This slide set is a work in progress and is embedded in my Principles of Finance course, which is also a work in progress, that I teach to computer scientists and engineers
http://financefortechies.weebly.com/
2. Learning
Objec-ves
¨ Lognormal
Distribu-ons
¨ Rela-ons
between
¤ Normal
&
lognormal
n Pdfs
n Sta-s-cs
¤ Simple
and
natural
log
rates
2
3. Hypotheses
and
Models
¨ Explana-ons
of
phenomenon
¤ Hypothesis
n A
proposed
explana-on
for
a
phenomena
¤ Law
n Statement
of
a
cause
and
effect
without
explana-on
n Newton’s
law
of
gravity
¤ Theory
n A
well-‐established
explana-on
for
a
phenomenon
n Einstein’s
theory
of
gravity
¨ A
model
is
a
mathema-cal
or
physical
representa-on
of
a
phenomenon
¤ The
“Bohr
atomic
model”
¤ Newton’s
inverse
square
law
of
gravity
¤ Einstein’s
Theory
of
General
Rela-vity
3
2
21
r
mm
GF
⋅
⋅=
5. SPX
Daily
Ln
Rate
Histogram:
More
Zoom
5
Again this histogram includes daily return rates from 1950
<-4.5% should happen less than once in a thousand years,
but there have been 31 such days since 1950 or about once
every two years
-22.9% day should not have happened (Oct 19, 1987)
7. SPX
Daily
Ln
Rate:
Mean
7
-‐350%
-‐250%
-‐150%
-‐50%
50%
150%
250%
1/5/51 11/9/57 9/13/64 7/19/71 5/23/78 3/27/85 1/30/92 12/4/98 10/8/05
Annualized
mean
22
day
annualized
tailing
mean
252
day
annualized
tailing
mean
Long
term
annualized
tailing
mean
8. SPX
Daily
Ln
Rate:
Mean
8
-‐80%
-‐60%
-‐40%
-‐20%
0%
20%
40%
60%
80%
1/2/90 3/12/92 5/21/94 7/29/96 10/7/98 12/15/00 2/23/03 5/3/05 7/12/07 9/19/09
Zoom
in
on
Annualized
mean
252
day
annualized
tailing
mean
Long
term
annualized
tailing
mean
9. 0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
1/3/1950 3/22/1958 6/8/1966 8/25/1974 11/11/1982 1/28/1991 4/16/1999 7/3/2007
SPX
Daily
Ln
Rate:
Standard
Devia-on
9
Annualized
standard
devia-ons
(‘vola-lity’)
22
day
annualized
trailing
vola-lity
252
day
annualized
trailing
vola-lity
Long
term
annualized
trailing
vola-lity
10. 0%
5%
10%
15%
20%
25%
30%
1/2/1990 9/28/1992 6/25/1995 3/21/1998 12/15/2000 9/11/2003 6/7/2006
SPX
Daily
Ln
Rate:
Standard
Devia-on
10
Zoom
in
on
Annualized
standard
devia-ons
(‘vola-lity’)
252
day
annualized
trailing
vola-lity
Long
term
annualized
trailing
vola-lity
12. SPX
Daily
Ln
Rate:
Autocorrelogram
12
Natural
log
daily
return
rates
for
SPX,
v
1950
–
2011
15471
days
Rates
do
look
rather
uncorrelated
13. SPX
Daily
Ln
Rate:
Autocorrelogram
13
Natural
log
daily
de-‐trended
squares
of
return
rates
(variance)
for
SPX,
(v-‐u)2
1950
–
2011
15471
days
There
is
some
posi-ve
autocorrela-on
(persistence)
Might
even
be
greater
persistence
over
shorter
periods
14. SPX
Daily
Ln
Rate:
Histogram
of
Annualized
Daily
Variance
14
Histogram
of
Annualized
Daily
Variance
15. SPX:
Annual
Accumula-on
of
Daily
Returns
15
10,000
annual
sums
of
252
day
(1
year)
con-guous
return
rates
randomly
selected
from
1950
to
2011
This
histogram
doesn’t
look
normal
at
all
as
the
addi-ve
CLT
would
indicate
So
the
rates
are
not
IID/
FV
16. SPX:
Ln
Rate
Q-‐Q
Plot
16
A
Q-‐Q
plot
compares
the
measured
rates
to
ideal
normal
rates
from
measured
mean
and
variance
17. Natural
Log
Rate
–
More
Tests
¨ Jarque
Bera
normality
test
¤ JB
is
a
Chi
Squared
sta-s-c
with
2
dof
¤ Normality
via
chi
squared
considera-on
of
s
skew,
S,
and
kurtosis,
K,
the
3rd
and
4th
moments
of
distribu-on
which
measure
asymmetry
17
440,657
4
3)(29.0600
1.05621
6
15471
4
3)(K
S
6
n
JB
2
2
2
2
=
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛ −
+−=
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛ −
+=
JB
(χ2
statistic)
If
normality
is
rejected,
what
is
the
probability
of
a
rejection
error
0.0000 100.00%
4.6051 10.00%
5.9914 5.00%
9.2103 1.00%
10.0000 0.67%
15.0000 0.06%
20.0000 0.00%
25.0000 0.00%
30.0000 0.00%
35.0000 0.00%
40.0000 0.00%
45.0000 0.00%
50.0000 0.00%
So
there
is
~0%
probability
of
incorrectly
rejec-ng
the
normal
hypothesis
19. Stock
Return
Rate
Summary
¨ Historical
stock
return
rates,
r
and
v,
are
characterized
by
¤ Leptokurtosis
n Fat
or
heavy
tails:
more
extreme
events
than
‘normal’
n More
return
rates
near
the
mean
than
‘normal’
¤ Nega-ve
skew
n More
extreme
downside
events
than
upside
¨ Dependence
in
return
rate
vola-lity
¤ Rate
vola-lity
clustering,
short
term
persistence
then
reversion
to
mean
¨ Less
frequent
sampling
e.g.,
weekly
and
monthly
would
show
some
smoothing,
but
s-ll
not
normal
¤ However,
quarterly
or
annual
sampling
would
ignore
important
rate
of
return
informa-on
19
20. Lognormal
Pdf
20
The
lognormal
pdf
is
asymmetric,
is
not
nega-ve,
over
-me
the
mean,
mode,
and
median
drii
further
apart,
and
the
distribu-on
skews
more
posi-vely.
21. Lognormal
Pdf
21
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0.018
Mode
Median
Mean
(expected)
22. 22
( )
( )
∞>>
⋅⋅
=
−
⋅
−
−
x
0
e
π2σx
1
σ
μ,
|x
f
1s,uNL~r
σ2
μ)(lnx
x
2
2
2
( )
( )
∞>>∞
⋅
= ⋅
−
−
x
-‐
e
π2σ
1
σ
μ,
|x
f
s,uN~v
σ2
μ)(x
x
2
2
2
u
is
mean,
median,
and
mode
The
parameters
is
the
normal
pdf
above
are
also
the
sta-s-cs
–
the
mean
and
variance
The
mean,
mode,
and
median
are
all
different
The
parameters
is
the
lognormal
pdf
are
the
same
as
for
the
normal
pdf,
but
they
are
not
the
sta-s-cs,
not
the
mean
or
variance
23.
¨ Why
simple
returns
can’t
really
be
normal
¤ Simple
returns
are
compounded
over
-me
increments,
but
normal
random
variables
are
mul-plied
¤ (1+r)n
¤ u·∙n
23
( )r1lnv +=
...
3
r
2
r
r)r1ln(v
)1x(
...
3
x
2
x
x)x1ln(
32
32
−+−=+=
−≠−+−=+
24. Variance
of
Simple
and
Log
Returns
24
[ ] [ ]
[ ] [ ]
( )
( )
( )
[ ]( ) ( )
1er1E
1ee
1ee
1ee
ee
ee
xExE
dr1VarrVar
2
2
2
2
2
22
22
222
s2
s
2
2
s
u
s2
s
u2
ssu2
s2us2u2
2
2
s
u
2
s2
u2
22
2
−⋅+=
−⋅
⎟⎟
⎟
⎠
⎞
⎜⎜
⎜
⎝
⎛
=
−⋅=
−=
−=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−=
−=
=+=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+⋅
+⋅
+⋅+⋅
+
⋅
+⋅
[ ]( ) ( )
( ) ( )
( )
( )
( )
1)s
1,(a
sd
1ed
d1e
d1lns
1)(a
1
a1
a1
d
1lns
1ea1
1er1E
d
22
s2
2s
22
2
2
2
2
s2
s22
2
2
2
2
<<<<≈≈
−≈
+≈
+≈
<<≈+
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
+
+=
−⋅+=
−⋅+=
( )
[ ]
[ ]
[ ]
[ ] 2
2
22
s2u22
2
s
u
2
sk
uk
k
2
v
e
xE
e
xE
e
xE
su,NL~
er1X
⋅+⋅
+
⋅
+⋅
=
=
=
=+=
25. Variance
of
Simple
and
Log
Returns
25
Future
Value
Factor:
1+r
=
ev
[ ] ( )
1eer1var
22
ssu2
−⋅=+ +⋅
[ ]
*
2
u2
s
u
eer1E ≡=+
+
[ ] u
er1M =+
26. 26
( ) ( )
[ ]
[ ] [ ]
[ ] ( )[ ]
[ ] ondistributi
normal
log
for
Median
er1MM[x]
ondistributi
lognormal
for
moment
2
er1E
xE
ondistributi
normal
log
for
(mean)
moment
1
er1E
xE
ondistributi
normal
log
for
moment
k
exE
su,NL~
er1
u
nds2u222
st2
s
u
th2
sk
uk
k
2v
2
2
22
=+=
=+=
=+=
=
=+
⋅+⋅
+
⋅
+⋅
27. Central
Limit
Theorem
27
( )
( )2n
n
n
1i
i
n
1i
i0n
s,uN~
n
y
u
n
y
n
v
vSln)Sln()Sln(
→=
=Δ=−
∑
∑
=
=
( )
1)x,x(NL~r
g1f)r(1
fr1
S
S
n
1
n
n
1
n
1i
i
n
n
1i
i
0
n
−
+→=⎥
⎦
⎤
⎢
⎣
⎡
+
≡+=
∏
∏
=
=
Assume
that
n
is
large
and
r
and
v
are
IID/FV
28. 28
( ) ( )
( )
∑∏
∑∏
∑∏
==
==
==
=⎥
⎦
⎤
⎢
⎣
⎡
+=⎥
⎦
⎤
⎢
⎣
⎡
+=⎥
⎦
⎤
⎢
⎣
⎡
+
n
1i
i
n
1i
v
n
1i
i
n
1i
v
n
1i
i
n
1i
i
veln
r1lneln
r1lnr1ln
i
i
n21
i
vvv
0
n
1i
v
0n
n210
n
1i
i0n
e
...e
e
S
eS
S
)r(1....)r(1)r(1
S
)r(1S
S
⋅⋅⋅⋅=
⋅=
+⋅⋅+⋅+⋅=
+⋅=
∏
∏
=
=
( ) ( )
( )
( ) ( )
( ) n210
n
1i
i0n
n210
n
1i
i0n
v...vvSln
vSln
Sln
)rln(1....)rln(1)rln(1Sln
)rln(1Sln
Sln
++++=
+=
+++++++=
++=
∑
∑
=
=
29. Mean
Natural
Log
Return
Rates
29
Example:
v
is
distributed
uniformly
from
-‐10%
to
+20%
Average
of
sums
of
vi
are
normal
(sum
of
n
rates)
( )2
n
1i
i
s,uN~
n
v∑=
30. Simple
Future
Value
Factors
30
Example:
r
is
distributed
uniformly
from
-‐10%
to
+20%
)x,x(NL~)r(1f
n
1
n
1i
i
n
1
n ⎥
⎦
⎤
⎢
⎣
⎡
+= ∏=
31. Simple
Future
Value
Factors
31
( ) )x,x(NL~r1f
n
1i
in ∏=
+=
Example:
r
is
distributed
uniformly
from
-‐10%
to
+20%
f
is
distributed
lognormal
32. We
did
plot
a
histogram
of
natural
return
rates,
v,
for
the
SPX.
It
did
have
the
general
appearance
of
normality.
But
a
Levy
stable
seems
like
a
bemer
fit,
but
has
disadvantages.
However,
the
typical
assump-on
in
finance
is
that
v
is
normally
distributed
which
has
a
number
of
advantages.
One
advantage
is
that
the
loca-on
sta-s-cs
are
iden-cal
–
mode,
median,
and
mean
–
it’s
a
symmetric
32
v)r1(ln
e)r1(
v)r1(ln
e)r1(
v
ii
v
i
i
=+=+
=+=+
For
stocks
or
other
financial
assets,
so
far
there
has
been
no
assump-on
on
the
distribu-ons
of
v
and
r
other
than
being
IID/FV
But
the
rela-onship
between
r
and
v
has
been
defined
as
( ) ( )2
s,uN~r1lnv +=
33. 33
( ) ( )
( ) ( ) ( )
( ) ( ) 1s,uNL1e~r
s,uNLe~r1
s,uN~r1lnv
2s,uN
2s,uN
2
2
2
−≡−
≡+
+=
Another
advantage
is
the
normal
distribu-on
scale
linearly
in
-me.
The
mean
driis
to
the
right
while
the
variance
increases.
( )2
sn,unN~vn ⋅⋅⋅
Another
advantage
is
the
normal
distribu-on
scale
linearly
in
-me.
The
mean
driis
to
the
right
while
the
variance
increases.
Yet
another
advantage
is
the
rela-on
between
the
normal
and
lognormal
distribu-ons
is
similar
to
the
rela-on
between
the
na-ral
log
rate
and
simple
rate
Therefore
the
simple
rate,
r,
is
lognormal
under
assump-on
that
the
natural
log
rate
is
lognormal
36. Natural
Log
Rate
Autocorrelogram
36
Natural
log
daily
absolute
return
rates
for
SPX,
|v|
Daily
range
1950
–
2011
15471
days
37. Common
PDFs
in
Finance
¨ Gaussian
/
Normal
¤ IID
/
FV,
two
parameters
¤ CLT
for
sums
of
IID/FV
random
variables
¤ Special
case
Levy
stable
and
ellip-c
distribu-ons
¨ Ellip-c
¤ IID
/
FV,
two
parameters
¤ unimodal,
no
skew,
no
kurtosis
other
than
Gaussian
case
¤ Linear
correla-on
defines
linear
dependence
¤ Used
in
MPT
and
CAPM
¤ Includes
Gauss,
Cauchy,
t-‐distr,
Laplace,
symmetric
Levy
Stable
¨ Lognormal
¤ IID
/
FV
¤ CLT
for
products
of
IID/FV
random
variables
¤ Posi-ve
¤ Mode,
median,
mean
non-‐coincident
¨ Levy
stable
¤ IID,
not
generally
FV,
4
parameters
¤ Unimodal,
skew,
kurtosis
other
than
Gaussian
case
¤ Central
limit
theorem
for
IID
and
stable
but
not
FV
random
variables
converges
to
a
Levy
stable
distribu-on
¤ Includes
Gaussian,
Cauchy,
Levy
37
38. More
on
Covar
&
Corre
38
[ ] [ ] [ ] [ ]
yExEyxEy,xCov ⋅−⋅=
39. Monthly
Idealized
PDFs
From
SPX
History
39
ln(1+r)
=
v
Normally
distributed
N(u,s2)
u
and
s2
are
normal
pdf
parameters
and
sta-s-cs
-‐
mean
and
variance
(1+r)=
ev
Lognormally
distributed
NL(u,s2)
Same
pdf
parameters,
but
different
mean
and
variance
40. Monthly
Idealized
PDFs
From
SPX
History
40
Future
value
factor,
(1+r)
=
ev
lei
shiied
by
-‐1,
r
41. Return
Rate
PDFs:
Sta-s-cs
Increase
With
Time
41
-‐75% -‐50% -‐25% 0% 25% 50% 75% 100% 125% 150% 175% 200% 225% 250% 275% 300%
Natural
log
rates
(v)
are
assumed
normal.
The
mean
and
variance
of
a
normal
distribu-on
scale
linear
in
-me
The
future
value
factors
(1+r)
are
assumed
log
normally
distributed.
The
mean
and
variance
do
not
scale
linearly
in
-me.
42. 42
Future
Value
Factor:
1+r
=
ev
[ ] ( )
1eer1var
22
ssu2
−⋅=+ +⋅
[ ]
*
2
u2
s
u
eer1E ≡=+
+
[ ] u
er1M =+
43. 43
[ ] [ ] [ ] [ ]
( )
( )
( )
[ ]( ) ( )
1er1E
1ee
1ee
ondistributi
normal
logof
Variance
1ee
ee
ee
xExEr1VarrVar
2
2
2
2
2
22
22
222
s2
s
2
2
s
u
s2
s
u2
ssu2
s2us2u2
2
2
s
u
2
s2
u2
22
−⋅+=
−⋅
⎟⎟
⎟
⎠
⎞
⎜⎜
⎜
⎝
⎛
=
−⋅=
−=
−=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−=
−=+=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+⋅
+⋅
+⋅+⋅
+
⋅
+⋅
46. Lognormal
Distribu-on
46
( ) ( )
[ ]
[ ] [ ]
[ ] ( )[ ]
[ ] ondistributi
normal
log
for
Median
er1MM[x]
ondistributi
lognormal
for
moment
2
er1E
xE
ondistributi
normal
log
for
(mean)
moment
1
er1E
xE
ondistributi
normal
log
for
moment
k
exE
su,NL~
er1
u
nds2u222
st2
s
u
th2
sk
uk
k
2v
2
2
22
=+=
=+=
=+=
=
=+
⋅+⋅
+
⋅
+⋅
( ) ( )
( )
( ) ( )
( ) n210
n
1i
i0n
n210
n
1i
i0n
v...vvSln
vSln
Sln
)rln(1....)rln(1)rln(1Sln
)rln(1Sln
Sln
++++=
+=
+++++++=
++=
∑
∑
=
=
S
SS
r
)r1(SS
rate
return
Simple
1i
1ii
i
i1ii
−
−
−
−
=
+⋅=
( ) ( )
)r1ln(
SlnSln
S
S
lnv
eSS
rate
return
log
Natural
i
1ii
1i
i
i
v
1ii
i
+=
−=⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
=
⋅=
−
−
−
47. Lognormal
Distribu-on
47
( ) ( )
[ ]
[ ] [ ]
[ ] ( )[ ]
[ ] ondistributi
normal
log
for
Median
er1MM[x]
ondistributi
lognormal
for
moment
2
er1E
xE
ondistributi
normal
log
for
(mean)
moment
1
er1E
xE
ondistributi
normal
log
for
moment
k
exE
su,NL~
er1
u
nds2u222
st2
s
u
th2
sk
uk
k
2v
2
2
22
=+=
=+=
=+=
=
=+
⋅+⋅
+
⋅
+⋅
48. GARCH
Time
Series
¨ Similar
to
historic
vola-lity
¤ Simple
condi-onal
dependence
in
the
second
moment
(vola-lity)
n Vola-lity
clustering
or
persistence
¨ The
GARCH
vola-lity
has
three
contribu-ons
¤ Long
term
average
vola-lity,
s2,
so
there’s
a
reversion
of
the
mean
¤ Short
term
dependence
on
recent
square
of
return
rate,
v2
¤ Short
term
dependence
on
recent
Garch
vola-lity,
h
¨ To
Do
n Is
there
a
probability
distribu-on?
Maybe
not
n Plot
the
resul-ng
rates
and
look
for
fat
tails
n So
it
looks
good
historically,
but
how
can
it
be
used
in
decision
making
?
48
49. GARCH
Time
Series
¨ The
GARCH(1,1)
vola-lity
model
with
the
natural
log
rate
process
model
vola-lity
has
three
contribu-ons
49
( )
0βλ,α,
1βαγ
β
,α
,γ
:weights
hβvαsβα1
hβvαsγh
zh
uv
1i
2
1i
2
1i
2
1i
2
i
iii
>
=++
⋅+⋅+⋅−−=
⋅+⋅+⋅=
⋅+=
−−
−−
The
Gaussian
rate
process
is
vi
=
u
+
s
·∙zi
s
is
the
(tradi-onal)
long
term
average
standard
devia-on
z
is
the
standard
normal
random
variable
h
is
the
Garch
variance
v
is
the
nat
log
return
rates
Example:
α = .85 , β = .1 , γ = .05
50. GARCH
Time
Series
50
Single
simulated
GARCH(1,1)
vola-lity
for
15,461
days
53. Adendum:
Nat
Log
&
Exp
53
( )
( ) ( )
y+xyxyxyx
32
32
x
)xln(
e
=
e
e
)(e
=
e
...
3
1x
2
1x
1x)xln(
)1x(
...
3
x
2
x
x)x1ln(
x
1
)xln(
dx
d
)yln()xln(
y
x
ln
)yln()xln()yxln(
x)eln(
)0x(
xe
⋅
−
−
+
−
−−=
−≠−+−=+
=
−=⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
+=⋅
=
>=
⋅
54. Addendum
54
Rate
Periodic
mean
Annual
mean
Periodic
standard
deviation
Annual
standard
deviation
a α
g γ
vi u µ s σ
d
=
Var(r)
=
Var(1+r)
ri d δ
55. Addendum
55
dwσdt
2
σ
μ
dln(S)
2
*
⋅+⋅⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−=( )
( )
Tσ
Tσ.5r
K
S
ln
d
Tσ
Tσ.5r
K
S
ln
d
2*0
2
2*0
1
⋅
⋅⋅−+⎟
⎠
⎞
⎜
⎝
⎛
=
⋅
⋅⋅++⎟
⎠
⎞
⎜
⎝
⎛
=
[ ] ( )
1eer1var
22
ssu22
−⋅=+=δ +⋅
[ ]
*
2
u2
s
u
eer1E ≡=+
+
( ) ( )
[ ]
[ ]
tsz
1ii
s,0N
1i
i
v
1ii
2
i
i1i-‐i
eSS
e~
S
S
eSS
s,0N~v
vSln
Sln
2
i
Δ⋅⋅
−
−
−
⋅=
⋅=
+=
tsz
i
i
tsz
i1ii
i
)rln(1v
ii
ii
i
ii
er
)r(1e
)r(1SS
)r(1ee
tszv
)rln(1
v
Δ⋅⋅
Δ⋅⋅
−
+
=
+=
+⋅=
+==
Δ⋅⋅=
+=
56. Addendum
56
( )
[ ] [ ]
( )
( ) ( )
−+−=+
+
Δ⋅⋅
+
Δ⋅⋅
+Δ⋅⋅+==
−+−=+
++++=
=⋅δ⋅+
=+≡δ
⋅δ⋅+⋅=
⋅δ⋅==
Δ⋅⋅
Δ⋅⋅
3
r
2
r
r)r1ln(
...
6
tsz
2
tsz
tsz1ee
3
x
2
x
x)x1ln(
6
x
2
x
x1e
e
ΔtZ1
rSDr)(1SD
ΔtZ1SS
ΔtZ?r
3
i
2
i
ii
3
i
2
i
i
tszv
32
32
x
tsz
i
itt
ii
ii
1-‐ii
Not
yet
ready
to
related
normal
and
lognormal
distribu-ons.
Need
lognormal
sta-s-cs
and
Ito’s
Lemma
Normal
Natural
log
rates
Natural
log
prices
Lognormal
Simple
rates
Future
value
factors
Prices
57. Levy
Stable
Distribu-on
¤ Bemer
fits
historical
rates
of
return
n Can
model
Leptokurtosis
and
skew
n Constant
parameters
n Generalized
Central
Limit
Theorem
n Normal
distribu-on
is
a
special
case
n Problems
included
n Infinite
variance
n Variance
cant
be
used
as
a
measure
of
risk
or
vola-lity
n CAPM,
MPT,
B-‐S
n PDF
models
not
applicable
n Generally
no
analy-c
representa-on
¤ To
Do
n Fit
data
to
a
distribu-on
and
graph
n Why
does
FMH
without
IID
invoke
this
model
n How
does
it
relate
to
power
law
model
(Has
an
α
>
2
?)
57
61. Power
Law
Method
¨ Coopera-on,
herding,
cri-cality
¤ How
Nature
Works
–
Bak
¤ Ubiquity
–
Buchanan
¨ The
ubiquity
of
scale-‐free
behavior
and
self-‐organiza-on
in
Nature
led
Bak,
Tang
and
Wiesenfeld
(BTW)
to
coin
the
term
Self-‐Organized
Cri-cality
(SOC)
to
explain
the
emergence
of
complexity
in
dynamical
systems
with
many
interac-ng
degrees
of
freedom
without
the
presence
of
any
external
agent
;
SOC
was
devised
to
be
a
sort
of
supergeneral
theory
of
complexity.
61
62. Power
Law
¨ Confusion
based
on
Fractal
Market
Hypothesis:
Is
it
stable
or
power
law??
¨ Hurst
soiware
shows
a
random
series
to
be
persistent
??
¨ Hurst
exponent
¤ 0.5
is
Brownian
t1/2
√t
¤ 0
<
H
<
0.5
:
an--‐persistent,
mean
rever-ng
¤ .5
<
H
≤
1.0
:
persistent
¨ Stability
parameter
¤ α
=
1
/
H,
example
Gaussian:
α =
2,
H
=
.5
¨ Correla-on
(?)
C
=
22H-‐1
–
1
¨ Example
¤ SPX:
3/1/1950
–
5/27/2005
daily
n
α
=
1.6735
β
=
.1064
µ
=
-‐.0002
σ
=
.0049
n
H
=
.5976
C=
.1448
¤ SPX:
1/3/1950
–
6/24/2011
daily
n H=
.562
α=
1.779
C=
.090
62
63. Reference:
Nat
Log
&
Exp
63
)1x(
...
3
x
2
x
x)x1ln(
x
1
)xln(
dx
d
)yln()xln(
y
x
ln
)yln()xln()yxln(
x)eln(
)0x(
xe
32
x
)xln(
−≠−+−=+
=
−=⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
+=⋅
=
>= ( )
( ) ( )
++++=
⋅
−
−
+
−
−−=
⋅
6
x
2
x
x1e
e
=
e
e
)(e
=
e
...
3
1x
2
1x
1x)xln(
32
x
y+xyxyxyx
32
Rate
Periodic
mean
Annual
mean
Annual
standard
deviation
Period
standard
deviation
Rate
pdf
a α
g γ
vi u µ s σ Normal
d
=
SD(r)
=
SD(1+r)
ri d δ Log
normal
64. Related
Concepts
¨ Expected
Rate
of
Return
On
Equity
¤ CAPM
requires
that
the
return
rate
is
normally
distributed
with
a
trend
¤ Ordinary
least
squares
¨ Theore-cal
basis
for
r
being
an
independent
random
variable
¤ Efficient
Market
Hypothesis
¨ Theore-cal
basis
for
r
being
an
independent
random
variable
with
a
trend
¤ Ra-onal
Market
Hypothesis
64
( ) ( )
( )
( )FMFEE
iE1ii
E1ii
rrβr
k
r
zsr1SS
r1SSE
−⋅+==
⋅++⋅=
+⋅=
−
−
65. Geometric
Brownian
Mo-on
65
( ) ( )
[ ]
[ ]
tsz
1ii
s,0N
1i
i
v
1ii
2
i
i1i-‐i
eSS
e~
S
S
eSS
s,0N~v
vSln
Sln
2
i
Δ⋅⋅
−
−
−
⋅=
⋅=
+=
tsz
i
i
tsz
i1ii
i
)rln(1v
ii
ii
i
ii
er
)r(1e
)r(1SS
)r(1ee
tszv
)rln(1
v
Δ⋅⋅
Δ⋅⋅
−
+
=
+=
+⋅=
+==
Δ⋅⋅=
+=
dwσdt
2
σ
μ
dln(S)
2
*
⋅+⋅⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−=( )
( )
Tσ
Tσ.5r
K
S
ln
d
Tσ
Tσ.5r
K
S
ln
d
2*0
2
2*0
1
⋅
⋅⋅−+⎟
⎠
⎞
⎜
⎝
⎛
=
⋅
⋅⋅++⎟
⎠
⎞
⎜
⎝
⎛
=
[ ] ( )
1eer1var
22
ssu22
−⋅=+=δ +⋅
[ ]
*
2
u2
s
u
eer1E ≡=+
+
66. Geometric
Brownian
Mo-on
66
( )
[ ] [ ]
( )
( ) ( ) ...
6
tsz
2
tsz
tsz1ee
3
x
2
x
x)x1ln(
6
x
2
x
x1e
e
ΔtZ1
rSDr)(1SD
ΔtZ1SS
ΔtZ?r
3
i
2
i
i
tszv
32
32
x
tsz
i
itt
ii
ii
1-‐ii
+
Δ⋅⋅
+
Δ⋅⋅
+Δ⋅⋅+==
−+−=+
++++=
=⋅δ⋅+
=+≡δ
⋅δ⋅+⋅=
⋅δ⋅==
Δ⋅⋅
Δ⋅⋅
Not
yet
ready
to
related
normal
and
lognormal
distribu-ons.
Need
lognormal
sta-s-cs
and
Ito’s
Lemma
Normal
Natural
log
rates
Natural
log
prices
Lognormal
Simple
rates
Future
value
factors
Prices
u,
s
µ, σ
r,
d
α, δ
g,
γ,
67. Alterna-ves
¨ Fat
Tail
Models
¤ Power
law
not
exponen-al
tails
¤ Leptokurtosis,
finite
variance
?
¤ Examples
n Student
t
–
no
skew
n Levy
stable
–
skew
¨ Non
IID
Models
–
non-‐sta-onary
process
¤ Correla-on
in
rate
vola-lity,
but
not
in
rate,
so
s-ll
‘unpredictable’
ARCH
models
¤ Used
with
normal
or
other
distribu-on
67
69. 69
[ ]
[ ]
[ ]
[ ]2
εii1i-‐i
2
εii1i-‐i
1i-‐i
2
εii1i-‐i
s0,N~ε
εSS
s0,IID~ε
εSS
SSE
s0,~ε
εSS
+=
+=
=
+=
( ) ( )
[ ]
( ) ( )
( )[ ] ( )
Δtσz
tt
tt
1i-‐1-‐ni
i
1ii
1ii
SlnSlnE
tszSlnSln
1,0N~z
nszSlnSln
⋅⋅
+
=
Δ⋅⋅+=
⋅⋅+=
−
−
Generally
Rate
Periodic
mean
Annual
mean
Annual
standard
deviation
Period
standard
deviation
Rate
pdf
a α
g γ
v u µ s σ Normal
d
=
SD(r)
=
SD(1+r)
r d δ Log
normal
70. 70
[ ]
[ ]
tΔBzSS
tΔB
,0N~ε
tttΔ
tΔ
,0N~ε
εSS
i1-‐ii
i
ii1-‐ii
ttt
2
t
1i-‐itttt
⋅⋅+=
⋅
−=+=
[ ]
[ ]
mszum
1i1mi
sm,umN
m
1i
v
2
m
1i
i
i
2
i
eSS
e~e
sm,umN~v
increment
d,multiperio
⋅⋅+⋅
−−+
⋅⋅
=
=
⋅=
⋅⋅
∏
∑
[ ] [ ]
it
1i
i
1ii
eS
eSS
tzt
,N~sm,umN~
t
tzt
tt
tt
22
t
µ
Δ⋅σ⋅+Δ⋅µ
⋅=
⋅=
Δ⋅σ⋅+Δ⋅µ=µ
σµ⋅⋅µ
−
−
( )
ΔtσzΔtμ
S
S
ΔtσzΔtμ1SS
t
*
i
*
tt 1-‐ii
⋅⋅+⋅=
Δ
⋅⋅+⋅+⋅=
ΔtzΔw
tttΔ
SSSΔ
1i-‐i
tt 1-‐ii
⋅=
−=
−=
71. 71
( )
1)r(1g
1fg
)r(1f
f
S
S
)r(1
S
S
n
1
n
1i
in
n
1
nn
n
1i
in
n
0
n
n
1i
i
0
n
−⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
+=
−=
+=
=
+=
∏
∏
∏
=
=
=
n
s
u
)r(1lns
v
S
S
ln
n
n
n
1i
in
n
1i
i
0
n
=
+=
=⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
∑
∑
=
=
75. Power
Law
¨ Power
law
with
rescaled
range
¨ Many
natural
phenomena
modeled
with
power
law
¨ Nonlinear
feedback
¨ Hurst
exponent
is
the
slope
¨ Fractal
and
self
similar
¨ Complexity
¨ How
can
it
be
used
in
decision
making?
¨ The
rescaled
range
follows
a
power
law
75
76. 76
[ ]
[ ]
tΔBzSS
tΔB
,0N~ε
tttΔ
tΔ
,0N~ε
εSS
i1-‐ii
i
ii1-‐ii
ttt
2
t
1i-‐itttt
⋅⋅+=
⋅
−=+=
Rate based process is Geometric
Brownian Motion
(GBM)
[ ]
[ ]
mszum
1i1mi
sm,umN
m
1i
v
2
m
1i
i
i
2
i
eSS
e~e
sm,umN~v
increment
d,multiperio
⋅⋅+⋅
−−+
⋅⋅
=
=
⋅=
⋅⋅
∏
∑[ ] [ ]
it
1i
i
1ii
eS
eSS
tzt
,N~sm,umN~
t
tzt
tt
tt
22
t
µ
Δ⋅σ⋅+Δ⋅µ
⋅=
⋅=
Δ⋅σ⋅+Δ⋅µ=µ
σµ⋅⋅µ
−
−
( )
ΔtσzΔtμ
S
S
ΔtσzΔtμ1SS
t
*
i
*
tt 1-‐ii
⋅⋅+⋅=
Δ
⋅⋅+⋅+⋅=
ΔtzΔw
tttΔ
SSSΔ
1i-‐i
tt 1-‐ii
⋅=
−=
−=
77. Appendix:
Exponen-als
and
Natural
Logs
77
( )
( )
dx
dy
y
1
dx
ln(y)d
e
dx
dy
dx
ed y
y
⋅=
⋅=
+++++=
⎟
⎠
⎞
⎜
⎝
⎛
+=
∞→
!4
x
!3
x
!2
x
x1e
n
1
1lime
432
x
n
n
xlndx
X
1
e
a
1
dxe xaxa
=
⋅=
∫
∫
⋅⋅
79. Price
as
a
Stochas-c
Diff
Eqn
79
( )
( )1eSSd
1eSS
eSSS
e
S
SS
e
S
S
eSS
dwtd
tzt
t
tzt
tt
tzt
t
t
tzt
t
t
tzt
tt
i
1i
i
1i1i
i
1i
1i
i
1i
i
i
1ii
−⋅=
−⋅=Δ
⋅=+Δ
=
+Δ
=
⋅=
⋅σ+⋅µ
Δ⋅σ⋅+Δ⋅µ
Δ⋅σ⋅+Δ⋅µ
Δ⋅σ⋅+Δ⋅µ
Δ⋅σ⋅+Δ⋅µ
Δ⋅σ⋅+Δ⋅µ
−
−−
−
−
−
−
( )SfF =
80. 80
( )[ ] ( )[ ] ...
1eS
S
F
2
1
1eS
S
F
dt
t
F
dF
...
dS
S
F
2
1
dS
S
F
dt
t
F
dF
2dwtd
2
2
dwtd
2
2
2
+−⋅
∂
∂
⋅+−⋅
∂
∂
+
∂
∂
=
+
∂
∂
⋅+
∂
∂
+
∂
∂
=
⋅σ+⋅µ⋅σ+⋅µ
( )
( )
( )
dx
dy
y
1
dx
dy
y
1
dx
ln(y)d
e
dx
yd
dx
ed
e
dx
dy
dx
ed
2
y
2
2
2
y2
y
y
⋅=⋅=
⋅=
⋅=
dx
dS
S
1
dx
dS
S
1
dx
d
dx
dS
S
1
dx
ln(S)
d
2
⋅−=⎟
⎠
⎞
⎜
⎝
⎛
⋅
⋅=
( )
n
0
nu
0
tμ
0t
*
*
n*
nnu
n
0
nu
0
)a1(SeSeS]E[S
)a1ln(u
)a1ln(n
n
1
u
)a1(lnnu
)a1(e
)a1(SeS
**
*
*
+⋅=⋅=⋅=
+=
+⋅⋅=
+=⋅
+=
+⋅=⋅
⋅⋅
⋅
⋅
81. ¨ Actually,
they
[power
laws]
aren’t
special
at
all.
They
can
arise
as
natural
consequences
of
aggrega-on
of
high
variance
data.
You
know
from
sta-s-cs
that
the
Central
Limit
Theorem
says
distribu-ons
of
data
with
limited
variability
tend
to
follow
the
Normal
(bell-‐shaped,
or
Gaussian)
curve.
There
is
a
less
well-‐known
version
of
the
theorem
that
shows
aggrega-on
of
high
(or
infinite)
variance
data
leads
to
power
laws.
Thus,
the
bell
curve
is
normal
for
low-‐
variance
data
and
the
power
law
curve
is
normal
for
high-‐variance
data.
In
many
cases,
I
don’t
think
anything
deeper
than
that
is
going
on.
81
