short sales constraint
TRANSCRIPT
8/3/2019 Short Sales Constraint
http://slidepdf.com/reader/full/short-sales-constraint 1/42
Market model
The investor
Method of solution
Examples
Summary
Convex duality in constrained mean-variance
portfolio optimization under a regime-switching
model
Catherine Donnelly1 Andrew Heunis2
1ETH Zurich, Switzerland
2University of Waterloo, Canada
26 June 2010
Catherine Donnelly, Andrew Heunis Convex duality in constrained portfolio optimization
8/3/2019 Short Sales Constraint
http://slidepdf.com/reader/full/short-sales-constraint 2/42
Market model
The investor
Method of solution
Examples
Summary
Motivation
Investor
Wishes to have $100 in 1 year’s time.
Starts with $90.
Invests money in stockmarket and bank account.No short-selling.
How to invest to minimize:
E(Investor’s wealth in 1 year − 100)
2
subject to satisfying the investment restrictions and
E (Investor’s wealth in 1 year) = 100.
Catherine Donnelly, Andrew Heunis Convex duality in constrained portfolio optimization
8/3/2019 Short Sales Constraint
http://slidepdf.com/reader/full/short-sales-constraint 3/42
Market model
The investor
Method of solution
Examples
Summary
Motivation
Investor
Wishes to have $100 in 1 year’s time.
Starts with $90.
Invests money in stockmarket and bank account.No short-selling.
How to invest to minimize:
E
(Investor’s wealth in 1 year − 100)
2
subject to satisfying the investment restrictions and
E (Investor’s wealth in 1 year) = 100.
Catherine Donnelly, Andrew Heunis Convex duality in constrained portfolio optimization
8/3/2019 Short Sales Constraint
http://slidepdf.com/reader/full/short-sales-constraint 4/42
Market model
The investor
Method of solution
Examples
Summary
Outline
1 Market model
2 The investor
3 Method of solution
4 Examples
Catherine Donnelly, Andrew Heunis Convex duality in constrained portfolio optimization
8/3/2019 Short Sales Constraint
http://slidepdf.com/reader/full/short-sales-constraint 5/42
Market model
The investor
Method of solution
Examples
Summary
Market model
(Ω,F ,P) and finite time horizon [0,T ].
Market consists of N traded assets and a risk-free asset.
Risk-free asset price process obeys
dS 0(t )
S 0(t )= r (t ) dt .
Price processes of stocks obeydS (t )
S (t )= µ(t ) dt + σ(t ) dW (t ).
Catherine Donnelly, Andrew Heunis Convex duality in constrained portfolio optimization
M k d l
8/3/2019 Short Sales Constraint
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Market model
The investor
Method of solution
Examples
Summary
Market model
Market subject to regime-switching modeled by Markov chainα.
Finite-state-space I = 1, . . . ,D .
Generator matrix G = (g ij ).Jump processes for i = j
N ij (t ) =
0<s ≤t
1[α(s −) = i ]1[α(s ) = j ]
Martingales for i = j
M ij (t ) = N ij (t ) −
t
0g ij 1[α(s −) = i ] ds .
Catherine Donnelly, Andrew Heunis Convex duality in constrained portfolio optimization
M k t d l
8/3/2019 Short Sales Constraint
http://slidepdf.com/reader/full/short-sales-constraint 7/42
Market model
The investor
Method of solution
Examples
Summary
Market model
Market subject to regime-switching modeled by Markov chainα.
Finite-state-space I = 1, . . . ,D .
Generator matrix G = (g ij ).Jump processes for i = j
N ij (t ) =
0<s ≤t
1[α(s −) = i ]1[α(s ) = j ]
Martingales for i = j
M ij (t ) = N ij (t ) −
t
0g ij 1[α(s −) = i ] ds .
Catherine Donnelly, Andrew Heunis Convex duality in constrained portfolio optimization
Market model
8/3/2019 Short Sales Constraint
http://slidepdf.com/reader/full/short-sales-constraint 8/42
Market model
The investor
Method of solution
Examples
Summary
The investor’s wealth process
Portfolio process π(t ) = π1(t ), . . . , πN (t ) at time t .
Wealth process X π(t ) at time t , given by
X π(t ) = π0(t ) +
N n =1
πn (t ).
Wealth equation: X π(0) = x 0, a.s. and
dX π
(t ) =
r (t )X π
(t ) + π
(t )σ(t )θ(t )
dt +σ
(t )π(t ) dW (t ),
where the market price of risk is
θ(t ) := σ−1(t ) (µ(t ) − r (t )1) .
Catherine Donnelly, Andrew Heunis Convex duality in constrained portfolio optimization
Market model
8/3/2019 Short Sales Constraint
http://slidepdf.com/reader/full/short-sales-constraint 9/42
Market model
The investor
Method of solution
Examples
Summary
The investor’s wealth process
Portfolio process π(t ) = π1(t ), . . . , πN (t ) at time t .
Wealth process X π(t ) at time t , given by
X π(t ) = π0(t ) +
N n =1
πn (t ).
Wealth equation: X π(0) = x 0, a.s. and
dX π
(t ) =
r (t )X π
(t ) + π
(t )σ(t )θ(t )
dt +σ
(t )π(t ) dW (t ),
where the market price of risk is
θ(t ) := σ−1(t ) (µ(t ) − r (t )1) .
Catherine Donnelly, Andrew Heunis Convex duality in constrained portfolio optimization
Market model
8/3/2019 Short Sales Constraint
http://slidepdf.com/reader/full/short-sales-constraint 10/42
Market model
The investor
Method of solution
Examples
Summary
The investor’s portfolio constraints
K ⊂ RN closed, convex set with 0 ∈ K .
Example: no short-selling
K := π = (π1, . . . , πN ) ∈ RN : π1 ≥ 0, . . . , πN ≥ 0.
Set of admissible portfolios
A := π ∈ L2(W ) |π(t ) ∈ K , a.e..
Catherine Donnelly, Andrew Heunis Convex duality in constrained portfolio optimization
Market model
8/3/2019 Short Sales Constraint
http://slidepdf.com/reader/full/short-sales-constraint 11/42
Market model
The investor
Method of solution
Examples
Summary
The investor’s risk measure
Risk measure J
J (x ) =1
2Ax 2 + Bx + C , ∀x ∈ R,
where A, B and C are random variables.
Example: J (x ) = (x − 100)2.
Catherine Donnelly, Andrew Heunis Convex duality in constrained portfolio optimization
Market model
8/3/2019 Short Sales Constraint
http://slidepdf.com/reader/full/short-sales-constraint 12/42
Market model
The investor
Method of solution
Examples
Summary
The investor’s problem
Does there exist π ∈ A such that
E(J (X π(T ))) = infπ∈A
E(J (X π(T )))?
Can we characterize π? Can we find π?
Example:
A := π ∈ L2
(W ) |π(t ) ≥ 0 a.e.and
E(X π(T ) − 100)2 = infπ∈A
E(X π(T ) − 100)2
Catherine Donnelly, Andrew Heunis Convex duality in constrained portfolio optimization
Market model
8/3/2019 Short Sales Constraint
http://slidepdf.com/reader/full/short-sales-constraint 13/42
The investor
Method of solution
Examples
Summary
The investor’s problem
Does there exist π ∈ A such that
E(J (X π(T ))) = infπ∈A
E(J (X π(T )))?
Can we characterize π? Can we find π?
Example:
A := π ∈ L2
(W ) |π(t ) ≥ 0 a.e.and
E(X π(T ) − 100)2 = infπ∈A
E(X π(T ) − 100)2
Catherine Donnelly, Andrew Heunis Convex duality in constrained portfolio optimization
Market model
8/3/2019 Short Sales Constraint
http://slidepdf.com/reader/full/short-sales-constraint 14/42
The investor
Method of solution
Examples
Summary
Convex duality approach
Restate MVO problem as primal problem.
Construct dual problem.
Necessary and sufficient conditions.
Existence of a solution to the dual problem.
Construct candidate primal solution.
Verification.
Catherine Donnelly, Andrew Heunis Convex duality in constrained portfolio optimization
Market model
8/3/2019 Short Sales Constraint
http://slidepdf.com/reader/full/short-sales-constraint 15/42
The investor
Method of solution
Examples
Summary
Convex duality approach
Restate MVO problem as primal problem.
Construct dual problem.Necessary and sufficient conditions.
Existence of a solution to the dual problem.
Construct candidate primal solution.
Verification.
Catherine Donnelly, Andrew Heunis Convex duality in constrained portfolio optimization
Market model
8/3/2019 Short Sales Constraint
http://slidepdf.com/reader/full/short-sales-constraint 16/42
The investor
Method of solution
Examples
Summary
Restate MVO problem as primal problem
Risk measure is minimized over portfolio processes.
infπ∈A
E(J (X π(T )))
Move this minimization to one over a space of processes.
infX ∈A
E(Φ(X ))
Key is the wealth equation.
Wealth processes X π embedded in space A.
Catherine Donnelly, Andrew Heunis Convex duality in constrained portfolio optimization
Market model
Th i
8/3/2019 Short Sales Constraint
http://slidepdf.com/reader/full/short-sales-constraint 17/42
The investor
Method of solution
Examples
Summary
Restate MVO problem as primal problem
Space A consists of square-integrable, continuous
processes.
If X ∈ A then a.s. X (0) = X 0 and
dX (
t ) = Υ
X
(t )
dt +
Λ
X (
t )
dW (
t ).
Catherine Donnelly, Andrew Heunis Convex duality in constrained portfolio optimization
Market modelTh i t
8/3/2019 Short Sales Constraint
http://slidepdf.com/reader/full/short-sales-constraint 18/42
The investor
Method of solution
Examples
Summary
Restate MVO problem as primal problem
Encode constraints as penalty functions l 0, l 1.
The initial wealth requirement X (0) = x 0 motivates
l 0(x ) := 0 if x = x 0
∞ otherwise,
for all x ∈ R.
The wealth equation and portfolio constraints motivate:
l 1(ω, t , x ,ν ,λ) :=
0 if ν = r (ω, t )x + λθ(ω, t )
and (σ(ω, t ))−1λ ∈ K
∞ otherwise,
for all (ω, t , x ,ν ,λ) ∈ Ω × [0,T ] × R×R× RN .
Catherine Donnelly, Andrew Heunis Convex duality in constrained portfolio optimization
Market modelThe investor
8/3/2019 Short Sales Constraint
http://slidepdf.com/reader/full/short-sales-constraint 19/42
The investor
Method of solution
Examples
Summary
Restate MVO problem as primal problem
Encode constraints as penalty functions l 0, l 1.
The initial wealth requirement X (0) = x 0 motivates
l 0(x ) := 0 if x = x 0
∞ otherwise,
for all x ∈ R.
The wealth equation and portfolio constraints motivate:
l 1(ω, t , x ,ν ,λ) :=
0 if ν = r (ω, t )x + λθ(ω, t )
and (σ(ω, t ))−1λ ∈ K
∞ otherwise,
for all (ω, t , x ,ν ,λ) ∈ Ω × [0,T ] × R×R× RN .
Catherine Donnelly, Andrew Heunis Convex duality in constrained portfolio optimization
Market modelThe investor
8/3/2019 Short Sales Constraint
http://slidepdf.com/reader/full/short-sales-constraint 20/42
The investor
Method of solution
Examples
Summary
Restate MVO problem as primal problem
Encode constraints as penalty functions l 0, l 1.
The initial wealth requirement X (0) = x 0 motivates
l 0(x ) := 0 if x = x 0
∞ otherwise,
for all x ∈ R.
The wealth equation and portfolio constraints motivate:
l 1(ω, t , x ,ν ,λ) :=
0 if ν = r (ω, t )x + λθ(ω, t )
and (σ(ω, t ))−1λ ∈ K
∞ otherwise,
for all (ω, t , x ,ν ,λ) ∈ Ω × [0,T ] × R×R× RN .
Catherine Donnelly, Andrew Heunis Convex duality in constrained portfolio optimization
Market modelThe investor
8/3/2019 Short Sales Constraint
http://slidepdf.com/reader/full/short-sales-constraint 21/42
The investor
Method of solution
Examples
Summary
Restate MVO problem as primal problem
Primal cost functional Φ : A → R ∪ ∞,
Φ(X ) := l 0(X 0)+E T
0 l 1(t ,X (t ), ΥX
(t ),ΛX
(t )) dt +E(J (X (T ))).
Primal problem: find X ∈ A such that
Φ(X ) = infX ∈AΦ(X ).
Use wealth equation to recover π.
Catherine Donnelly, Andrew Heunis Convex duality in constrained portfolio optimization
Market modelThe investor
8/3/2019 Short Sales Constraint
http://slidepdf.com/reader/full/short-sales-constraint 22/42
The investor
Method of solution
Examples
Summary
Restate MVO problem as primal problem
Primal cost functional Φ : A → R ∪ ∞,
Φ(X ) := l 0(X 0)+E T
0 l 1(t ,X (t ), ΥX
(t ),ΛX
(t )) dt +E(J (X (T ))).
Primal problem: find X ∈ A such that
Φ(X ) = infX ∈AΦ(X ).
Use wealth equation to recover π.
Catherine Donnelly, Andrew Heunis Convex duality in constrained portfolio optimization
Market modelThe investor
8/3/2019 Short Sales Constraint
http://slidepdf.com/reader/full/short-sales-constraint 23/42
The investor
Method of solution
Examples
Summary
Convex duality approach
Restate MVO problem as primal problem.
Construct dual problem.Necessary and sufficient conditions.
Existence of a solution to the dual problem.
Construct candidate primal solution.
Verification.
Catherine Donnelly, Andrew Heunis Convex duality in constrained portfolio optimization
Market modelThe investor
8/3/2019 Short Sales Constraint
http://slidepdf.com/reader/full/short-sales-constraint 24/42
Method of solution
Examples
Summary
Construct dual problem
Look for solutions to dual problem in the space B.
Space B consists of square-integrable, right-continuousprocesses.
If Y ∈ B then a.s. Y (0) = Y 0 and
dY (t ) = ΥY (t ) dt +
ΛY (t ) dW (t ) +
i = j
ΓY ij (t ) dM ij (t ).
Catherine Donnelly, Andrew Heunis Convex duality in constrained portfolio optimization
Market modelThe investor
8/3/2019 Short Sales Constraint
http://slidepdf.com/reader/full/short-sales-constraint 25/42
Method of solution
Examples
Summary
Construct dual problem
Take convex conjugates of l 0, l 1 and J .
m 0(y ) := supx ∈R
xy − l 0(x ), ∀y ∈ R.
Dual cost functional Ψ : B → R ∪ ∞,
Ψ(Y ) := m 0(Y 0) + E
T
0
m 1(t , Y (t ), ΥY (t ),ΛY (t )) dt
+ E(m J
(−Y (T ))).
Dual problem: find Y ∈ B such that
Ψ(Y ) = infY ∈B
Ψ(Y ).
Catherine Donnelly, Andrew Heunis Convex duality in constrained portfolio optimization
Market modelThe investor
8/3/2019 Short Sales Constraint
http://slidepdf.com/reader/full/short-sales-constraint 26/42
Method of solution
Examples
Summary
Construct dual problem
Take convex conjugates of l 0, l 1 and J .
m 0(y ) := supx ∈R
xy − l 0(x ), ∀y ∈ R.
Dual cost functional Ψ : B → R ∪ ∞,
Ψ(Y ) := m 0(Y 0) + E
T
0
m 1(t , Y (t ), ΥY (t ),ΛY (t )) dt
+ E(m J
(−Y (T ))).
Dual problem: find Y ∈ B such that
Ψ(Y ) = infY ∈B
Ψ(Y ).
Catherine Donnelly, Andrew Heunis Convex duality in constrained portfolio optimization
Market modelThe investor
8/3/2019 Short Sales Constraint
http://slidepdf.com/reader/full/short-sales-constraint 27/42
Method of solution
Examples
Summary
Construct dual problem
Take convex conjugates of l 0, l 1 and J .
m 0(y ) := supx ∈R
xy − l 0(x ), ∀y ∈ R.
Dual cost functional Ψ : B → R ∪ ∞,
Ψ(Y ) := m 0(Y 0) + E
T
0
m 1(t , Y (t ), ΥY (t ),ΛY (t )) dt
+ E(m J
(−Y (T ))).
Dual problem: find Y ∈ B such that
Ψ(Y ) = infY ∈B
Ψ(Y ).
Catherine Donnelly, Andrew Heunis Convex duality in constrained portfolio optimization
Market modelThe investor
M h d f l i
8/3/2019 Short Sales Constraint
http://slidepdf.com/reader/full/short-sales-constraint 28/42
Method of solution
Examples
Summary
Convex duality approach
Restate MVO problem as primal problem.
Construct dual problem.Necessary and sufficient conditions.
Existence of a solution to the dual problem.
Construct candidate primal solution.
Verification.
Catherine Donnelly, Andrew Heunis Convex duality in constrained portfolio optimization
Market modelThe investor
M th d f l ti
8/3/2019 Short Sales Constraint
http://slidepdf.com/reader/full/short-sales-constraint 29/42
Method of solution
Examples
Summary
Necessary and sufficient conditions
For (X , Y ) ∈ A× B,
X ∈ A solves the primal problem and Y ∈ B solves the dualproblem
if and only if
(X , Y ) ∈ A× B satisfy necessary and sufficient conditions, eg
X (T ) = −Y (T ) + B
A, a.s.,
ΥY (t ) = −r (t )Y (t ), a.e.
Catherine Donnelly, Andrew Heunis Convex duality in constrained portfolio optimization
Market modelThe investor
Method of solution
8/3/2019 Short Sales Constraint
http://slidepdf.com/reader/full/short-sales-constraint 30/42
Method of solution
Examples
Summary
Convex duality approach
Restate MVO problem as primal problem.
Construct dual problem.Necessary and sufficient conditions.
Existence of a solution to the dual problem.
Construct candidate primal solution.
Verification.
Catherine Donnelly, Andrew Heunis Convex duality in constrained portfolio optimization
Market modelThe investor
Method of solution
8/3/2019 Short Sales Constraint
http://slidepdf.com/reader/full/short-sales-constraint 31/42
Method of solution
Examples
Summary
Existence of a solution to the dual problem
There exists Y ∈ B such that
Ψ(Y ) = infY ∈B
Ψ(Y ).
Catherine Donnelly, Andrew Heunis Convex duality in constrained portfolio optimization
Market modelThe investor
Method of solution
8/3/2019 Short Sales Constraint
http://slidepdf.com/reader/full/short-sales-constraint 32/42
Method of solution
Examples
Summary
Convex duality approach
Restate MVO problem as primal problem.
Construct dual problem.Necessary and sufficient conditions.
Existence of a solution to the dual problem.
Construct candidate primal solution.
Verification.
Catherine Donnelly, Andrew Heunis Convex duality in constrained portfolio optimization
Market modelThe investor
Method of solution
8/3/2019 Short Sales Constraint
http://slidepdf.com/reader/full/short-sales-constraint 33/42
Method of solution
Examples
Summary
Construct candidate primal solution
State price density process
H (t ) = exp−
t
0r (s ) ds E (−θ • W )(t )
X πππ(t )H (t ) = E (X π
ππ(T )H (T ) | F t )
From necessary and sufficient conditions,
X (T ) = −Y (T ) + B
A.
Candidate primal solution X ∈ B
X (t ) := −1
H (t )E
Y (T ) + B
A
H (T )
F t
.
Catherine Donnelly, Andrew Heunis Convex duality in constrained portfolio optimization
Market modelThe investor
Method of solution
8/3/2019 Short Sales Constraint
http://slidepdf.com/reader/full/short-sales-constraint 34/42
Method of solution
Examples
Summary
Construct candidate primal solution
State price density process
H (t ) = exp−
t
0r (s ) ds E (−θ • W )(t )
X πππ(t )H (t ) = E (X π
ππ(T )H (T ) | F t )
From necessary and sufficient conditions,
X (T ) = −Y (T ) + B
A.
Candidate primal solution X ∈ B
X (t ) := −1
H (t )E
Y (T ) + B
A
H (T )
F t
.
Catherine Donnelly, Andrew Heunis Convex duality in constrained portfolio optimization
Market modelThe investor
Method of solution
8/3/2019 Short Sales Constraint
http://slidepdf.com/reader/full/short-sales-constraint 35/42
Examples
Summary
Construct candidate primal solution
State price density process
H (t ) = exp−
t
0r (s ) ds E (−θ • W )(t )
X πππ
(t )H (t ) = E (X πππ
(T )H (T ) | F t )From necessary and sufficient conditions,
X (T ) = −Y (T ) + B
A.
Candidate primal solution X ∈ B
X (t ) := −1
H (t )E
Y (T ) + B
A
H (T )
F t
.
Catherine Donnelly, Andrew Heunis Convex duality in constrained portfolio optimization
Market modelThe investor
Method of solution
8/3/2019 Short Sales Constraint
http://slidepdf.com/reader/full/short-sales-constraint 36/42
Examples
Summary
Convex duality approach
Restate MVO problem as primal problem.
Construct dual problem.Necessary and sufficient conditions.
Existence of a solution to the dual problem.
Construct candidate primal solution.
Verification.
Catherine Donnelly, Andrew Heunis Convex duality in constrained portfolio optimization
Market modelThe investor
Method of solution
8/3/2019 Short Sales Constraint
http://slidepdf.com/reader/full/short-sales-constraint 37/42
Examples
Summary
Examples
Market parameters F αt -previsible.
J (x ) = (x − d )2, some d ∈ R.
Use necessary and sufficient conditions.
Catherine Donnelly, Andrew Heunis Convex duality in constrained portfolio optimization
Market modelThe investor
Method of solution
8/3/2019 Short Sales Constraint
http://slidepdf.com/reader/full/short-sales-constraint 38/42
Examples
Summary
No portfolio constraints
K := RN
π(t ) = −X π(t ) − d R (t )
S (t )σ(t )
−1θ(t ),
for
R (t ) = E
exp
T
t r (u ) − θ(u )2
du
α(t )
,
S (t ) = E
exp
T
t
2r (u ) − θ(u )2
du
α(t )
.
Catherine Donnelly, Andrew Heunis Convex duality in constrained portfolio optimization
Market modelThe investor
Method of solution
8/3/2019 Short Sales Constraint
http://slidepdf.com/reader/full/short-sales-constraint 39/42
Examples
Summary
Restricted investment portfolio constraints
Example: K = π = (π1, . . . , πN ) ∈ RN : π1 = 0, . . . , πM = 0.
π(t ) = −
X π(t ) − d
R (t )
S (t )
σ(t )
−1ξ(t ),
forξ(t ) = θ(t ) − proj
θ(t )
σ−1(t )K
,
R (t ) = Eexp T
t r (u ) − θ(u )ξ(u ) du α(t ) ,
S (t ) = E
exp
T
t
2r (u ) − θ(u )ξ(u )
du
α(t )
Catherine Donnelly, Andrew Heunis Convex duality in constrained portfolio optimization
Market modelThe investor
Method of solution
E l
8/3/2019 Short Sales Constraint
http://slidepdf.com/reader/full/short-sales-constraint 40/42
Examples
Summary
Convex conic portfolio constraints
K closed convex cone containing the origin.
Further assume: r deterministic and x 0 ≤ d exp− T
0 r (u ) du .
π(t ) = −
X π(t ) − d exp−
T
t
r (u ) du
σ(t )
−1ξ(t ),
forξ(t ) = θ(t ) − proj
θ(t )
σ−1(t )K
.
Catherine Donnelly, Andrew Heunis Convex duality in constrained portfolio optimization
Market modelThe investor
Method of solution
Examples
8/3/2019 Short Sales Constraint
http://slidepdf.com/reader/full/short-sales-constraint 41/42
Examples
Summary
Summary
Showed existence and characterized the solution for MVO
problem with
general convex portfolio constraints; and
random market coefficients
in a regime-switching model.
Solutions in feedback form.
Catherine Donnelly, Andrew Heunis Convex duality in constrained portfolio optimization
Market modelThe investor
Method of solution
Examples
8/3/2019 Short Sales Constraint
http://slidepdf.com/reader/full/short-sales-constraint 42/42
Examples
Summary
Convex duality in constrained mean-variance
portfolio optimization under a regime-switching
model
Catherine Donnelly1 Andrew Heunis2
1ETH Zurich, Switzerland
2University of Waterloo, Canada
26 June 2010
Catherine Donnelly, Andrew Heunis Convex duality in constrained portfolio optimization