Proceedings of the 29th Conference
on Decision and Control
Honolulu, Hawall December 1990
TP-11 = 5:20
Investment-Consumption Models w i t h T r a n s a c t i o n C o s t s and Markov-Chain
Parameters
T h a l e i a Zariphopoulou
Department of M a t h e m a t i c a l S c i e n c e s
Worcester P o l y t e c h n i c I n s t i t u t e
W o r c e s t e r , MA 01609
Abstract
T h i s paper c o n s i d e r s an i n f i n i t e h o r i z o n i n v e s t ment-consumption model i n which a s i n g l e a g e n t consumes
and d i s t r i b u t e s h i s w e a l t h i n two a s s e t s , a bond and a
s t o c k . The problem of m a x i m i z a t i o n o f t h e t o t a l
u t i l i t y from consumption i s t r e a t e d .
S t a t e (amount
a l l o c a t e d i n a s s e t s ) and c o n t r o l ( c o n s u m p t i o n , r a t e s o f
It i s shown t h a t t h e
trading) constraints are present.
v a l u e f u n c t i o n i s t h e u n i q u e v i s c o s i t y s o l u t i o n of a
s y s t e m o f v a r i a t i o n a l i n e q u a l i t i e s w i t h g r a d i e n t constraints.
1. INTRODUCTION
I n t h i s p a p e r we examine a g e n e r a l i n v e s t m e n t and
consumption d e c i s i o n problem f o r a s i n g l e a g e n t . The
i n v e s t o r consumes a t a n o n n e g a t i v e r a t e and h e d i s t r i b u t e s h i s c u r r e n t w e a l t h between two a s s e t s c o n t i n u o u s l y
i n time.
One a s s e t i s a bond, i . e . a r i s k l e s s s e c u r i t y
w i t h i n s t a n t a n e o u s r a t e o f r e t u r n r. The o t h e r a s s e t
i s a s t o c k , which r a t e o f r e t u r n
is a continuous
zt
t i m e Markov c h a i n . I n o u r v e r s i o n o f t h e model t h e
i n v e s t o r cannot borrow money t o f i n a n c e h i s i n v e s t m e n t
i n bond and h e cannot s h o r t - s e l l t h e s t o c k . I n o t h e r
words, t h e amount o f money a l l o c a t e d i n bond and s t o c k
must s t a y n o n n e g a t i v e .
When t h e i n v e s t o r makes a t r a n s a c t i o n , h e p a y s
t r a n s a c t i o n f e e s which a r e assumed t o b e p r o p o r t i o n a l
t o t h e amount t r a n s a c t e d .
The c o n t r o l o b j e c t i v e i s t o
maximize, i n a n i n f i n i t e h o r i z o n , t h e e x p e c t e d d i s c o u n t e d u t i l i t y which comes o n l y from consumption.
Due
t o t h e presence of t h e t r a n s a c t i o n f e e s , t h i s is a
s i n g u l a r c o n t r o l problem.
11.
The F i n a n c i a l Model w i t h T r a n s a c t i o n F e e s
dyt
where
0.
r
s a l e s o f stock r e s p e c t i v e l y .
u ( z s : O ~ s i t ) and
of the stock s a t i s f i e s
= z(t)Ptdt
(2)
P
0
=
p.
z i s a f i n i t e s t a t e continuous
time Markov chain d e f i n e d on some u n d e r l y i n g p r o b a b i l -
The r a t e of r e t u r n
i t y space
(Q,F,P)
w i t h jumping r a t e
qzz,
from
s t a t e z t o s t a t e z’.
The s t a t e s p a c e i s d e n o t e d by
2.
The a s s o c i a t e d g e n e r a t o r L o f t h e Markov c h a i n
h a s t h e form
L e t K = maxz.
A n a t u r a l assumption is
K
5 r.
The
xt
and
yt,
i n v e s t e d a t time
t
.
The c o n t r o l s
(ii)
Mt,Nt
Ct
50
are
a.e.
in
=
F
t 20.
F -measurable,
r i g h t continuous
t 2 0 , where x
t’Yt
a r e t h e t r a j e c t o r i e s g i v e n by t h e s t a t e e q u a t i o n ( 3 )
u s i n g t h e c o n t r o l s (Ct,Mt,Nt).
We d e n o t e by A t h e s e t of a d m i s s i b l e c o n t r o l s .
The t o t a l e x p e c t e d d i s c o u n t e d u t i l i t y J coming
from consumption i s g i v e n by
r-m
J(x,y,z,C,M.N)
=
E
1
e-’tU(Ct)dt
’0
w i t h (C,M,N) E A and z ( 0 ) = z , where t h e u t i l i t y
f u n c t i o n U: [O,+-)+[O,+-)
i s assume t o h a v e t h e
following properties:
i s s t r i c t l y i n c r e a s i n g , bounded, c o n c a v e , C1
=
0 , limU’(c)=
4
+ m ,
l i m U’(c)
C++m
=
0.
The d i s c o u n t f a c t o r B > 0 w e i g h t s consumption now
v e r s u s consumption l a t e r , l a r g e 5 d e n o t i n g i n s t a n t
gratification.
Note t h a t t h e c o n t r o l s M and N a r e
a c t i n g i m p l i c i t l y through the constraint ( i i i ) .
The v a l u e f u n c t i o n U i s g i v e n by
ie-’U(Ct)dt.
u ( x , y , z ) = sup E
A
j0
/
Our g o a l i s t o d e r i v e t h e Bellman e q u a t i o n
a s s o c i a t e d w i t h t h i s s i n g u l a r c o n t r o l problem and t o
c h a r a c t e r i z e u a s i t s unique s o l u t i o n .
It t u r n s
o u t t h a t t h e Bellman e q u a t i o n h e r e i s a s y s t e m o f
variational inequalities.
T r a n s a c t i o n c o s t s are a n e s s e n t i a l f e a t u r e o f
some economic t h e o r i e s . I n [ Z ] , [3] C o n s t a n t i n i d e s
assumes t h a t t h e t r a n s a c t i o n c o s t s d e p l e t e o n l y t h e
r i s k l e s s a s s e t and t h a t t h e s t o c k p r i c e i s a l o g a r i t h m i c Brownian m o t i o n . H e shows t h a t i f a n o p t i m a l
p o l i c y e x i s t s , i t i s c h a r a c t e r i z e d by two r e f l e c t i n g
with X 5 7,s u c h t h a t t h e i n v e s t o r
barriers
lies in
does n o t t r a d e a s l o n g a s t h e r a t i o y t / x
[&,TI
and t r a n s a c t s t o t h e c l o s e s t boundary of t h e
r e g i o n of no t r a n s a c t i o n s
[&,XI,
whenever t h i s r a t i o
C o n s t a n t i n i d e s ’ s work
l i e s outside t h i s interval.
was g e n e r a l i z e d by D a v i s and Norman [51
bond and s t o c k r e s p e c t i v e l y , a r e t h e s t a t e v a r i a b l e s
and t h e y e v o l v e ( s e e 1171) a c c o r d i n g t o t h e e q u a t i o n s
.
CH2917-3/90/0000-2354$1.OO @ 1990 IEEE
(Ct,Mt,Nt)
l,x
ZEZ
amount o f w e a l t h
(3)
and n o n d e c r e a s i n g p r o c e s s e s .
( i i i ) xt L 0, yt
0 a.e.
C
= PO’
dP
dNt
are ahissible i f :
( i ) C t i s F - m e a s u r a b l e where
U(0)
P
-
f u n c t i o n and
= rP:dt
The p r i c e
dMt
The numbers A and LI r e p r e s e n t t h e p r o p o r t i o n a l
t r a n s a c t i o n f e e s ; t h e y a r e assumed t o b e n o n n e g a t i v e
and one o f them must a l w a y s b e p o s i t i v e .
For s i m p l i c i t y w e assume h e r e t h a t a l l f i n a n c i a l c h a r g e s a r e p a i d
from t h e h o l d i n g s i n bond.
The i n v e s t o r c a n n o t borrow
money o r s h o r t s e l l t h e s t o c k . The c o n t r o l p r o c e s s e s
a r e t h e consumption r a t e Ct
and t h e p r o c e s s e s
Mt
and N t which r e p r e s e n t t h e cwmctative purchases and
U
P;
+
xo = x , y o = y , z ( O ) = z
We c o n s i d e r a market w i t h two a s s e t s : a bond and
a s t o c k . The p r i c e Po of t h e bond i s g i v e n by
dP:
z(t)ytdt
=
2354
D i f f e r e n t c r i t e r i a were used by T a k s a r , Klass and
Assaf I161 and, u n d e r more g e n e r a l a s s u m p t i o n s by
Fleming, Grossman, V i l a and Z a r i p h o p o u l o u [ 6 ] .
S i n g l e - p e r i o d models w i t h f i x e d t r a n s a c t i o n c o s t s
a r e d i s c u s s e d i n L e l a n d [ I l l , Mukherjee and Z a b e l [141,
Brennan [l] , Goldsmith [ 8 ] , Levy [12] and Mayshar 1131.
F i n a l l y , Kandel and Ross [lo] i n t r o d u c e q u a s i - f i x e d
transaction costs.
We now - c n s i d e r a s i m i l a r c o n t r o l problem i n which
the c o n t r o l s , which r e p r e s e n t t h e r a t e s of t r a d i n g , a r e
assumed t o b e a b s o l u t e l y c o n t i n u o u s p r o c e s s e s .
More
p r e c i s e l y , w e c o n s i d e r a market which o f f e r s a bond and
a stock with p r i c e s evolving according t o equations
(1) and ( 2 ) r e s p e c t i v e l y . The s t a t e v a r i a b l e s x t and
yt,
which are t h e amount of money i n v e s t e d i n bond and
Q,
i i ) U i s a v i s c o s i t y s u p e r s o l u t i o n of (5) i n
i . e . f o r each Z E Z
IV. Results
I n t h e s e q u e l , we c h a r a c t e r i z e t h e v a l u e f u n c t i o n
U
as t h e u n i q u e c o n s t r a i n e d v i s c o s i t y s o l u t i o n of t h e
a s s o c i a t e d Bellman e q u a t i o n .
Some r e s u l t s a b o u t
uL a r e f i r s t s t a t e d .
Theorem 3 . 1 :
The v a l u e f u n c t i o n
v i s c o s i t y s o l u t i o n of
is a constrained
uL
s t o c k , obey t h e s t a t e e q u a t i o n s
= ( r x -2
dx
t
t
) d t - (l+h)m d t
+ mtdt
dy, = z ( t ) y d t
x
0
= xty0 = y,z(O)
=
+
(l-u)ntdt
- ntdt
z
(4)
.
(XrY)EEr ZEZ
The c o n t r o l s of t h e i n v e s t o r a r e t h e consumption rate
Ct and t h e rates of trading mt and n t .
The s e t o f
4.
admissible c o n t r o l s
such t h a t
( i ) Ct
is
C
(ii)
c o n s i s t s of c o n t r o l s
F
F -measurable where
.O
t-
are
o(z
:OLsLt),
tFO.
a.e.
mt,nt
=
(c,m,n)
F -measurable r i g h t c o n t i n u o u s and
nonnegative processes.
Ozmt,ntLL a . e .
t 2 0 f o r some p o s i t i v e c o n s t a n t
(iii)
L.
(iv)
xg0, ytO
a.e.
to,
(7)
where
xtyt
are the
s o l u t i o n s of ( 4 ) u s i n g t h e c o n t r o l s (C,m,n).
The c o n t r o l o b j e c t i v e i s t o maximize t h e e x p e c t e d
d i s c o u n t e d u t i l i t y from consumption o v e r t h e s e t o f
admissible controls.
For e a c h f i x e d L
0, t h e v a l u e
f u n c t i o n i s g i v e n by
UL ( x , y , z )
The p r o o f f o l l o w s a l o n g t h e r e s u l t s of Fleming, S e t h i ,
and Soner [ 7 ] .
It i s e s s e n t i a l l y b a s e d on t h e dynamic
programming p r i n c i p l e and Dynkin’s f o r m u l a .
Theorem 3.2:
The v a l u e f u n c t i o n uL i s t h e u n i q u e
c o n s t r a i n e d v i s c o s i t y s o l u t i o n o f (6) i n t h e c l a s s of
bounded and u n i f o r m l y c o n t i n u o u s f u n c t i o n s .
P r o o f : W e show t h a t if U and v -are r e s p e c t i v e l y
__
a v i s c o s i t y s u b s o l u t i o n o f (6) on fi and a - v i s c o s i t y
s u p e r s o l u t i o n of (6) i n Q , t h e n u<v on 61- We a r g u e
by c o n t r a d i c t i o n , i . e . w e assume t h a t
e
which i m p l i e s t h a t f o r s u f f i c i e n t l y small
max s u p [ u ( x , z )
-
V(X,Z) -
2
elx[ 1
> 0
> 0.
(8)
Z E Z XES2
We can f i n d p o i n t s
= sup E j r e - ’ t L J ( C t ) d t ,
Z ~ E Z and
FEE
such t h a t
4.
where U i s t h e u s u a l u t i l i t y f u n c t i o n and
the discount f a c t o r .
111.
B > 0
is
I n t h e s e q u e l we o m i t z
consider the auxiliary functioi
. Next,
for
$:Ex61 +lR
> 0 we
g i v e n by
E
Preliminaries
P r o p o s i t i o n 2.1:
The v a l u e f u n c t i o n s U and uL are
i n c r e a s i n g , concave and u n i f o r m l y c o n t i n u o u s f u n c t i o n s
on fi = [0,+=1 x io,+).
I n t h e s e q u e l w e w i l l need t h e f o l l o w i n g d e f i n i tion:
D e f i n i t i o n 2.1: We c o n s i d e r a n o n l i n e a r p a r t i a l d i f f e r e n t i a l e q u a t i o n o f t h e form
We show t h a t i f i t s maximum i s a c h i e v e d a t (X Y ) t h e n
0’ 0
Y E Q and
0
\Yo
- xol 5 R E .
(9)
W e now c o n s i d e r t h e f u n c t i o n s
F(X,z,u(X,z),
where
z EZ,
(5)
Du(X,z)) = 0
X = (x,y)
au(x,z)) and
with
(x,y)
E
5, Du(x,z)
i s continuous,
ax
ay
f o r each Z E Z . A c o n t i n u o u s f u n c t i o n u:$xZ+R
is a
c o n s t r a i n e d v i s c o s i t y s o l u t i o n of (5) i f
i ) U i s a v i s c o s i t y s u b s o l u t i o n of ( 5 ) on 5 ,
i.e. f o r each ZEZ
0
P(X,z,u(X,z),r) I
= (au(x,z),
X = (x,Y)E;
where
and
F:zxZxRxR2 + R
r~D;~,~)u(X,z)
We o b s e r v e t h a t
U
- 4 h a s a maximum a t
X O ~ c and
v - $ h a s a minimum a t Y E R . Applying t h e d e f i n i t i o n
o f v i s c o s i t y s o l u t i o n a n d o u s i n g (9) w e g e t
2 2
2
CL
B[U(X,z)-v(X,z)-O/X( 1 <BO,
XE;
and
zcz.
Sending 0 + 0 , c o n t r a d i c t s ( 8 ) .
W e now s t a t e t h e mafn t h e o r e m s .
Theorem 3.3:
The v a l u e f u n c t i o n
v i s c o s i t y s o l u t i o n of
2355
U
is a constrained
m i n [ ( l + X ) u -uy,
- max
-(l-u)u
Ou-rxu -zyu
+uy,
-
[-cu +U(c)l
X
Cu(z)l
where
Y
(10)
0
=
c>o
The p r o o f i s b a s e d on t h e Dynamic Programming
P r i n c i p l e and t h e g e n e r a l i z e d D y n k i n ' s f o r m u l a .
The
p r e s e n c e o f s i n g u l a r c o n t r o l s and t h e f a c t t h a t t h e
Bellman e q u a t i o n i s a c t u a l l y a V a r i a t i o n a l I n e q u a l i t y
make t h e p r o o f r a t h e r t e c h n i c a l .
Theorem 3.4:
The v a l u e f u n c t i o n U i s t h e u n i q u e
c o n s t r a i n e d v i s c o s i t y s o l u t i o n of (10) i n t h e c l a s s o f
bounded and u n i f o r m l y c o n t i n u o u s f u n c t i o n s .
U are
P r o o f : W e a r e g o i n g t o show t h a t i f U - a n d
r e s p e c t i v e l y a s u b s o l u t i o n o f (10) on R Cnd a s u p e r s o l u t i o n o f (10) i n $2, t h e n U 5
on R . W e follow
t h e s t r a t e g y of I s h i i [ 9 ] . L e t $ : S I + R b e d e f i n e d by
$ ( x , y ) = C1x + C2y + k , where C1, C2 and k a r e
(l-v)Cl< C2
positive constants satisfying
Bk > r C l
and
X
= (x,y)
E
+
+ max
KC
+ U(c)].
[-cC1
crp
R, P = (p,q)eRXR
and
2
-(l-p)p+q,
- max [-cp+U(c)l
-
U,
=
-
Ue
M
+lR
W
e can f i n d p o i n t s
,Z
0
,U (Y z 1 , P E ) .
a
(Po
6U - r x o p E - zoyoqE
a
H(Y
o ,z o 'Ue (Y o ,z o ) , P E )
- max[-cp + U(c)]-fU (Y z )
e 0' o
c,'
E
If
Cl
> 0.
Sending
BJ.0
In
and u s i n g (13) we
5+1.
C o n s t a n t i n i d e s , G.M., M u l t i p e r i o d Consumption and
I n v e s t m e n t B e h a v i o r w i t h Convex T r a n s a c t i o n s Costs,
Management S c i . 25 ( 1 9 7 9 ) , 1127-1137.
3.
C o n s t a n t i n i d e s , G . M . , C a p i t a l Market E q u i l i b r i u m
w i t h T r a n s a c t i o n C o s t s , J o u r n a l of P o l i t i c a l
Economy 9 4 , No. 4 ( 1 9 8 6 ) , 842-862.
4.
C r a n d a l l , M.G. and P.-L. L i o n s , V i s c o s i t y s o l u t i o n s o f H a m i l t o n - J a c o b i e q u a t i o n s , T r a n s . AMs
277 ( 1 9 8 3 ) , 1-42.
5.
D a v i s , M.H.A. and A.R. Norman, P o r t f o l i o s e l e c t i o n
w i t h t r a n s a c t i o n c o s t s , s u b m i t t e d t o Math of
Operations Research, (1987).
6.
Fleming, W.H., S. Grossman, J.-L. V i l a and T.
Zariphopoulou, Optimal P o r t f o l i o r e b a l a n c i n g w i t h
t r a n s a c t i o n c o s t s , s u b m i t t e d t o Econometrica.
such t h a t
2
-eIx/ 1
and
z EZ
0
We now l o o k a t
2.
Bu-rxp-zyq-
>.U.
e
XEQ
PE = (p,,q,).
Brennan, M . J . , The o p t i m a l number o f S e c u r i t i e s i n
a Risky A s s e t P o r t f o l i o when t h e r e a r e f i x e d C o s t s
o f T r a n s a c t i n g : Theory and some e m p i r i c a l r e s u l t s ,
J. F i n a n c i a l and Q u a n t i t a t i v e A n a l y s i s 1 0 , ( 1 9 7 5 ) ,
483-496.
given
X E ~ Z,E Z .
0
max s u p [ u ( ~ , z ) - U (x,z)
ZEZ
and
4(1,1)).
1.
We work a s i n Theorem 3 . 1 ,
we assume t h a t f o r s u f f i c i e n t l y s m a l l R > 0 ,
W
e n e x t show t h a t
-
E
References
O ~ ( 0 , l ) . Then t h e r e
for
e x i s t s a p o s i t i v e constant
>
-x
-
(1-O)O
H(X,z,U,,VUe) Z M ( 1 - 9 )
(xo,y0)
=
f o r some
- Lu(z)l.
+
@U
Y
E
we work a s i n Theorem 3.2 and w e c o n t r a d i c t ( 3 . 6 ) .
t h e o t h e r c a s e s (14) y i e l d s
C O
Let
0
=
by
H(X,z,v,P)=min[(l+h)p-q,
-
d i f f e r e n t c a s e s d e p e n d i n g on t h e form o f
H(Y
Let
H:?&X%xRxR
Y0
Let
=
E
a g a i n c o n t r a d i c t (11). F i n a l l y we s e n d
(l+X)C,
c
P
0.
(11)
__
X ~ s 2 such t h a t
F l e m i n g , W.H., S e t h i , S. and M. H . S o n e r , An o p t i m a l s t o c h a s t i c p l a n n i n g problem w i t h randomly
f l u c t u a t i n g demand, SIAM J. C o n t r o l and
O p t i m i z a t i o n 25, No. 6 ( 1 9 8 7 ) .
G o l d s m i t h , D . , T r a n s a c t i o n C o s t s and t h e Theory o f
P o r t f o l i o S e l e c t i o n , J . F i n a n c e 3 1 ( 1 9 7 6 ) , 1127 1139.
I s h i i , H., A s i m p l e , d i r e c t proof of uniqueness
f o r s o l u t i o n s of t h e H a m i l t o n - J a c o b i e q u a t i o n s o f
E i k o n a l t y p e , P r o c e e d i n g s o f t h e American Mathem a t i c a l S o c i e t y , 1 0 0 , n o . 2 ( 1 9 8 7 ) , 247-251.
10.
W e now c o n s i d e r t h e f u n c t i o n s
Y-x
-
$(x)
Y
=
ue(YO) -
We o b s e r v e t h a t
U,
- IJJ
0
I
$U)= NX,) -
U
1%
-
h a s a minimum a t
property we get
F-
-x
4(1,1)1
2
-oIxo/ 2
- 4(1,1)1 2 + AIX/ 2
h a s a maximum a t
X0
and
Yo. U s i n g t h e v i s c o s i t y
Kandel, S . and S.A. Ross, Some I n t e r t e m p o r a l Models
of Portfolio Selection with Transaction Costs,
Working P a p e r no. 1 0 7 , U n i v e r s i t y o f Chicago, Grad.
S c h o o l B u s . , C e n t e r R e s . S e c u r i t y P r i c e s , 1983.
11. L e l a n d , H . E . , On Consumption and P o r t f o l i o c h o i c e s
w i t h T r a n s a c t i o n C o s t s , i n E s s a y s on Economic
B e h a v i o r u n d e r U n c e r t a i n t y , M. B a l c h , D. McFadden,
S . Wu e d s , Amsterdam, North-Holland ( 1 9 7 4 ) .
12.
Levy, H . , E q u i l i b r i u m i n an I m p e r f e c t Market: A
C o n s t r a i n t on t h e number of S e c u r i t i e s i n t h e P o r t -
13.
Mayshar, J., T r a n s a c t i o n C o s t s i n a Model of
C a p i t a l Market E q u i l i b r i u m , J.P.E. 87 (1979),
673-700.
14.
Mukherjee, R. and E. Z a b e l , Consumption and P o r t f o l i o choices with Transaction Costs, i n Essays
on Economic Behavior u n d e r U n c e r t a i n t y , M. Balch,
D. McFadden, S . Wu e d s , Amsterdam, North-Holland
(1974).
15.
Soner, M.H.,
1984.
16.
T a k s a r , M . , Klass, M.J. and D. A s s a f , A. D i f f u s i o n Model f o r Optimal P o r t f o l i o S e l e c t i o n i n t h e
p r e s e n c e of Brokerage F e e s , Dept. o f S t a t i s t i c s ,
F l o r i d a S t a t e U n i v e r s i t y (1986).
17.
Zariphopoulou,
1988.
Ph.D.
T.,
T h e s i s , Brown U n i v e r s i t y ,
Ph.D.
T h e s i s , Brown U n i v e r s i t y ,
2357