 l1 k
So we conclude that for any T, the sum of Q (ωi ) is one, and this defines a probability. Now we try to prove that the probability is a risk neutral probability in the market. Consider an arbitrary time t, and arbitrary event A corresponding to t−1 (i.e. for ω ∈ A , ( e1 , , et −1 ) = ω have been known). Suppose et −1 = ζ , we
know that ∀ω ∈ A , et ∈ K ζ ( t ) ={ζ , ζ + 1, , ζ + n} . We denote the subset of A as A1 , , An+1 , where all element in Ai has et = ζ + i − 1 ( A = A1 A2 An+1 ).
( i = 1, , n )
For ith
risky security we have
EQ St* = ( i ) t −1 EQ σ t ( i ) + St −1 ( i ) κ t −1 ( i ) t −1 =
∑Q (ω ) σ t ( i, ω ) + St −1 ( i, ω ) κ t −1 ( i, ω )
ω∈ A
= Q ( w ζ
n +1
)∑
( −1)
l =1
l
Dtζ ( l ) + ∑ n D tζ ( l , j ) St −1 ( j , ζ ) κ t −1 ( j , ζ ) j =1 n +1 j ζ ∑ j =1 ( −1) Dt ( j )
× σ t ( i, l ) + St −1 ( i, ζ ) κ t −1 ( i, ζ ) n +1 ( −1)l Dζ ( l ) σ ( i, l ) + n D ζ ( l , j ) S ( j , ζ ) κ ( j , ζ ) σ ( i, l ) ∑ j =1 t t t −1 t −1 t t = Q ( w ζ ) ∑ n +1 j ζ l =1 ∑ j =1 ( −1) Dt ( j ) n +1
+∑ l =1
( −1)
l
Dtζ ( l ) + ∑ n D tζ ( l , j ) St −1 ( j , ζ ) κ t −1 ( j , ζ ) j =1 × S i, ζ κ i, ζ ) t −1 ( ) t −1 ( n +1 j ∑ j =1 ( −1) Dtζ ( j )
l ζ n +1 ∑ l =1 ( −1) Dt ( l ) σ t ( i, l ) = Q ( w ζ ) n +1 j ζ ∑ j =1 ( −1) Dt ( j ) n +1 l ( −1) ∑ j ≠i D tζ ( l , j ) St −1 ( j, ζ ) κ t −1 ( j, ζ ) σ t ( i, l ) ∑ l =1 + n +1 j ∑ j =1 ( −1) Dtζ ( j ) n +1 l ( −1) ∑ +
n +1 l D ζ ( l , j ) σ ( i, l ) + ∑ ( −1) Dtζ ( l )
t t =l 1 = l 1 n +1 j ζ t j =1
∑ ( −1)
D
( j)
St −1 ( i, ζ ) κ t −1 ( i, ζ )
n +1 n l ∑ ( −1) ∑ D tζ ( l , j ) St −1 ( j, ζ ) κ t −1 ( j, ζ ) + S i , ζ κ i , ζ ( ) ( ) t − 1 t − 1 n +1 j ∑ j =1 ( −1) Dtζ ( j )
=l 1 =j 1
There are four part. For part (1), n +1
∑ ( −1) l =1
l
Cζ − tζ = Dtζ ( l ) σ t ( i, l ) = 0, Dt
where Ctζ is the ith row of Dtζ . 759
J. Qi, L. Yi
For part (2), because n +1
∑ ( −1)
D tζ ( l , j ) St −1 ( j , ζ ) κ t −1 ( j , ζ ) σ t ( i, l )
l
l =1
Cζ = − ζ t 0 St −1 ( j , ζ ) κ t −1 ( j , ζ ) = Dt ( l , j ) for j ≠ i , where Ctζ is the ith row of Dtζ . Hence, n +1
∑∑ ( −1)
D tζ ( l , j ) St −1 ( j , ζ ) κ t −1 ( j , ζ ) σ t ( i, l ) = 0.
l
l =1 j ≠i
For part (3), n +1
∑ ( −1)
n +1
l D tζ ( l , i ) σ t ( i, l ) + ∑ ( −1) Dtζ ( l )
l
=l 1 =l 1
Cζ 11×n = ( −1) ζ t + ( −1) ζ Dt ( l ) Dt ( l , i ) Cζ 2 = ( −1) ζ t + ( −1) (l, i ) D t
Ctζ ζ = 0 Dt ( l , i )
For part (4), n +1
n
∑ ( −1) ∑ D tζ ( l , j ) St −1 ( j, ζ ) κ t −1 ( j, ζ ) l
=l 1 =j 1
=
n
At ( j ) St −1 ( j , ζ ) κ t −1 ( j , ζ ) ∑=
0.
j =1
Hence, EQ σ t ( i ) + St −1 ( i ) κ t −1 ( i ) t −1 = 0, ∀i, t.
So this probability is a risk neutral probability of the market. The dual problem is equivalent to 2
n i i min ∑∏ Dtet −1 eti + ∑ D tet −1 eti , j St −1 j , eti−1 κ t −1 j , eti−1 . κ∈ i t =1 j =1 T
( )
(
)
(
)
(
)
To simplify our problem, we give the following notations: atl ( i ) = Dtl ( i ) ,
κ t −1 ( l ) = (κ t −1 (1, l ) , , κ t −1 ( n, l ) )′ , btl ( i, j ) = D tl ( i, j ) St −1 ( j , l ) ,
btl ( i ) = btl ( i,1) , , btl ( i, n ) ′ .
(
)
So the problem can be rewrite as: T i i min ∑∏ atet −1 eti + btet −1 eti ′ κ t −1 eti−1 κ∈ i t =1
( )
( )
( )
2
There have some special properties of the objective function, which makes the calculus much easy. Denote 760
J. Qi, L. Yi
= ϕτ ( i )
∏ ate+1 ( et +1 ) + bte+1 ( et +1 )′ κ t ( et ) T
∑
t
2
t
{( eτ ,,eT );eτ = i∈K (τ )} t =τ
.
Proposition 5.1. Under the assumption 2.1, ϕτ ( i ) have the following property:
ϕτ −1 ( l ) = = =
∏ ate+1 ( et +1 ) + bte+1 ( et +1 )′ κ t ( et ) T
∑
t
2
t
l∈K (τ −1)}t = τ −1 {( eτ −1 ,eτ ,,eT );eτ −1= l +n
∑
∏ ate+1 ( et +1 ) + bte+1 ( et +1 )′ κ t ( et ) T
∑
t
2
t
i =l {( eτ −1 ,eτ ,,eT );eτ −1 = l∈K (τ −1),eτ = i∈K (τ )}t = τ −1
l +n
∑ aτl ( i ) + bτl ( i )′ κτ −1 ( l )
2
i =l
ϕτ ( i )
So we have
= f0 :
i i ∑∏ atet −1 eti + btet −1 eti ′ κ t −1 eti−1 i t =1
( )
T
( )
( )
2
2 a l1 ( l ) + bl1 ( l )′ κ ( l ) ∑ 2 2 2 2 1 1 l2∈K ( 2 ) 2 l l ′ ∑ ∑ aTT −1 ( lT ) + bTT −1 ( lT ) κT −1 ( lT −1 ) . l3∈K ( 3) lT ∈K (T )
= ∑ a11 ( l1 ) + b11 ( l1 )′ κ 0 (1) l1∈K (1)
2
The problem is separable and can be solved by dynamic programming. m+ n Theorem 5.2. If ∑ i=m ϕt +1 ( i ) ⋅ btm+1 ( i ) btm+1 ( i )′ has full rank for any t and m ∈ K ( t ) , then the optimal solution of the dual problem is
κ t* ( m ) = (κ t* (1, m ) , , κ t* ( n, m ) )′ = for t 0,1, 2, , T − 1 = and m 1, 2, , nt + 1 , with κt ( j , m ) if κt ( j , m ) ≥ 0 κ t* ( j , m ) = if κτ −1 ( j , m ) < 0 0
where κt ( m ) = (κt (1, m ) , , κt ( n, m ) )′ is:
m+ n
−1
m+ n
κt ( m ) = − ∑ϕt +1 ( i ) ⋅ btm+1 ( i ) btm+1 ( i )′ ⋅ ∑ϕt +1 ( i ) ⋅ btm+1 ( i ) ⋅ atm+1 ( i ) ,
=i m=i m
ϕt +1 ( m ) =
m+ n
∑ atm+2 ( i ) + btm+2 ( i )′ κ t*+1 ( m ) i =m
2
ϕt + 2 ( i ) .
Proof. Let ft −1 ( lt −1 ) : =
=
2 a lt −1 ( l ) + blt −1 ( l )′ κ ( l ) ∑ 1 1 t t t t t t − − lt +1∈K lt ( t +1) lt ∈K lt −1 ( t ) 2 l l ′ ∑ aTT −1 ( lT ) + bTT −1 ( lT ) κT −1 ( lT −1 ) lT ∈K lT −1 (T )
∑
∑
lt ∈K lt −1 ( t )
2
a lt −1 ( l ) + blt −1 ( l )′ κ ( l ) f ( l ) t t t t t −1 t −1 t t
761
J. Qi, L. Yi
At s= T − 1 , for lT −1 ∈ K = (T − 1)
= fT −1 ( lT −1 )
∑
lT ∈K lT −1 (T )
{1, 2, , n (T − 1) + 1} , 2
a lT −1 ( l ) + blT −1 ( l )′ κ ( l ) . T T T T T −1 T −1
We separately solve the following problem for each lT −1 = m :
max
∑ i=m aTm ( i ) + bTm ( i )′ κT −1 ( m ) m+ n
2
κT −1 ( m ) ≥ 0
s.t
We solve the unconstrained problem, and the optimal κT −1 ( m ) = (κT −1 (1, m ) , , κT −1 ( n, m ) )′ is:
m+ n
−1
m+ n
κT −1 ( m ) = − ∑bTm ( i ) ⋅ bTm ( i )′ ⋅ ∑bTm ( i ) ⋅ aTm ( i )
=i m=i m
The optimal solution of the constrained problem is κT −1 ( m ) = (κT −1 (1, m ) , , κT −1 ( n, m ) )′ where
κ ( j , m ) if κT −1 ( j , m ) ≥ 0 κT* −1 ( j , m ) = T −1 if κT −1 ( j , m ) < 0 0
ϕ= So f= T −1 ( m ) T −1 ( m ) fT −2 ( lT −2 ) =
∑ i=m aTm ( i ) + bTm ( i )′ κT*−1 ( m ) m+ n
2
, and 2
a lT −2 ( l ) + blT −2 ( l )′ κ ( l ) ϕ ( l ) . T −1 T −1 T −2 T −2 T −1 T −1 T −1 T −1 lT −1∈K lT − 2 (T −1)
∑
Now we come to s= T − 2 . We separably to solve the following problem for
lT −2 = m : m+ n
2
∑ aTm−1 ( i ) + bTm−1 ( i )′ κT −2 ( m ) ϕT −1 ( i ) i =m
max
κ T −2 ( m ) ≥ 0
s.t
We solve the unconstrained problem, and the optimal κT −2 ( m ) = (κT −2 (1, m ) , , κT −2 ( n, m ) )′ is:
m+ n
−1
m+ n
κT −2 ( m ) = − ∑bTm−1 ( i ) ⋅ bTm−1 ( i )′ ϕT −1 ( i ) ⋅ ∑bTm−1 ( i ) ⋅ aTm−1 ( i ) ϕT −1 ( i )
=i m=i m
The optimal solution of the constrained problem is
κT −2 ( m ) = (κT −2 (1, m ) , , κT −2 ( n, m ) )′ where
κ ( j , m ) if κT −2 ( j , m ) ≥ 0 κT* −2 ( j , m ) = T −2 if κT −2 ( j , m ) < 0 0 So f= T −2 ( m ) ϕ= T −2 ( m ) fT −3 ( lT −3 ) =
∑ i=m aTm−1 ( i ) + bTm−1 ( i )′ κT*−2 ( m ) m+ n
2
ϕT −1 ( i ) , and 2
a lT −3 ( l ) + blT −3 ( l )′ κ ( l ) × ϕ ( l ) . T −2 T −2 T −3 T −3 T −2 T −2 T −2 T −2 lT − 2∈K lT −3 (T −2 )
∑
Generally, at stage s = τ , we suppose is fτ ( lτ ) =
762
2
a lτ ( l ) + blτ ( l )′ κ ( l ) ϕ ( l ) τ +1 τ +1 τ τ τ +1 τ +1 τ +1 τ +1 lτ lτ +1∈K (τ +1)
∑
J. Qi, L. Yi
We separably to solve the following problem for each lτ= m ∈ K (τ ) :
max
∑ i=m aτm+1 ( i ) + bτm+1 ( i )′ κτ ( m ) m+ n
2
ϕτ +1 ( i )
κτ ( j , m ) ≥ 0
s.t
We solve the unconstrained problem, and the optimal
κτ ( m ) = (κτ (1, m ) , , κτ ( n, m ) )′ is: m+ n
−1
m+ n
κτ ( m ) = − ∑ϕτ +1 ( i ) ⋅ bτm+1 ( i ) bτm+1 ( i )′ ⋅ ∑ϕτ +1 ( i ) ⋅ bτm+1 ( i ) ⋅ aτm+1 ( i )
=i m=i m
The optimal solution of the constrained problem is
κτ ( m ) = (κτ (1, m ) , , κτ ( n, m ) )′ where
κ ( j , m ) if κτ ( j , m ) ≥ 0 κτ* ( j , m ) = τ if κτ ( j , m ) < 0 0
Hence, the optimal solution of the dual problem is
κ t* ( m ) = (κ t* (1, m ) , , κ t* ( n, m ) )′ = and m 1, 2, , nt + 1 , with for t 0,1, 2, , T − 1 = κ ( j , m ) if κt ( j , m ) ≥ 0 κ t* ( j , m ) = t if κτ −1 ( j , m ) < 0 0
where κt ( m ) = (κt (1, m ) , , κt ( n, m ) )′ is: m+ n
−1
m+ n
κt ( m ) = − ∑ϕt +1 ( i ) ⋅ btm+1 ( i ) btm+1 ( i )′ ⋅ ∑ϕt +1 ( i ) ⋅ btm+1 ( i ) ⋅ atm+1 ( i ) ,
=i m=i m
ϕt +1 ( m ) =
m+ n
∑ atm+2 ( i ) + btm+2 ( i )′ κ t*+1 ( m )
2
i =m
ϕt + 2 ( i ) .
Hence, the risk neutral probability with optimal κ * is T
Q (ωi ) = ∏ t =1
( −1)
eti
( )
(
)
(
) (
)
* i i D eti−1 ei + n D eti−1 ei , j S ∑ j=1 t t t t −1 j , et −1 κ t −1 j , et −1 t , n+1 j eti−1 ∑ j=1 ( −1) Dt ( j )
and the optimal terminal wealth is
* * v − β E Lκ BTκ * vBκ * − β * * Lκ Bκ = Lκ β + β + T VTκ (ωi ) = T E Lκ * 2 Bκ * 2 E Lκ * 2 T * 1 vBTκ − β = β+ Q (ωi ) . T ( n + 1) E Q (ωi )2 *
( ) ( )
( )
Hence, the expectation and variance of terminal wealth are: E (VT= ) aβ + bv,
763
J. Qi, L. Yi
(
Var= (VT ) c vBTκ + β *
), 2
where
a =1
b=
c=
( n + 1)
( n + 1)
T
,
2 E Q ( ω )
T
1
E Q (ω )
BTκ , *
2 E Q ( ω )
2 E Q (ω ) − E Q (ω ) .
1
( n + 1)
E Q (ω )
1
( E Q (ω ) )
2T
2
2
The optimal β must satisfy the optimality condition of
dU = 0 , that is, dβ
a + 2ω cvBTκ . β = 2ω c *
*
6. Optimal Trading Strategy Let α =
κ* vBT − β , so V (ω = β + α Q (ωi ) . T i) E Q ω 2 ) ( i
1
( n + 1)
T
Proposition 6.1. For any t ∈ {0,1, , T } , Vt (ω ) are equivalent for
= ω ∈Ω lt
e , , e ) ; ( e , , e ) {(= T
1
t
1
ω lt } ,
{
}
where ω lt is belong to the t period sample space, and lt ∈ 1, 2, , ( n + 1) . We denote t
( )
Vt (ω ) = Vt ω lt . We have the relationship between Vt and Vt +1 for any t : Vt (ω ) ⋅ 1 + rt +1 + π t′+1 ( Rt +1 + Bt +1κ t − rt +11) = Vt +1 (ω ) .
So we can iteratively derive Vt and corresponding trading strategy π t +1 for each t . For t= T − 1 , we consider ω ∈ ΩlT −1 (= = e1 , , eT ) ; ( e1 , , eT −1 ) ω lT −1
(
{
{
})
}
, and Ω = Ω1 Ω n+1 T −1 . For each i there are n + 1 lT −1 ∈ 1, 2, , ( n + 1) ( ) element in the set Ωi . For an arbitrary set ΩlT −1 , we notate the n + 1 element T −1
in it as ω1lT −1 , , ωnlT+−11 . So
( ) (ω ) ⋅ 1 + r
( ( ) + π ′ ( R (ω ) + B κ
) ( ) − r 1) = V (ω )
VT −1 ω lT −1 ⋅ 1 + rT + π T′ RT ω1lT −1 + BT κT −1 − rT 1 = V ω lT −1 T 1 VT −1
lT −1
T
T
T
lT −1 2
T
(
)
( (
)
T −1
T
lT −1 2
T
)
(
VT −1 ω lT −1 ⋅ 1 + rT + π T′ RT ωnlT+−11 + BT κT −1 − rT 1 = V ω lT −1 T n+1
b , where Rewrite the above equations as the matrix form as G ⋅ X = 764
)
J. Qi, L. Yi
lT −1 VT ω1 G= VT ωnlT+−11
′ − rT 1 ′ + BT κT −1 − rT 1
(
) ( R (ω ) + B κ
(
) ( R (ω )
lT −1 1
T
lT −1 n +1
T
T
)
(
1 VT −1 ω l T −1 X = πT b=
)
T −1
)
(1 + rT ) 1
If G is full rank, X = G −1b .
{
}
τ Generally, for t = τ , we consider ω ∈ Ω = e1 , , eτ +1 ) ; ( e1 , , eτ ) ω lτ (= lτ τ τ + 1 τ τ = Ω Ω . For each i there are n + 1 lτ ∈ 1, 2, , ( n + 1) , and Ω 1 ( n+1)τ τ τ element in the set Ω . For an arbitrary set Ω , we notate the n + 1 element in
})
( {
i
lτ
it as ω1lτ , , ωnlτ+1 . So
( ) ( )
( ( ( (
) )
) )
( (
) )
( )
( (
)
)
(
)
Vτ ω lτ ⋅ 1 + rτ +1 + π τ′+1 Rτ +1 ω1lτ +1 + Bτ +1κτ − rτ +1 1 = V ω lτ +1 τ +1 1 Vτ ω lτ ⋅ 1 + rτ +1 + π τ′+1 Rτ +1 ω2lτ +1 + Bτ +1κτ − rτ +1 1 = V ω lτ +1 τ +1 2 Vτ ω lτ ⋅ 1 + rτ +1 + π τ′+1 Rτ +1 ωnlτ++11 + Bτ +1κτ − rτ +1 1 = V ω lτ +1 τ +1 n +1
Rewrite the above equations as the matrix form as Gτ ⋅ X τ = bτ , where
lτ Vτ +1 ω1 Gτ = Vτ +1 ωnlτ+1
( ) ( R (ω ) + B
(
τ +1
lτ 1
)′
κ − rτ +1 1
τ +1 τ
′ + Bτ +1κτ − rτ +1 1
) ( R (ω ) τ +1
lτ n +1
)
( )
1 Vτ ω l τ Xτ = π τ +1 bτ=
(1 + rτ +1 ) 1
If Gτ is full rank, X τ = Gτ−1bτ .
7. Numerical Example We consider a market with one risky security and one bond, and the investment horizon is T = 3 . Suppose the bond price is constant. The prices of the risky security are: ω
S0
S1
S2
S3
ω1
5
8
9
12
ω2
5
8
9
6
ω3
5
8
6
8
ω4
5
8
6
5
ω5
5
4
6
8
ω6
5
4
6
5
ω7
5
4
3
5
ω8
5
4
3
2
765
J. Qi, L. Yi
The risk neutral probability in market Mκ are: Qκ (ω )
ω
(1 − κ (1) ) ( 2 − 8κ (1) ) (1 − 9κ (1) )
ω1
0
1
(1 − κ (1) ) ( 2 − 8κ (1) ) ( 3 + 9κ (1) )
ω2
0
1
2
48
(1 − κ (1) ) (1 + 8κ (1) ) (1 − 6κ ( 2 ) )
ω3
0
1
2
36
(1 − κ (1) ) (1 + 8κ (1) ) ( 2 + 6κ ( 2 ) )
ω4
0
1
2
36
( 3 + κ (1) ) (1 − 4κ ( 2 ) ) (1 − 6κ ( 2 ) )
ω5
0
1
2
36
( 3 + κ (1) ) (1 − 4κ ( 2 ) ) ( 2 + 6κ ( 2 ) )
ω6
0
1
2
36
( 3 + κ (1) ) ( 2 + 4κ ( 2 ) ) (1 − 3κ ( 3) )
ω7
0
1
2
36
( 3 + κ (1) ) ( 2 + 4κ ( 2 ) ) ( 2 + 3κ ( 3) )
ω8
0
1
36
We solve the dual problem: max
∑ i =1Qκ (ωi )
s.t
1, 2,3 κ t ≥ 0 for t =
8
2
The optimal solution of this is easily found to be:
κ= κ= κ= 0 2 (1) 2 ( 2) 2 ( 3) = κ1 (1)
5 = , κ1 ( 2 ) 0 68
κ 0 (1) = 0 The optimal value is
766
2
48
ω
V (ω )
ω1
1.846853464
ω2
1.540560391
ω3
1.770280196
ω4
1.540560391
ω5
1.566084814
ω6
1.131648721
ω7
1.131648721
ω8
0.263818349
2
J. Qi, L. Yi
Finally, we get the optimal trading strategy of the original problem by solving some linear equations. The optimal strategy are ω
π
ω1
= π 1 1.0287,= π 2 0,= π 3 0.4262
ω2
= π 1 1.0287,= π 2 0,= π 3 0.4262
ω3
= π 1 1.0287,= π 2 0,= π 3 0.2841
ω4
= π 1 1.0287,= π 2 0,= π 3 0.2841
ω5
= π 1 1.0287, = π 2 1.2144, = π 3 0.6807
ω6
= π 1 1.0287, = π 2 1.2144, = π 3 0.6807
ω7
= π 1 1.0287, = π 2 1.2144, = π 3 1.569
ω8
= π 1 1.0287, = π 2 1.2144, = π 3 1.569
8. Conclusion Optimal mean-variance multiperiod portfolio selection with no shorting constraints problem is studied in the paper. We connect the original mean-variance problem to an auxiliary problem by using an embedding technique. Since the auxiliary problem is difficult to solve directly, we extend the literature by using duality theory and martingale approach to do the analysis. Finally, the derived analytical optimal multiperiod portfolio strategy provides investors with the best strategy to follow in a no-short selling dynamic investment environment. The limitation in this paper is that we derive the optimal portfolio policy by maximizing the quadratic utility function. A future research subject is investigation of an optimal solution using different utility objective function.
Acknowledgements Jinan University scientific research cultivation and Innovation Fund, Number: 17JNQN025.
References [1]
Markowiz, H. (1983) Portfolio Selection. The Journal of Finance, 7, 77-91.
[2]
Merton, R.C. (1972) An Analytic Derivation of the Efficient Portfolio. The Journal of Financial and Quantitative Analysis, 7, 13-15. https://doi.org/10.2307/2329621
[3]
Duan, L. and Ng, W.L. (2000) Optimal Dynamic Portfolio Selection: Multiperiod Mean-Variance Formulation. Mathematical Finance, 10, 387-406. https://doi.org/10.1111/1467-9965.00100
[4]
Li, X., Zhou, X.Y. and Lim, A.E.B. (2002) Dynamic Mean-Variance Portfolio Selection with No-Shorting Constraints. Journal on Control and Optimization, 40, 15401555. https://doi.org/10.1137/S0363012900378504
[5]
Xu, G.L. and Shreve, S.E. (1992) A Duality Method for Optimal Consumption and Investment under Short-Selling Prohibition. i. General Market Coefficients. Annals of Applied Probability, 2, 87-112. https://doi.org/10.1214/aoap/1177005772 767
J. Qi, L. Yi [6]
Xu, G.L. and Shreve, S.E. (1992) A Duality Method for Optimal Consumption and Investment under Short-Selling Prohibition. ii. Constant Market Coefficients. Annals of Applied Probability, 2, 314-328. https://doi.org/10.1214/aoap/1177005706
[7]
Pliska, S.R. (1986) A Stochastic Calculus Model of Continuous Trading: Optimal Portfolios. Mathematics of Operations Research, 11, 371-382. https://doi.org/10.1287/moor.11.2.371
Submit or recommend next manuscript to SCIRP and we will provide best service for you: Accepting pre-submission inquiries through Email, Facebook, LinkedIn, Twitter, etc. A wide selection of journals (inclusive of 9 subjects, more than 200 journals) Providing 24-hour high-quality service User-friendly online submission system Fair and swift peer-review system Efficient typesetting and proofreading procedure Display of the result of downloads and visits, as well as the number of cited articles Maximum dissemination of your research work
Submit your manuscript at: http://papersubmission.scirp.org/ Or contact
[email protected] 768
