I J
> ,
A
SUBORDINATED
STOCHASTIC
PROCESS
MODEL
WITH
FINITE
VARIANCE
FOR
SPECULATIVE
PRICES
by
Peter
King
Clark
Discussion
Paper
No.1,
April
1971
Center
for
Economic
Research
Department
of
Economics
University
of
Minnesota
Minneapolis,
Minnesota
55455
c>
, .
Acknowledgements
Thanks
are
due
to
Hendrik
Houthakker
and
Christopher
Sims,
for
both
encouragement
and
advice
in
developing
this
paper.
As
usual,
all
remaining
errors
are
my
own.
This
research
was
supported
by
a
Harvard
Dissertation
Fellowship,
NSF
grant
33-708,
and
the
Boston
College
Department
of
Economics.
, I
r I
A
SUBORDINATED
STOCHASTIC
PROCESS
MODEL
WITH
FINITE
VARIANCE
FOR
SPECULATIVE
PRICES
I.
INTRODUCTION
The
past
seventeen
years
have
seen
a
large
amount
of
research
by
academic
economists
on
prices
in
speculative
markets,
an
area
which
was
formerly
studied
almost
exclusively
by
financial
speculators
and
their
advisors.
l
Considering
the
time
series
of
prices
at
short
inter-
vals
on a
speculative
market
such
as
that
for
futures
in
commodities,
or
corporation
shares,
one
primary
characteristic
is
evident.
If
~
denotes
price
at
time
t
and
~~ =
~
-
~-l
,
examination
of
the
data
suggests
that:
and
The
increments
in
the
price
process
are
stationary
in
the
mean
and
un-
correlated;
a
random
walk
model
(1)
( 1 )
~
=
~_
1 +
Ct,
E
(Ct)
= O. E
(Ct
E.
) = 0 I t J s
explains
these
empirical
facts
well.
Besides
empirical
realism,
the
random
walk
model
has
a
theoretical
basis.
Z
If
price
changes
were
correlated,
then
alert
speculators
should
notice
the
correlation
and
trade
in
the
right
direction
until
the
relation-
ship
was
removed.
This
was
first
shown by
Bachelier
in
1900,
when
he
derived
the
diffusion
equation
from
a
condition
that
speculators
should
receive
no
information
from
past
prices.
Equation
(1)
is,
of
course,
a
solution
to
a
discrete
formulation
of
the
diffusion
problem.
1.
See
Clark
[6J
for
a
comprehensive
bibliography,
or
Cootner
[8J
for
a
collection
of
these
articles.
2.
Bachelier
[3J
I'
( ,
- 2 -
It
is
also
empirically
evident
that
the
price
changes
AXt
' however
independent,
are
not
normally
distributed.
Instead
of
having
the
normal
shape,
which would
be
the
case
if
the
components
in
A~
were
almost
independent
and
almost
identically
distributed,3
AX
has
too
many
small
and
too
many
large
observations,
as
pictured
in
Figure
1.
One way
to
express
this
is
to
say
that
the
distribution
of
AX
-is
leptokurtic,
for
the
sample
kurtosis:
is
much
greater
than
1
--
4
-
~
(AX!
-
AX)
= n 1
1
~
[(AX
1
_
AX)~]2
n 1
3,
the
value
for
a
normal
population.
It
is
evident,
then,
that
conditions
sufficient
for
the
Central
Limit
Theorem
are
not
met by
the
influences
which
make
up
AX.
The
violation
of
these
conditions
and
the
reason
for
the
leptokurtic
distribution
of
AX
is
the
subject
of
the
present
article.
In
1963,
Mandelbrot
set
out
to
explain
this
non-normality
in
price
changes
that
had
been
observed
by Kendal1
4
and
many
others.
s
The
ob-
served
distribution
of
price
changes
clearly
indicates
that
the
Central
Limit
Theorem
does
not
apply
to
them. But
what
condition
is
being
vio-
lated?
Mandelbrot
decided
that
the
individual
effects
making up a
price
change
did
not
have
finite
variance,
but
were
still
independent.
The
distribution
of
price
change
should
then
belong
to
the
stable
family
3.
Feller
[9J,
Gnedenko
and
Kolmogorov
[10],
and
Loeve
[13J
contain
good
expositions
on
the
conditions
under
which
the
Central
Limit
Theorem
is
satisfied.
4.
Kendall
[llJ
5.
Mandelbrot
[14J
lists
many
references
to
the
problem
of
non-
normality,
one
as
early
as
1915.
Distribution
of
6X
Figure
1
Normal
distribution
with
the
same mean
and
variance
f(6X)
the
daily
changes
in
price
w
6X
