Machine Learning and Finance III
-
Upload
tradercat-solaris -
Category
Documents
-
view
217 -
download
0
Transcript of Machine Learning and Finance III
-
8/9/2019 Machine Learning and Finance III
1/31
M a c h i n e l e a r n i n g a n d p o r t f o l i o s e l e c t i o n s .
L s z l G y r
D e p a r t m e n t o f C o m p u t e r S c i e n c e a n d I n f o r m a t i o n T h e o r y
B u d a p e s t U n i v e r s i t y o f T e c h n o l o g y a n d E c o n o m i c s
g y o r f i @ s z i t . b m e . h u
A b s t r a c t
I n t h i s p a p e r t h e s t a t i c p o r t f o l i o a n d t h e d y n a m i c l o g - o p t i m a l p o r t -
f o l i o f o r m e m o r y l e s s a n d f o r s t a t i o n a r y m a r k e t p r o c e s s e s i s s t u d i e d .
W e i l l u s t r a t e t h e i r p e r f o r m a n c e ( a v e r a g e a n n u a l y i e l d ) f o r N Y S E
d a t a .
1 N o t a t i o n s .
g o n d m k o n n o d F o l o n o m k n
m n d n o o sI
; sP
; : : : PR
d
C
D
sn
a @ s
@ I A
n
; : : : ; s
@ d A
n
A
j E o m o n n s
@ j A
n
o sn
d n o o j E
o n n E d n o d F s n o d o n o m l z D s
@ j A
H
a I F f sn
o n n l n d X
s
@ j A
n
a e
n W
( j )
n
% e
n W
( j )
;
o @ l d A
W
@ j A
n
X a
I
n
l n s
@ j A
n
n d m o o
W
@ j A
X a l i m
n 3 I
I
n
l n s
@ j A
n
:
I
-
8/9/2019 Machine Learning and Finance III
2/31
s n o d o l l d o n n o m n l
o n o n o m n o f sn
n o m o o l
o n n o n o f xn
o l l o X
xn
a @ x
@ I A
n
; : : : ; x
@ d A
n
A
x
@ j A
n
a
s
@ j A
n
s
@ j A
n I
:
D j E o m o n n x
@ j A
n
o n o xn
d n o m o n
o n d n n n l n j E o n n E d n
o d F
2 S t a t i c p o r t f o l i o .
o o l o n l o d n m n F e o o l o
o d n o d b a @ b @ I A ; : : : b @ d A A F j E o m o n n b @ j A o b
d n o o o o n o n o 9 l n d n j F
m o o l o o b n o n n o m o n n m o
I F o o o l o o d n o d
d
a
8
-
8/9/2019 Machine Learning and Finance III
3/31
s b
@ j A
> H o l l j a I ; : : : ; d n o n d m l
W X a l i m
n 3 I
I
n
l n S
n
a l i m
n 3 I
m x
j
I
n
l n s
@ j A
n
a m x
j
W
@ j A
:
D n o o l o m m l o m x
j
W
@ j A
F
3 D y n a m i c p o r t f o l i o s e l e c t i o n : s p e c i a l c a s e
y n n n o o l o n n n m n F
o n n l l n d o o l o @ g A l d n m o o l o l E
o n F d n m o o l o l o n m l E o d n m n E
D n n n o d n o d n l
m o n F o n o g n o d d n d d d
u l l P R D v n P S D f m n U D w k o z P V D p n k l n n d E
l I Q D w Q H D w n d z k l Q Q n d f o n n d g o R F l o
g T n d I S n g o n d o m I P D n d g I S n v n
P T F
p o o l o o b
P
d
D F F D o n n d o l
n o o n o n m n o d o n n o o o l o D
n o o l o d n o d F
v S
H
d n o n o 9 n l l F n n n n o
( d n o d S
H
b
@ j A
n d n o j D n d l n n
S
H
b
@ j A
x
@ j A
I
D o n d o ( d n o d n o 9
l o m
S
I
a S
H
d
X
j a I
b
@ j A
x
@ j A
I
a S
H
h b ; xI
i ;
h ; i d n o n n o d F p o o n d d n o d D S
I
n n l l
S
P
a S
I
h b ; xP
i a S
H
h b ; xI
i h b ; xP
i :
f n d o n D o d n o d n n l l S
n I
D o
S
n
a S
n I
h b ; xn
i a S
H
n
Y
i a I
h b ; xi
i :
Q
-
8/9/2019 Machine Learning and Finance III
4/31
m o o o o o l o l o n
l i m
n 3 I
I
n
l n S
n
a l i m
n 3 I
I
n
l n S
H
C
I
n
n
X
i a I
l n h b ; xi
i
!
a l i m
n 3 I
I
n
n
X
i a I
l n h b ; xi
i ;
o o l o o n l o n n m n l
n l l S
H
a I F
4 L o g - o p t i m a l p o r t f o l i o f o r m e m o r y l e s s m a r k e t
p r o c e s s .
s m k o fX
i
m m o l D F F D n o n d E
n d n n d d n l l d d @ F F d F A n d o m n o n
o o n n l l n d o o l o @ f g A l o E
o m l o o l o X
b X a r g m x b
E
f l n h b ; XI
i :
o m l m n S
n
a S
n
@ b A d n o l d
n d l o E o m m o o l o b D n o n o o l o
b ( n E f @ l n h b ; XI
i A
P
n d l S
n
a S
n
@ bA n d
o n m m o l m k o f Xn
I
I
D
l i m
n 3 I
I
n
l n S
n
l i m
n 3 I
I
n
l n S
n
l m o l
n d m m l m o o
l i m
n 3 I
I
n
l n S
n
a W
X aE
f l n h b ; XI
i l m o l F
o o o o m l m l o n n o o n l o
l n m F s n o d n o o n
W @ bA a E f l n h b ; XI
i :
R
-
8/9/2019 Machine Learning and Finance III
5/31
n
I
n
l n S
n
a
I
n
n
X
i a I
l n h b ; Xi
i
a
I
n
n
X
i a I
E
f l n h b ; Xi
i C
I
n
n
X
i a I
@ l n h b ; Xi
i E
f l n h b ; Xi
i A
a W @ bA CI
n
n
X
i a I
@ l n h b ; Xi
i E
f l n h b ; Xi
i A :
o n l o l n m m l
I
n
n
X
i a I
@ l n h b ; Xi
i E
f l n h b ; Xi
i A 3 H l m o l D
o
l i m
n 3 I
I
n
l n S
n
a W @ bA a E f l n h b ; XI
i l m o l F
m l l D
l i m
n 3 I
I
n
l n S
n
a W @ b A a m x b
W @ bA l m o l F
o m z o n d o n o m o d l X m
@ A l d l D n d l l o n o
o l l n n n o n d d n n n n
d n o d D
@ A n o n o n o D
@ A o o m k n o ' d o n o n o
n n d n o n F
g o n n n o n d o n @ A D f l m n d u l S D s n n d g o P Q D
s n P I D P P D f o k n d n T D n d Q T o n d d
o l m o n o n o F p o m m o l o w k o n m k o D
o m l n n o d d d o n o m k
S
-
8/9/2019 Machine Learning and Finance III
6/31
o k n o n F n o m o l l o m l D m l l o m
k n n o o n n o n o F g o n d o n @ A m n
m k n 0 n F
n l o l o E o m l m o n o n n
S
n
@ bA n o l o o E f Sn
@ bA :
o m o F o m l o o d o m n D
o n > H D n
-
8/9/2019 Machine Learning and Finance III
7/31
f o
E
f S
n
@ bA a h b ; E f XI
i
n
;
n o l n o m
r g m x
b
E
f S
n
@ bA a r g m x b
h b ; E f XI
i ;
l l d m n o F s o r g m x b h b ; E f XI i
o o l o o n I o o n D E
f XI
l
o m o n n F
s n m n l w k o z P U l z d m n o
n d D F F D d n o o o l o F s n o d o o d d 0 l
d d ( o n D l l d m n E n o o l o
~b a r g m x bX Var@ h b ; X
1
i A
h b ; E f XI
i ;
> H k o n m F
p o o o o D o m n @ o A o
~b n l o o o m n o o m l
bD o D o o d
o o d n d o n @ n k n o n A d o n o n o XF
l l o n o
~b n o n l n o m m n @ x v A o l m D
l n n o n m m z d n d d o n n F
s n o d o l l l o E o m l o o l o b D o n o k n o
d o n o XI
F s d o n n k n o n n m l l o E
o m l o o l o n d ( n d
bn
a r g m x
b
I
n
n
X
i a I
l n h b ; Xi
i
l n o n n
d
X
j a I
b
@ j A
a I n d H b
@ j A
I j a I ; : : : ; d :
o o m l o o l o bn
n d m o d ( o n d d
w Q I D Q P n d w o Q R D Q S F
U
-
8/9/2019 Machine Learning and Finance III
8/31
l l o n o bn
x v o l m D o o F g o I H n o d d n
l o m o l l n b
n
F e n l n o l o o E
n d o n l p 2 o l l Q V F o n d o n n l d
o m m n m o d D o m n l l l o l o l o n o x v
o l n d o m m n o l m n d m l o l
m m m o d n o l o l m m m F
5 E x a m p l e s f o r c o n s t a n t l y r e b a l a n c e d p o r t f o -
l i o .
E x a m p l e 1 . g o n d m l o d a P n d X a @ X @ I A ; X @ P A A
( o m o n n X
@ I A
o n o X n ( l o k X
X
@ I A
a
(
P o l I = P D
I = P o l I = P D
@ I A
n d o n d o m o n n X
@ P A
X
X
@ P A
a I :
y o l D z o o F n o n o (
o m o n n
E
f X
@ I A
a I = P @ P C I = P A a S = R > I ;
n d F F d F o o m k o D
E
f S
@ I A
n
aE
(
n
Y
i a I
X
@ I A
i
)
a @ S = R A
n
; @ P A
o E
f S
@ I A
n
o o n n l l F r o D d o n o m l
n d o m l S
@ I A
n
o o n n l l D o o F v 9 l l o
W
@ I A
X
W
@ I A
X a l i m
n 3 I
I
n
l n S
@ I A
n
a l i m
n 3 I
I
n
n
X
i a I
l n X
@ I A
i
aE
f l n X
@ I A
a I = P l n P C I = P l n @ I = P A a H ;
V
-
8/9/2019 Machine Learning and Finance III
9/31
m n ( o m o n n z o o D o o F v 9
l l l o E o m l o o l o o n o D o o m E
o n n z o o F o o l o o o m
b a @ b ; I b A :
n
W @ bA a E f l n h b ; Xi
a I = P @ l n @ P b C @ I b A A C l n @ b = P C @ I b A A A
a I = P l n @ I C b A @ I b = P A :
y n n k W @ bA m m m o b a I = P D o l o E o m l
o o l o
b a @ I = P ; I = P A ;
n d m o o
W
aE
f l n h b ; Xi a I = P l n @ W = V A a H : H S W ;
o o F
E x a m p l e 2 . g o n d m l o d a Q n d X a @ X @ I A ; X @ P A ; X @ Q A A
( n d o n d o m o n n o n o X
( l o k o o m @ I A D l d o m o n n F y n
n o l o E o m l o o l o
b a @ H : R T ; H : R T ; H : H V A ;
n d m m l m o o
W
aE
f l n hb
;X
i a H : I I P :
E x a m p l e 3 . g o n d m l o d > Q n d X a @ X@ I A
; X
@ P A
; : : : ; X
@ d A
A
( d I o m o n n o n o X
( l
o k o o m @ I A D l l o m o n n F y n n o
l o E o m l o o l o
b a @ I = @ d I A ; : : : ; I = @ d I A ; H A ;
W
-
8/9/2019 Machine Learning and Finance III
10/31
m n D o d > Q D o d n o l o E o m l o o l o
z o F v N d n o n m o o m o n n o X l
o P D n N n o m l l d d m @ d I ; I = P A D n d
l n hb
;X
i a l n
P N C @ d I N A = P
d I
!
a l n
Q N
P @ d I A
C
I
P
!
;
o
W
aE
f l n h b ; Xi a E
(
l n
Q N
P @ d I A
C
I
P
! )
:
p o d a R D o m l m l m m l m o o
W
aE
f l n h b ; Xi a H : I S P ;
l o d 3 I D
W
aE
f l n hb
;X
i 3 l n @ S = R A a H : P P Q ;
m n
S
n
% e
n W
a @ S = R A
n
;
o m n o k
S
n
%E
f S
n
@ F @ P A A F
E x a m p l e 4 . g o n d m l o o n d o n F
e m o j n o l p
j
F o ' d n o d o
j
D
m n n n I 6 o n o j l n o
j
n D o
H 6 F n n o o o m
X a @ H ; : : : ; H ; oj
; H ; : : : ; H A
o j n F p o d D o n n l l n d o o l o
o l m F v 9 l l d l o E n X
W @ bA a E f l n h b ; Xi ad
X
j a I
p
j
l n @ b
@ j A
o
j
A a
d
X
j a I
p
j
l n b
@ j A
C
d
X
j a I
p
j
l n o
j
;
I H
-
8/9/2019 Machine Learning and Finance III
11/31
o
r g m x
b
E
f l n hb
;X
i a r g m x
b
d
X
j a I
p
j
l n b
@ j A
:
s n o d o o l o m z o n o l m
r g m x
b
d
X
j a I
p
j
l n b
@ j A
;
n o d u l l k E v l d n o d o n p
n db
X
K L @ p; bA ad
X
j a I
p
j
l n
p
j
b
@ j A
:
o o u l l k E v l d n
K L @ p; bA ! H ;
n d l o z o ' o d o n l F o o o
o m l X
K L @ p; bA a d
X
j a I
p
j
l n
b
@ j A
p
j
!
d
X
j a I
p
j
b
@ j A
p
j
I
!
a
d
X
j a I
b
@ j A
C
d
X
j a I
p
j
a H :
n l m l
r g m x
b
d
X
j a I
p
j
l n b
@ j A
a p:
n l D l o E o m l o o l o n d n d n o o ' D n d
W
a
d
X
j a I
p
j
l n @ p
j
o
j
A :
l o o o '
o
j
a
I
p
j
;
n d n
W
a H :
s m n D o o o o ' D n m l n n
o F
I I
-
8/9/2019 Machine Learning and Finance III
12/31
6 S e m i - l o g - o p t i m a l p o r t f o l i o .
j d R H d n o m o n o b n d bn
n
h @ z A X a z I
I
P
@ z I A
P
;
o n d o d l o n o n o n o n l n z z a I F
n D m E l o E o m l o o l o l o n
"b a r g m x b
E
f h @ h b ; xI
i A ;
n d m l m E l o E o m l o o l o
"bn
a r g m x
b
I
n
n
X
i a I
h @ h b ; xi
i A :
s n o d o o m bn
D o n o m k n o m z o n o bF s n
o m z o n o m o n l o m l o o o n l o n F p o
"bn
D o m l n d d F
I
n
n
X
i a I
h @ h b ; xi
i A a
I
n
n
X
i a I
@ h b ; xi
i I A
I
P
I
n
n
X
i a I
@ h b ; xi
i I A
P
:
s 1 d n o l l I o D n
I
n
n
X
i a I
h @ h b ; xi
i A a h b ; mi h b ; Cbi ;
m aI
n
n
X
i a I
@ xi
1A
n d
Ca
I
P
I
n
n
X
i a I
@
xi
1A @
xi
1A
T
:
s l l o m n d m C o n d n n
o m z o n o m l d o n o d n d o n n D o n n n
m o l l n
"bn
m m l l n o bn
F o d n
o m E l o E o m l o o l o n l l d d
I P
-
8/9/2019 Machine Learning and Finance III
13/31
o m m n D d o l D F F D n o n Q u a d P r o g + + o h
q o I R F o m q o l d E s d n n d l m o d o o l n
d o m m n o l m I S F s o m C
o m E d ( n D o D o m n o n d d l m o d o n l
l C o d ( n F d ' n n o d n o l m
n m n D n o l m l o m m C
l l d o m d D n d o C n o l l E n k m D m l
C o n l o m E d ( n F
7 N u m e r i c a l c o m p a r i s o n o f l o g - o p t i m a l a n d s e m i -
l o g - o p t i m a l p o r t f o l i o s .
l I m m z n m l l o N Y S E d F e
w w w F s z i t F m e F h u G o t i G p o r t f o l i o o n m k d o m
N Y S E X
( d o n o d l d o Q T o k l n P P
@ S T S I d n d n d n n I W V S A F d n d
o n o o l o l o n n g o I I D n n Q U D n q ( D
v o D d n I U D n q ( D d n D l k I W n d n q ( D n D
j d P H F
o n d d o n n P Q o k n d l n R R @ I I I U V
d n d n d n n P H H T A F
y m n o n o n d d F ( o l m n o l I
l o k 9 n m D o n d o l m n o n n l l d
@ e e A F d n d o o l m n n o o k
@ o m o n n o o o l o o A n l o E o m l n d m E
l o E o m l l o m F n l D o o o l o o l m o
m D o d n o n E o E E l o o m D
n d o n l o o k n o n E l @ n o n E z o A F
e n d d o n l n n o l u s x e
m l l o @ S F W 7 A n l d d n o o l o F l
o o n n n m o l o m F n o d ' n
I Q
-
8/9/2019 Machine Learning and Finance III
14/31
S t o c k ' s n a m e A A Y B C R P
l o g - N L P w e i g h t s s e m i - l o g - N L P w e i g h t s
A H P 1 8 . 6 % 0 0
A L C O A 1 3 . 1 % 0 0
A M E R B 2 0 . 2 % 0 0
C O K E 2 0 . 9 % 0 0
C O M M E 2 6 . 1 % 0 . 3 0 2 8 0 . 2 9 6 2
D O W 1 7 . 0 % 0 0
D U P O N T 1 2 . 4 % 0 0
F O R D 1 3 . 7 % 0 0
G E 1 8 . 7 % 0 0
G M 9 . 9 % 0 0
H P 2 1 . 8 % 0 . 0 1 0 0 0 . 0 3 1 7
I B M 1 4 . 1 % 0 0
I N G E R 1 6 . 0 % 0 0
J N J 2 3 . 8 % 0 0
K I M B C 1 8 . 9 % 0 0
K I N A R 5 . 9 % 0 . 2 1 7 5 0 . 2 1 3 0
K O D A K 8 . 9 % 0 0
M E R C K 2 1 . 7 % 0 0
M M M 1 5 . 8 % 0 0
M O R R I S 3 0 . 1 % 0 . 4 6 9 6 0 . 4 5 9 0
P A N D G 1 8 . 8 % 0 0
S C H L U M 2 2 . 2 % 0 0
S H E R W 1 8 . 2 % 0 0
A A Y 3 5 . 2 % 3 5 . 2 %
r u n n i n g t i m e ( s e c ) 9 0 0 2 3
l I X g o m o n o o l o m o g F
n o n d n n l l d o m o d D o D
n n n m n ( n l o o m E l o E o m l l o m F
I R
-
8/9/2019 Machine Learning and Finance III
15/31
8 D y n a m i c p o r t f o l i o s e l e c t i o n : g e n e r a l c a s e
p o n l d n m o o l o l o n D o o l o o m d n d
o n d F e o D xi
a @ x
@ I A
i
; : : : x
@ d A
i
A d n o n o
o n d n o d i F v b a bI
o o l o o o ( d n
o d F p o n l l S
H
D
S
I
a S
H
h bI
; xI
i :
p o o n d d n o d D S
I
n n l l D o o l o o
bP
a b@ xI
A D n d
S
P
a S
H
h bI ; xI i h b@ xI A ; xP i :
p o n d n o d D o o l o o bn
a b@ xI
; : : : ; xn I
A a
b@xn I
I
A n d
S
n
a S
H
n
Y
i a I
D
b@ xi II
A ; xi
E
a S
H
e
n W
n
@ BA
o
W
n
@ BA aI
n
n
X
i a I
l n
D
b@ xi II
A ; xi
E
:
9 L o g - o p t i m a l p o r t f o l i o f o r s t a t i o n a r y m a r k e t
p r o c e s s .
n d m n l l m D d m n d n w P W D n e l o n d g o
Q D n d n e l o I D P D l o E l l d l o g - o p t i m u m p o r t f o l i o
B a f b @ A o l o F w o l D o n d n o d
n l b
@ A
E
n
l n
D
b @ Xn II
A ; Xn
E
Xn II
o
a m x
b @ AE
n
l n
D
b@ Xn II
A ; Xn
E
Xn II
o
:
s S
n
a S
n
@ B A d n o l d l o E o m m o o l o
B
D n d n o d D n o n o n m n
B l Sn
a S
n
@ BA n d
s u p
n
E
n
@ l n
D
bn
@ Xn II
A ; Xn
E
A
P
o
-
8/9/2019 Machine Learning and Finance III
16/31
n d o n o n n d o d o f Xn
I
I
D
l i m s u p
n 3 I
I
n
l n
S
n
S
n
H l m o l @ Q A
n d
l i m
n 3 I
I
n
l n S
n
a W
l m o l D
W
X aE
(
m x
b@ AE
n
l n
D
b@ X I I
A ; XH
E
X I I
o
)
m m l o l o o n n m n F @ x o
o m m o l m k W
a m x b E f l n h b ; XH i o n
l o E o m l o o l o o n n l l n d o o l o F A
p o o o o o m l o n o m n l d ' E
n X
D e n i t i o n 1 T h e r e a r e t w o s e q u e n c e s o f r a n d o m v a r i a b l e s f Z
n
a n d
f X
n
s u c h t h a t
Z
n
i s a f u n c t i o n o f X
I
; : : : ; X
n
,
E
f Z
n
j X
I
; : : : ; X
n I
a H a l m o s t s u r e l y .
T h e n f Z
n
i s c a l l e d m a r t i n g a l e d i e r e n c e s e q u e n c e w i t h r e s p e c t t o
f X
n
.
p o m n l d ' n n D o n l o l n m E
X s f Z
n
m n l d ' n n o f X
n
n d
I
X
n a I
E
f Z
P
n
n
P
-
8/9/2019 Machine Learning and Finance III
17/31
s n o d o l E o n n d D o m n l d ' n D o
k l o l n m F o f Z
n
m n l d ' n
n o f X
n
n f Z
n
n o l d F i < j D n
E
f Z
i
Z
j
aE
fE
f Z
i
Z
j
j X
I
; : : : ; X
j I
aE
f Z
i
E
f Z
j
j X
I
; : : : ; X
j I
aE
f Z
i
H a H :
s m l
E
8
-
8/9/2019 Machine Learning and Finance III
18/31
f o d ( n o n o l o E o m l o o l o
E
f l n
D
b@ Xi II
A ; Xi
E
j Xi II
E
f l n
D
b @ Xi II
A ; Xi
E
j Xi II
;
n d o o ( n d F
1 0 E m p i r i c a l p o r t f o l i o s e l e c t i o n
o m l l o n o d o o o l l o n d ( n o n X
D e n i t i o n 2 A n e m p i r i c a l ( d a t a d r i v e n ) p o r t f o l i o s t r a t e g y B
i s c a l l e d
u n i v e r s a l l y c o n s i s t e n t w i t h r e s p e c t t o a c l a s s g o f s t a t i o n a r y a n d
e r g o d i c p r o c e s s e s f Xn
I
I
, i f f o r e a c h p r o c e s s i n t h e c l a s s ,
l i m
n 3 I
I
n
l n S
n
@ BA a W a l m o s t s u r e l y .
s n o l l o o n l l o n n o o l o E
F n D n l
o l o l l o n n d o d o o d
e l o I F
w o o d l n o o l o l o n m
d o n o m k o k n o n F s d o n
n k n o n n o n n l o l n m F
I X s n ( m o d n o o l l d D n d m
o o n d n d o n F s n o d n o n n m n F
P X s n o n d m o d n o d o m
d o n m n d o m n m n F
s n l o n o n d o m k n l n D
o n n o n n l l o m n o n m k
d n d n o l m o d D F F D m o n n d o o l o
l o n m d o n o l m o d F
v 9 l d ( n o n o l o E o m l o o l o X
E
f l n
D
b @ Xn II
A ; Xn
E
j Xn II
a m x
b@ AE
f l n
D
b@ Xn II
A ; Xn
E
j Xn II
I V
-
8/9/2019 Machine Learning and Finance III
19/31
p o ( d n k > H l n o D
E
f l n
D
b@Xn I
I
A ;X
n
E
jXn I
I
%E
f l n
D
b@Xn I
n k
A ;X
n
E
jXn I
n k
n d
b @ Xn II
A % bk
@ Xn In k
A a r g m x
b@ A
E
f l n
D
b@ Xn In k
A ; Xn
E
j Xn In k
:
f o o n
bk
@ xkI
A a r g m x
b@ A
E
f l n
D
b@ Xn In k
A ; Xn
E
j Xn In k
a xkI
a r g m x
b@ A
E
f l n
D
b@
xk
I
A ;
Xk C IE
j
Xk
I
a
xk
I
a r g m x
b
E
f l n h b ; Xk C I
i j XkI
a xkI
;
m m z o n o o n n o n
m b @ xk
I
A aE
f l n h b ; Xk C I
i j XkI
a xkI
:
D o l o m o l l o m l m l o o l o l E
o n D d o n d D n l l m o n
n o n m b @ xk
I
A D n d o o o o l o o D m m z
o n n o n m F
1 1 R e g r e s s i o n f u n c t i o n e s t i m a t i o n
f ) m m z o n o n m o n n o n m E
o n F g o n n n d l o o o k o q ( D u o l D u z z k
n d l k I T F v Y l l d n d o m l D n d l X d n o
n d o m o F o n n o n o n d o n l o n o
Y n X X
m @ x A aE
f Y j X a x :
s d o n o @ X ; Y A n k n o n n o n o m E
o n n o n o m d F d n o F F d F o o
@ X ; Y A X
D
n
a f @ X
I
; Y
I
A ; : : : ; @ X
n
; Y
n
A :
I W
-
8/9/2019 Machine Learning and Finance III
20/31
o n n o n m o o m
m
n
@ x A a m
n
@ x ; D
n
A :
e n m o n l o m l o l n m
m
n
@ x A a
n
X
i a I
W
n i
@ x Y X
I
; : : : ; X
n
A Y
i
;
l l W
n i
@ x Y X
I
; : : : ; X
n
A n o n E n n d m
o I F w o o D W
n i
@ x Y X
I
; : : : ; X
n
A l l l x l o o X
i
D
o z o F
o l l o n o o n d n n n l o n
m o n n d o o l o l o n X
X $ XkI
;
Y $ l n h b ; Xk C I
i ;
m @ x A aE
f Y j X a x $ m b @ xk
I
A aE
f l n h b ; Xk C I
i j XkI
a xkI
:
1 2 H i s t o g r a m b a s e d p o r t f o l i o s e l e c t i o n
o m o o n n o n m l o l n E
m F v
n
a f A
n ; I
; A
n ; P
: : : o n o o D X
k l F A
n ; j
l l d l l F l m l o l l F s
A
n
@ x A l l o o n
n
n o x l l n o n n
o n m d ( n d
m
n
@ x A a
P
n
i a I
Y
i
I
X
i
P A
n
@ x A
P
n
i a I
I
X
i
P A
n
@ x A
;
I
n d o n d o n o n F
v G
n
n z o o n d n o o n
n
X G
n
@ x A a j
x P A
n ; j
F s o m @ m l A
I
n
@ x A a f i n X G
n
@ x A a G
n
@ X
i
A
P H
-
8/9/2019 Machine Learning and Finance III
21/31
n o m m o m
m
n
@ x A a
P
i P I
n
@ x A
Y
i
j I
n
@ x A j
:
x d h i s t o g r a m b a s e d p o r t f o l i o s e l e c t i o n n o d d
q ( n d I V d n o BH F BH o n d o l l o F
( d ( n n n ( n o l m n @ o E l l d e x p e r t s A
B@ k ; A a f b@ k ; A @ A D n d d o n k ; a I ; P ; : : : F i
B@ k ; A d m n d o d l n k n d o n
a
f A
` ; j
D j a I ; P ; : : : ; m
o R
d
C
n o m
d j o n l l F o d m n
o o l o o n n d n o d D
B@ k ; A
l o o k m k o
xn k
; : : : ; xn I
o l k o d D d z k d E d m n o n l o
m n o o n
D n d d m n o o l o o
o m l o o d n o d o d n k d n o d
d n l d z d m k o o n o n F p o m l l D l
G
d z o n n o n o o n d n o o n
D D
G
@ xA a j ; x P A` ; j
:
o m o n o o n D o n n n d xnI
PR
d n
D G
@ xnI
A o
n G
@ xI A ; : : : ; G @ xn A F n d ( n B@ k ; A
a f b@ k ; A
@ A
n D o n > k C I D
b@ k ; A @ xn II
A a r g m x
bP d
X
f
k < i < n X G
@ xi 1
i k
A a G
@ xn 1
n k
A
l n h b ; xi
i ; @ R A
m n o n E o d D n d n o m b
H
a @ I = d ; : : : ; I = d A o F
D h@ k ; An
d z n xn II
o d n o o n
D n d
o o l l n o l n d z d n
G
@ xn In k
A o l n k F n d n ( d o o l o o o m z n
n o d n o d o l l o n o n o n F
o l m l o o o o k ; F o m X
m l l k o m l l m l o o n d n o n m
l D
P I
-
8/9/2019 Machine Learning and Finance III
22/31
l k n d l m l l l m n D
l n l n F
o o d D d d n d n o o k n d d o l o o n n
n o m m n l n n F s n m n l n n k n d
o n d d m o m D l l d F d
o m n l n n o m n o n o F o m n o n
n d m D n l
o m n o o d @ F g E f n n d v o V A F
m o l o m n o n o n n l n F g o m n
l m n o o l o B@ k ; A a f b@ k ; An
o l l o X l f q
k ;
o l d o n o n o l l @ k ; A o l l k ; D
q
k ;
> H F
p o > H D n o d o n n l
w
n ; k ; `
a q
k ;
e
l n S
n 1
@ B( k ; ) A:
p o a I D m n
w
n ; k ; `
a q
k ;
e
l n S
n 1
@ B( k ; ) Aa q
k ;
S
n I
@ B@ k ; A A
n d
v
n ; k ; `
a
w
n ; k ; `
P
i ; j
w
n ; i ; j
:
o m n d o o l o b d ( n d
bn
@ xn II
A a
X
k ;
v
n ; k ; `
b@ k ; An
@ xn II
A :
P P
-
8/9/2019 Machine Learning and Finance III
23/31
o m n o n m l n o n X
S
n
@ BA an
Y
i a I
D
bi
@ xi II
A ; xi
E
a
n
Y
i a I
P
k ;
w
i ; k ; `
D
b@ k ; A
i
@ xi II
A ; xi
E
P
k ;
w
i ; k ; `
a
n
Y
i a I
P
k ;
q
k ;
S
i I
@ B@ k ; A AD
b@ k ; A
i
@ xi II
A ; xi
E
P
k ;
q
k ;
S
i I
@ B@ k ; A A
a
n
Y
i a I
P
k ;
q
k ;
S
i
@ B@ k ; A AP
k ;
q
k ;
S
i I
@
B@ k ; A
A
a
X
k ;
q
k ;
S
n
@ B@ k ; A A :
B a BH n o m n l m n o o l o
B@ k ; A a f b@ k ; An
n o 9 l o m
S
n
@B
A a
X
k ;
q
k ;
S
n
@B@ k ; A
A : @ S A
s o n n I V BH n l l o n n
o l o l l o d o E
f j l n X
@ j A
j
-
8/9/2019 Machine Learning and Finance III
24/31
W
n
@ BA aI
n
l n S
n
@ BA
a
I
n
l n
0
@
X
k ;
q
k ;
S
n
@ B@ k ; A A
1
A
!
I
n
l n
s u p
k ;
q
k ;
S
n
@ B@ k ; A A
!
a
I
n
s u p
k ;
l n q
k ;
C l n S
n
@ B@ k ; A A
a s u p
k ;
W
n
@
B@ k ; A
A C
l n q
k ;
n
!
:
l i m i n f
n 3 I
W
n
@ BA ! l i m i n f n 3 I
s u p
k ;
W
n
@ B@ k ; A A Cl n q
k ;
n
!
! s u p
k ;
l i m i n f
n 3 I
W
n
@ B@ k ; A A Cl n q
k ;
n
!
a s u p
k ;
l i m i n f
n 3 I
W
n
@ B@ k ; A A
a s u p
k ;
k ;
:
n o n
n d D n o
s u p
k ;
k ;
a l i m
k 3 I
l i m
l 3 I
k ;
a W
:
1 3 K e r n e l b a s e d p o r t f o l i o s e l e c t i o n
k n l o n m d ( n d k n l n o n K @ x A ! H
n d n d d h > H
m
n
@ x A a
P
n
i a I
Y
i
K
x X
i
h
P
n
i a I
K
x X
i
h
:
p o n @ n d o A k n l n o n K @ x A a I
f k x k I g
D
m
n
@ x A a
P
n
i a I
Y
i
I
f k x X
i
k h g
P
n
i a I
I
f k x X
i
k h g
:
P R
-
8/9/2019 Machine Learning and Finance III
25/31
q ( D v o D d n I U n o d d k e r n e l - b a s e d p o r t f o l i o s e l e c t i o n
F h ( n n n ( n o B@ k ; A
a f b@ k ; A
@ A D
k ; o n F p o ( d o n k ; D o o d
r
k ;
> H o n ( d k D
l i m
3 I
r
k ;
a H :
n D o n > k C I D d ( n b@ k ; A
b@ k ; A @ xn II
A a r g m x
b P d
X
f
k < i < n X k xi 1
i k
xn 1
n k
k r
k ;
l n h b ; xi
i ;
m n o n E o d D n d bH
a @ I = d ; : : : ; I = d A o F
m d m n o o m E d @ F
@ S A A F
q ( D v o D d n I U o d o o l o m BK
n E
l l o n n o l o l l o d o
E
f j l n X
@ j A
j H D k E n n o @ x x A o n m
l o l n o n m D o o D
m
n
@ x A a
n
X
i a I
W
n i
@ x Y X
I
; : : : ; X
n
A Y
i
;
W
n i
I = k X
i
o n o k n n o o x m o n
X
I
; : : : ; X
n
D n d W
n i
H o F
q ( D d n D l k I W n o d d n e a r e s t - n e i g h b o r - b a s e d p o r t f o -
l i o s e l e c t i o n
BN N
o l l o F h ( n n n ( n o
B@ k ; A
a
f b@ k ; A @ A D H < k ; ` n F t l k o D k n d o
l n o n D n d o o o p
P @ H ; I A
l i m
3 I
p
a H : @ T A
P S
-
8/9/2019 Machine Learning and Finance III
26/31
a p
n :
e n m n n n D o
n n o
m n F p o ( d o n k ; @ n > k C
C I A n o d
o
n n o m X
J
@ k ; A
n
a f i Y k C I i n xi Ii k
m o n
x x o xn In k
n xkI
; : : : ; xn Pn k I
:
h ( n b@ k ; A
b@ k ; A
@ xn I
I
A a r g m x
bP d
X
n
i P
J
( k ; )
n
o
l n h b ; xi
i ;
m n o n E o d D n d bH
a @ I = d ; : : : ; I = d A o F D b@ k ; An
( d o o l o o o d n o n o l l o n n
n o F m d m o @ F @ S A A F
o o l z o o n o s a skI
n d o m l
k XkI
sk
o n n o d o n n o n F
q ( D d n D l k I W o d o l l o n X e m @ T A n d
o o l z o F o o l o m BN N
n E
l l o n n o l o l l o d o
E
f j l n X
@ j A
j
-
8/9/2019 Machine Learning and Finance III
27/31
k I P Q R S
I P W F R 7 P U F V 7 P P F W 7 P Q F W 7 P Q F U 7
P P H I F H 7 I P R F T 7 W V F Q 7 Q T F I 7 W H F V 7
Q I I R F P 7 T P F W 7 Q V F V 7 W I F R 7 Q I F T 7
R I U P F S 7 I S S F H 7 I H H F T 7 I T P F Q 7 S H F V 7
S P Q Q F S 7 I U H F P 7 I T T F T 7 I U I F I 7 I H U F U 7
T P R S F R 7 P I T F P 7 I U T F W 7 I V P F H 7 I R Q F H 7
U P T I F V 7 P I I F H 7 I V I F P 7 I T S F T 7 I S V F U 7
V P P W F R 7 I V W F P 7 I U I F H 7 I Q V F V 7 I Q I F Q 7
W P I W F Q 7 I U P F V 7 I T P F R 7 I I V F U 7 I I T F H 7
I H P I H F T 7 I S I F S 7 I Q I F V 7 I H Q F R 7 I I H F W 7
l P X n n l l d o n d d l F
o o d m l o o l o l o n l o m n n ( n
o F s n k ( n o z K L F s n o
m n l d K a S n d L a I H F g o o n o m d o n
f q
k ;
a I = @ K L A o n D n d d
r
P
k ; l
a H : H H H I d k ` ;
@ k a I ; : : : ; K n d a I ; : : : ; L A F
l P m m z n n l l d d
l o d n n n o n n o k n l E d m E l o E
o m l o o l o F i n d d k a I : : : S n o l m n n d
a I : : : I H n o F n n l l d o k n l d m E l o E
o m l o o l o P P P : U 7 F e o d n o l I D w y s d
n n l l d D Q H : I 7 D l f g d n n l l d
Q S : P 7 D o k n l d m E l o E o m l o o l o E
l m o m n F
e n o n n o l P o n ( d D
k l o I D o m l o o l o o n n d w k o n
m o d l l n o F
P U
-
8/9/2019 Machine Learning and Finance III
28/31
R e f e r e n c e s
I e l o D F n l m o d o n D m l n D n d o o l o
l o n F A n n a l s o f P r o b a b i l i t y D P H X W H I ! W R I D I W W P F
P e l o D F o n l o l n m o n l d o n
n d n n F I E E E T r a n s a c t i o n s o n I n f o r m a t i o n T h e o r y D
R H X T H W ! T Q R D I W W R F
Q e l o D F n d g o D F w F e m o o m l m o
o n o o l o E o m m n m n F A n n a l s o f P r o b -
a b i l i t y D I T X V U T ! V W V D I W V V F
R f o n D e F F n d g o D F w F e o n d o n ( n n l l o
n o m o n F I E E E T r a n s a c t i o n s o n I n f o r m a t i o n T h e o r y D Q R X I H W U !
I I H H D I W V V F
S f l m D e F n d u l D e F n l o o l o n d o n E
o n o F M a c h i n e L e a r n i n g D Q S X I W Q ! P H S D I W W W F
T f o k D F F n d n D v F h m o o l o l o n
o o o n l n o n o F P r o b a b i l i t y a n d M a t h e m a t i c a l S t a t i s -
t i c s D I W X P Q S E P R V D I W W W F
U f m n D v F y m l m l n m o o l m F P r o c .
F o u r t h B e r k e l e y S y m p . M a t h . S t a t i s t . P r o b . D I X T S ! U V D n F g l E
o n D f k l D I W T I F
V g E f n D x F n d v o D q F P r e d i c t i o n , L e a r n i n g , a n d G a m e s F
g m d n D g m d D P H H T F
W g o D F F v o l o n n o m n l n d l o l
n m F A n n a l s o f M a t h e m a t i c a l S t a t i s t i c s D Q T X S S P ! S S V D I W T S F
I H g o D F e n l o m o m m z n d l o n m n E
n F I E E E T r a n s a c t i o n s o n I n f o r m a t i o n T h e o r y D Q H X Q T W ! Q U Q D I W V R F
I I g o D F n l o o l o F M a t h e m a t i c a l F i n a n c e D I X I ! P W D I W W I F
P V
-
8/9/2019 Machine Learning and Finance III
29/31
I P g o D F n d o m D t F E l e m e n t s o f I n f o r m a t i o n T h e o r y F t o n
l n d o n D I W W I F
I Q p n k l n D w F n d l D F y m l o d m F
A d v a n c e s i n A p p l i e d P r o b a b i l i t y D I Q X R I S ! R P V D I W V I F
I R h q o D v F d o C C F L u c a D i G a s p e r o ' s h o m e s i t e D P H H T D
h t t p X G G w w w F d i e g m F u n i u d F i t G d i g s p e r o G
I S q o l d D h F n d s d n n D e F e n m l l l d l m o d o
o l n l d o m F M a t h e m a t i c a l P r o g r a m m i n g D
P U X I ! Q Q D I W V Q F
I T q ( D v F D u o l D w F D u z k D e F n d l k D r F A D i s t r i b u t i o n - F r e e
T h e o r y o f N o n p a r a m e t r i c R e g r e s s i o n . n l D P H H P F
I U q ( D v F D v o D q F n d d n D p F x o n m k n l d
n l n m n F M a t h e m a t i c a l F i n a n c e D I T X Q Q U ! Q S U D
P H H T F
I V q ( D v F n d D h F x o n m d o n F s n t F e F u F
k n D q F r o D F f D g F w l l D n d t F n d l l D d o D
A d v a n c e s i n L e a r n i n g T h e o r y : M e t h o d s , M o d e l s a n d A p p l i c a t i o n s D
F Q Q W ! Q S R F s y D x e y n D P H H Q F
I W q ( D v F D d n D p F n d l k D r F x o n m n E n o E
d n l n m n D @ m d o l o n A D
P H H U F
X G G F z F m F G o G o o l o G l G x x F d
P H q ( D v F D n D e F n d j d D s F u n l E d m E l o E o m l m E
l o o l o l o n F I n t e r a n t i o n a l J o u r n a l o f T h e o -
r e t i c a l a n d A p p l i e d F i n a n c e D I H X S H S E S I T D P H H U F
P I s n D q F h m o o m l n m n o F
W o r k i n g P a p e r D P H H P F
X G G F o l m F d G I H G G o F d
P W
-
8/9/2019 Machine Learning and Finance III
30/31
P P s n D q F n l n m n n m k n o n o F
M a t h e m a t i c a l F i n a n c e D I S X Q S W E Q U I D P H H S F
P Q s n D q F n d g o D F q o o m l n m n n o
m k o F I E E E T r a n s a c t i o n s o n I n f o r m a t i o n T h e o r y D
R T X P T U S E P T V Q D P H H H F
P R u l l D t F v F e n n o n o n o m o n F B e l l S y s t e m
T e c h n i c a l J o u r n a l D Q S X W I U ! W P T D I W S T F
P S v n D r F e F g o o m o n k n F J o u r n a l o f
P o l i t i c a l E c o n o m y D Q V X I R S ! I S S D I W S W F
P T v n D h F q F I n v e s t m e n t S c i e n c e F y o d n D I W W V F
P U w k o z D r F o o l o l o n F J o u r n a l o f F i n a n c e D U X U U ! W I D I W S P F
P V w k o z D r F s n m n o l o n n X n d n o n o l d l F
J o u r n a l o f F i n a n c e D Q I X I P U Q ! I P V T D I W U T F
P W w D F p F y n o l o m F A n n a l e s U n i v . S c i .
B u d a p e s t . R . E t v s N o m . , S e c t . C o m p u t . D Q X W W ! I H Q D I W V P F
Q H w D F p F s E d n o m o d o n n d o
m F s n P r o c . o f t h e 3 r d P a n n o n i a n S y m p . o n M a t h . S t a t . D i d F
t F w o o d l D d l D h o d D P Q I ! P Q V D I W V R F
Q I w D F p F e m o o o m l n
o l o m F s n L i m i t T h e o r e m s i n P r o b a b i l i t y a n d
S t a t i s t i c s D i d F F z l D x o E r o l l n d D U U U ! U W H D I W V R F
Q P w D F p F s m l o m l c S t a t i s t i c s a n d D e c i -
s i o n s D R X R S ! T H D I W V T F
Q Q w D F p F n d z k l D q F t F r o o n o n c s n L i m i t
T h e o r e m s i n P r o b a b i l i t y a n d S t a t i s t i c s D i d F F z l D x o E
r o l l n d D U W I ! V H T D I W V R F
Q H
-
8/9/2019 Machine Learning and Finance III
31/31
Q R w o D q F i m l l o E o m l o o l o l o n F P r o b l e m s o f
C o n t r o l a n d I n f o r m a t i o n T h e o r y D P H X R S Q ! R T Q D I W W I F
Q S w o D q F o o l o o d o n m l d o n F K y -
b e r n e t i k a D P V X R V R ! R W Q D I W W P F
Q T D h F N o n p a r a m e t r i c E s t i m a t i o n f o r F i n a n t i a l I n v e s t m e n t
u n d e r L o g - U t i l i t y . h h o n D w m l s n D n E
D e n X k l D P H H P F
X G G F z F m F G o G o o l o G l G d o m n k F d
Q U n D F n o o l o F I n t e r n a t i o n a l J o u r n a l o f N e u r a l
S y s t e m s D V R X R R S ! R S S D I W W U F
Q V l l D F h y x v P D g o n U G W W F N e t l i b D I W W W D
h t t p X G G w w w F n e t l i F o r g G
Q W o F F p F A l m o s t s u r e c o n v e r g e n c e F e d m D x o k D
I W U R F
R H j d D s F e n l o m E l o E o m l n m n D n P r a g u e
S t o c h a s t i c s 2 0 0 6 D w F r k o D w F t n z @ i d F A D U I W E U P U D
w e p i D D P H H T F
Q I