This slide set is a work in progress and is embedded in my Principles of Finance course site (under construction) that I teach to computer scientists and engineers http://awesomefinance.weebly.com/

1. 8/28/14
Dynamic
Equity
Models
Learning
Objec>ves
¨ Simula>on
¤ Daily,
monthly,
annual
sta>s>cal
rela>onships
¨ Lognormal
probability
density
¨ Stochas>c
differen>al
equa>on
¨ Con>nuous
>me
price
process
¨ Exact
solu>on
¨ Price
and
return
probabili>es
in
con>nuous
>me
¨ Probability
basics
for
op>on
deriva>ves
2
More
Simula>on
3
Perform
a
stock
price
simula>on
for
which
current
stock
price,
S0
=
$40.00,
the
expected
monthly
con>nuously
compounded
mean
rate
of
return,
u,
is
1%,
and
the
expected
standard
devia>on,
s,
is
5%.
Perform
the
simula>on
with
daily
>me
increments
for
one
year.
Use
floa>ng
point
>me,
annualized,
μ
and
σ,
sta>s>cs.
Run
the
simula>on
10,000
>mes.
u 12 12.000%
μ = ⋅ =
s 12 17.321%
σ = ⋅ =
.004
years
t 1
Δ = =
252
T =
1.000
years
μ ⋅Δ t
+ z ⋅ σ ⋅ Δ
t
.12 .004
z .17321 .004
S
=
S e
+Δ ⋅
t t t
S
=
S e
+ ⋅
t .004 t
⋅ + ⋅ ⋅
Simula>on:
4
$60
$55
$50
$45
$40
$35
$30
$25
$20
$15
$10
$5
$0
μ ⋅Δ t
+ z ⋅ σ ⋅ Δ
t
.12 .004
z .17321 .004
S
=
S e
+Δ ⋅
t t t
S
=
S e
+ ⋅
t .004 t
⋅ + ⋅ ⋅
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00
Stock
Price
Time
[years]
Dynamic
Equity
Models
1

2. 8/28/14
Simula>on:
5
From
Simulation Daily
Mean
rate:
u 0.04859%
Standard
deviation:
s
1.09460%
-­‐6% -­‐5% -­‐4% -­‐3% -­‐2% -­‐1% 0% 1% 2% 3% 4% 5% 6%
Natural
Log
Daily
Return
Rate
Simula>on:
6
From
simula>on
M[ST] $
45.09
E[ST] $
45.91
Min[ST] $
23.95
Max[ST] $
93.91
From
input
M[S ] S e
μ T
= ⋅
T 0
⋅
$40.00 e
= ⋅
$45.10
=
E[S ] S e
*
μ T
= ⋅
T 0
⋅
$40.00 e
= ⋅
$45.78
.12 ⋅
1.0
.135 ⋅
1.0
The
median
price
is
the
5,000th
in
an
ordered
list
of
10,000
simulated
prices
at
T=1.0
years.
The
expected
price
is
the
average
of
the
10,000
prices.
$20 $25 $30 $35 $40 $45 $50 $55 $60 $65 $70 $75 $80 $85 $90 $95
Stock
Price
At
1
Year
=
Lognormal
PDF
7
The
lognormal
pdf
is
• Asymmetric
• Mode,
median,
and
mean
not
equal
• Never
nega>ve
• Over
>me
the
mode,
median,
and
mean
driZ
further
apart
• Over
>me
the
distribu>on
skews
more
posi>vely
In
the
standard
price
theory
Simple
rates,
future
value
factors,
and
asset
prices
are
distributed
lognormal
Return
Rate
and
Future
Value
Factor
PDFs
8
[ ]
E[v] u
M v
= =
v
~N(u,s2 )
~N( ,
s2 )
μ
[ ]
= [μ]
M
μ
E
[ v ]
u
E [ e v ] e
u*
Me e
=
=
e v ~ e N ( u,s2
)
Me12⋅[ v ] = eμ
E [ e12⋅v ] eμ * =
e12⋅v ~ eN(μ,σ2 )
Dynamic
Equity
Models
2

3. 8/28/14
Exact
Solu>on
The
differen>al
equa>on
for
dln(S)
is
dln(S)
= μ⋅ dt + σ ⋅
dw
The
solu>on
with
ini>al
condi>on
is
ln(S ) = ln(S ) + μ ⋅ t + σ ⋅
dw
t 0
ln(S ) = ln(S ) + μ ⋅ t + z ⋅ σ ⋅
t
t 0 t
μ t z σ t
⎞
⋅ ⋅ + ⋅ = ⎟⎟⎠
~N[μ t, t]
ln S
S
t
0
⎛
⎜⎜⎝
ln S
S
2
t
0
⎞
⋅ σ ⋅ ⎟⎟⎠
⎛
⎜⎜⎝
9
At
>me
t
the
natural
log
of
price
ln(St)
is
distributed
normally
as
ln(S ) ~Nln(S ) μ t,
σ t
t 0
Therefore
[ + ⋅ ⋅
]
[ *
]
N ln(S ) + μ ⋅ t,
σ ⋅
t
[ *
]
0
N μ t,
σ t
S ~ e
t
S ~ S e
t 0
⋅
[ ]
[ ] t
MS =
e
E S e
t
t
t
μ⋅
*
μ ⋅
⋅ ⋅
=
Simula>on
10
( )
[ ]
[ ]
[ ]
2
v N(u,s )
f 1 r e ~ e
2 2
≡ + =
k u k ⋅
s k
2 th
⋅ +
E f e
k
moment
for
f
u s
2
2 st
=
+
E f e
1
moment
for
f
2
=
2 2 u 2 s nd
E f
e
⋅ + ⋅
2
moment
for
f
u
=
M[f] =
e
Median
for
f
Var[f] E[f ] (E[f])
Variance
for
f
2 2
= −
u u s
2
2
*
*
= +
u = ln(1 +
a)
u*
a = e −
1
u = ln(1 +
g)
u = −
g e 1
d2 = Var[r] = Var[f] = E[f2 ] −
(E[f])2
Monthly
Statistics
Specified
Rate
of
return:
u 1.0%
Standard
deviation,
s 5.0%
Annual
frequency,
m
12
Computed
Variance,
s2
0.00250
Expected
rate
of
return,
u* 1.12500%
Expected
first
moment
of
f
1.01131
Expected
second
moment
of
f
1.02532
Simple
mean
rate,
a
1.13135%
Geometric
rate,
g
1.00502%
Simple
standard
deviation,
d 5.05973%
Simula>on
11
k k k
2 2
σ ⋅
[ ]
⋅μ+
σ
2
[ ]
2
[ ]
[ ] [ 2 ] ( [ ])2
E f =
e
E f =
e
μ+
2 2 2
2
E f
e
2
=
⋅μ+ ⋅σ
Var f = E f −
E f
2
σ
2
μ = μ +
ln(1 )
μ = + α
*
e 1
*
*
μ
α = −
ln(1 )
μ = + γ
e μ
1
γ = −
2 Var[ ] Var[f] E[f2 ] (E[f])2
δ = α = = −
Monthly
Statistics
Specified
Rate
of
return:
u 1.0%
Annual
Statistics
Computed
μ 12.00000%
Standard
deviation,
s 5.0% σ 17.32051%
Annual
frequency,
m
12
Computed
Computed
Variance,
s2
0.00250 σ2
0.03000
Expected
rate
of
return,
u* 1.12500%
μ* 13.50000%
Expected
first
moment
of
f
1.01131
1.14454
Expected
second
moment
of
f
1.02532
1.34986
Simple
mean
rate,
a
1.13135% α 14.45368%
Geometric
rate,
g
1.00502% γ 12.74969%
Simple
standard
deviation,
d 5.05973% δ 19.97357%
Simula>on
12
Monthly
Statistics
Annual
Statistics
Daily
Statistics
Computed
Specified
Computed
Rate
of
return:
u 1.0%
μ 12.00000%
μ Δt 0.04762%
Standard
deviation,
s 5.0% σ 17.32051% σ √Δt 1.09109%
Annual
frequency,
m
12 m 252
Computed
Computed
Computed
Variance,
s2
0.00250 σ2
0.03000 σ2 t
0.00012
Expected
rate
of
return,
u* 1.12500%
μ* 13.50000%
μ∗ Δt 0.05357%
Expected
first
moment
of
f
1.01131
1.14454
1.00054
Expected
second
moment
of
f
1.02532
1.34986
1.00119
Simple
mean
rate,
a
1.13135% α 14.45368% 0.05359%
Geometric
rate,
g
1.00502% γ 12.74969% 0.04763%
Simple
standard
deviation,
d 5.05973% δ 19.97357% 1.09171%
Dynamic
Equity
Models
3

4. 8/28/14
Daily
Sta>s>cs
13
m
( )
E[S ] $40 1 a $45.78
T
= ⋅ + =
a .05357%
* *
u m u 252
=
E[S ] = S ⋅ e ⋅ = $40 ⋅ e ⋅
=
$45.78
T 0
*
u .05356%
um u 252
=
M[S ] = S ⋅ e ⋅ = 40 ⋅ e ⋅
=
$45.10
T 0
u .04762%
252
( )
=
M[S ] $40 1 g $45.10
T
= ⋅ + =
g =
.04763%
Price
as
a
Stochas>c
Diff
Eqn
14
Difference
eqn
for
price
as
geometric
Brownian
mo>on
with
posi>ve
expected
rate
of
return
S S St t t
= ⋅
Δ = − +Δ
ΔS * = ⋅ + ⋅ Δw z Δt
μ Δt σ Δw
S
Transform
to
a
differen>al
eqn
as
Δt
-­‐>
dt
with
the
goal
to
solve
the
eqn
for
price,
S
dS S μ* dt S σ dw = ⋅ ⋅ + ⋅ ⋅ dw= z ⋅ dt
μ:
con>nuously
compounded
natural
log
mean
rate
of
return
μ*:
con>nuously
compounded
simple
mean
or
expected
rate
of
return
To
understand
stochas>c
differen>al,
dS,
introduce
F
which
is
a
func>on
of
stochas>c
process,
S.
S
is
dependent
on
Weiner
process,
w.
F = f(S)
Stochas>c
Differen>al,
dF
15
Write
dF
as
a
Taylor
series
expansion
2
dF F F
2
dS
higher
order
terms
S
dt F
t
dS 1
S
+ ⋅
2
2
+
∂
∂
∂
+
∂
∂
=
∂
Subs>tute
dS
into
dF
F 1
∂
F
⋅ ⋅ + ⋅ ⋅
* (μ S dt σ S dw)
* 2
2
2
S
⋅ ⋅ + ⋅ ⋅ + ⋅
2
(μ S dt σ S dw)
F
∂
+
S
dt
t
dF
∂
∂
∂
=
∂
Ignore
dt2
and
dw·∙dt
terms
and
subs>tute
dw2
=
dt
which
will
be
explained
on
the
next
slide.
2
dF F 2 2
σ S ⎞
dt F
∂
+ ⋅ ⎟⎟⎠
∂
∂
* ⋅ ⋅
σ S dw
S
F
S
1
+ ⋅
∂
2
μ S F
S
t
2
∂
⎜⎜⎝⎛
⋅ ⋅
∂
∂
⋅ ⋅ +
∂
=
Stochas>c
Differen>al,
dF
16
Determine:
E[dW],
E[dW2],
VAR[dW],
VAR[dW2]
to
resolve
dw2
=
dt
E[dw]= E[z ⋅ dt]= dt ⋅E[z]= 0 E[dw2 ] E[(z dt)2 ] dt E[z2 ] dt = ⋅ = ⋅ =
= − [( ) ] ( [ ])
[ 2 ] ( [ ])
2
[ ]
dt E[z ] dt 1 dt
VAR(dw) E dw E dw
2
E z dt 0
= ⋅ −
2
= ⋅ = ⋅ =
2 2 2 2 2
VAR
(dw ) = E dw −
E dw
[ ] ( )
[ ]
dt 3 dt 0
4 2 2
E z dt dt
= ⋅ −
2 4 2
dt E z dt
= ⋅ −
2 2
= ⋅ − =
dw
∼
N(0,dt)
dw2
∼
N(dt,0)
Stochas>c
Determinis>c
Dynamic
Equity
Models
4

5. 8/28/14
Probability
Distribu>ons
Related
to
dw
and
dw2
17
Z
distribu>on
-­‐4 -­‐3 -­‐2 -­‐1 0 1 2 3 4
Z2
distribu>on
0 1 2 3 4 5 6 7 8 9 10
Z4
distribu>on
0 5 10 15 20 25
Solve
For
Price
.
18
2
Price
differen>al
eqn
dF F 2 2
μ S F
S
⎞
∂
+ ⋅ ⎟⎟⎠
∂
∂
* ⋅ ⋅
∂
⋅ ⋅ +
∂
This
differen>al
equa>on
cannot
be
solved
analy>cally,
but
can
be
solved
under
a
change
of
variable,
S.
Ln(S)
can
be
solved
for
⎛
⎜⎜⎝
∂
1
+ ⋅
∂
t
ln(S)
σ S dt F
F
S
1
2
2
⋅ ⋅
∂
μ *
S ∂
lnS
⋅ +
S
∂
μ S 0 1
S
t
+ ⋅
∂
( 1
2
F =ln(S)
=
= ⎛ ⋅ ⋅ + + − ⋅
S
2
dt σ dw
*
μ σ
2
2
⎜⎝
⎛
= *
−
⎞
⋅ + ⋅ ⎟⎟⎠
⎜⎜⎝
μ dt σ dw
σ S dw
S
2 2
⎞
∂
+ ⋅ ⎟⎟⎠
σ S dt
1
σ S dw
S
2
∂
ln(S)
S
1
2
∂
∂
2 2
2
⎞
)σ S dt
σ S dw
ln(S)
S
⎜⎜⎝⎛
=
dln(S)
= ⋅ + ⋅
⋅
⋅ ⋅ + ⋅ ⎟⎠
⋅ ⋅
∂
Solu>on
For
Price
19
μ t
z σ t
S
S e
*
⋅Δ + ⋅ ⋅ Δ
=
+Δ ⋅
μ ⋅ t
+ z ⋅ σ ⋅
t
[ ]
t t
S
S e
t
⋅
=
S
~ S e
t
[ ]
t
μ σ
2
2
*
N μ ⋅ t
,
σ ⋅
t
⋅
E S = S ⋅ e = S ⋅
e
0
*
μ t
0
0
t 0
t
⎞
⋅ ⎟⎟
⎠
⎛
⎜⎜
⎝
+
⋅
ln(S ) = ln(S ) + μ ⋅Δ t + z ⋅ σ ⋅ Δ
t
t +Δ
t t
ln(S ) = ln(S ) + μ ⋅ t + z ⋅ σ ⋅
t
t 0
μ t z σ t
⎞
⋅ ⋅ + ⋅ = ⎟⎟⎠
[ ]
t
μ σ
2
2
~Nμ t, t
ln S
S
⎛
⎜⎜⎝
ln S
S
⎞
⋅ σ ⋅ ⎟⎟⎠
⎛
⎜⎜⎝
M[S ] = S ⋅ e = S ⋅
e
0
μ t
t
0
t
0
t 0
2
*
⎞
⋅ ⎟⎟
⎠
⎛
⎜⎜
⎝
−
⋅
Log
and
Expecta>on
Operators
20
Note
nonlinearity
of
expecta>on
and
natural
log
Start
with
natural
log
of
price,
Start
with
price
expecta>on,
then
take
expected
value
then
take
natural
log
⎛ [ ]
μ t z σ t
⎞
⋅ ⋅ + ⋅ = ⎟⎟⎠
μ t
ln S
S
t
0
⎜⎜⎝
⎡
E ln S
S
t
0
t
⎤
⋅ = ⎥⎦
⎢⎣
⎞
⎟⎟⎠
⎛
⎜⎜⎝
E[ln(S )] = ln(S ) + μ ⋅
t
t 0
E S S e
*
μ t
= ⋅
t 0
*
⎤
t μ t
e
E S
S
0
= ⎥⎦
⎡
⎢⎣
⋅
⋅
ln(E[S ]) ln(S ) μ *
t
t
= + ⋅
0
( [ ]) *
[ ]
ln E S + μ ⋅ t = E ln(S ) + μ ⋅
t
t
t
( [ ]) [ ] ( *
)
ln E S − E ln(S ) = μ -­‐μ ⋅
t
t t
ln(E[S ])
E[ln(S )]
2
σ ⋅
t
2
>
t t
=
μ μ σ
2
2
*
= +
μ μ σ
2
2
*
− =
Dynamic
Equity
Models
5

6. 8/28/14
Simula>on:
Probability
of
Median
and
Mean
Price
21
ln S
MED
T
S
0
σ⋅
⎛
⎜⎜⎝
0.0011
ln $45.09
⎞
⋅ μ − ⎟⎟⎠
T
T
=
= −
=
[ ] % 4995 . 0 Sˆ
z
0
Pr S
.12 1.0
$40.00
⎞
⋅ − ⎟⎠
.12 1
< =
T MED
⋅
⎛
⎜⎝
ln S
EXP
T
S
0
σ⋅
⎛
⎜⎜⎝
=
0.1022
ln $45.91
⎞
⋅ μ − ⎟⎟⎠
T
T
=
=
[ ] % 071 . 54 Sˆ
z
0
Pr S
.12 1.0
$40.00
⎞
⋅ − ⎟⎠
.12 1
≤ =
T EXP
⋅
⎛
⎜⎝
Simula>on:
Probability
of
Min
and
Max
Price
22
ln S
min
T
S
0
σ⋅
⎞
⋅ μ − ⎟⎟⎠
⎛
⎜⎜⎝
3.6547
T
T
=
= −
=
[ ] % 013 . Sˆ
z
0
Pr S
.12 1.0
ln $23.95
$40.00
⎞
⋅ − ⎟⎠
.12 1
≤ =
T MIN
⋅
⎛
⎜⎝
ln S
MAX
T
S
0
σ⋅
⎛
⎜⎜⎝
=
4.2347
⎞
⋅ μ − ⎟⎟⎠
T
T
=
=
[ ] % 001 . Sˆ
z
0
Pr S
.12 1.0
ln $93.91
$40.00
⎞
⋅ − ⎟⎠
.12 1
≤ =
T MAX
⋅
⎛
⎜⎝
Probability
of
a
Price
Decline
23
Using
the
IBM
equity
price
sta>s>cs
of
μ=8%
and
the
probability
of
the
drop
in
IBM
price
during
the
week
ending
October
10,
2008?
IBM
stock
opened
Monday
October
6th
at
$101.21,
10th
at
$87.75,
S0.
January
1962
to
September
2008.
μ T
ln S
S
T
⎛
⎜⎜⎝
z 0
⎞
⋅ − ⎟⎟⎠
σ ⋅
T
4.16064
Recall
that
the
IBM
return
sta>s>cs
were
computed
from
.08 1
ln 87.75
101.21
⎞
⋅ − ⎟⎠
0.25 1
52
52
0
=
= −
⋅
⎛
⎜⎝
=
σ
=
25%
(Topic
9)
,
what
was
ST,
and
closed
Friday
October
) z ( N ~
Pr S S T ≤ 0 = 0 = − =
[ ] % 00159 . ) 16064 . 4 ( N ~
That
weekly
decline
was
expected
once
in
1,212
years
[ ]
( ) 0
Pr S ≤ S =
T 0
z N ~
Probability
of
Not
Exceeding
a
Cri>cal
Value
24
An
investor
owns
100
shares
of
an
equity
with
a
current
price
per
share
of
$40.00.
The
equity
has
an
expected
rate
of
return
μ*=16%
and
annual
standard
devia>on
σ
=
20%.
What
is
the
probability
that
the
investor’s
$4,000,
S0,
will
grow
to
no
more
than
$6,000,
K,
aZer
5
years?
14.0%
2 2
16.0% 20%
* = − = − =
2
μ μ σ
2
⎞
⋅ μ − ⎟⎟⎠
T
ln K
S
0
σ⋅
⎛
⎜⎜⎝
0.65860
T
z
0
=
= −
) z ( N ~
Pr S K
.14 5.0
ln $6,000
$4,000
⎞
⋅ − ⎟⎠
.2 ⋅
5.0
⎛
⎜⎝
=
[ ≤ ] = = ( − 0.65860 ) =
25 . 51 % ~
N T 0
[ ]
( ) 0
Pr S ≤K = [ ]
T
z N ~
Pr S >K =
T
z N ~
( ) 2
Dynamic
Equity
Models
6

7. 8/28/14
Probability
of
a
Loss
of
Value
25
What
is
the
probability
that
the
investor
will
have
a
loss
aZer
5
years?
(
S0
=
K
=
$4,000
)
μ T
ln K
S
0
⎞
⋅ − ⎟⎟⎠
σ ⋅
T
⎛
⎜⎜⎝
1.56525
z
0
=
= −
) (z N ~
Pr
S K
.14 5.0
ln $4,000
$4,000
⎞
⋅ − ⎟⎠
.2 ⋅
5.0
⎛
⎜⎝
=
[ ≤ ] = = ( − 1.56525) =
~
5.88% N T 0
The
probability
of
a
loss
is
5.88%
Pr S ≤K = [ ]
[ ]
( ) 0
T
z N ~
Pr S >K =
T
z N ~
( ) 2
Probability
of
Exceeding
a
Cri>cal
Value
26
An
investor
owns
100
shares
of
an
equity
with
a
current
price
per
share
of
$40.00.
The
equity
has
an
expected
rate
of
return
μ*=16%
and
annual
standard
devia>on
σ
=
20%.
What
is
the
probability
that
the
investor’s
$4,000,
S0,
will
grow
to
more
than
$6,000,
K,
aZer
5
years?
μ T
ln K
S
⎛
⎜⎜⎝
= [ ]
Z 0
0
= −
⎞
⋅ − ⎟⎟⎠
σ ⋅
T
0.65860
.14 5.0
ln $6,000
$4,000
⎞
⋅ − ⎟⎠
.2 ⋅
5.0
⎛
⎜⎝
=
) Z ( N ~
) Z ( N ~
Pr S K 1 T > = − 0 = − 0 = 2 =
[ ] % 49 . 74 ) Z ( N ~
The
probability
that
the
value
of
the
shares
exceeds
$6,000
is
74.49%
( )
.14 5.0
ln $4,000
$6,000
⎞
⋅ + ⎟⎠
.2 5.0
⎛
⎜⎝
0.65860
⎞
0 * 2
μ .5 σ T
⋅ ⋅ − + ⎟⎠
σ T
ln S
K
Z
2
=
=
⋅
⋅
⎛
⎜⎝
≡
Pr S ≤K = [ ]
T
z N ~
( ) 0
Pr S >K =
T
z N ~
( ) 2
Simple
Binary
Op>on
27
A
security,
C,
is
offered
as
follows:
If
an
equity,
S,
currently
priced
at
$40,
S0,
exceeds
$45,
$K,
aZer
one
year
(T=1.0),
then
the
buyer
of
this
security,
C,
will
receive
$K,
if
the
equity,
S,
is
less
than
or
equal
to
K,
then
the
buyer
will
receive
nothing.
The
annual
standard
devia>on
of
the
equity,
σ,
is
20%
and
the
annual
expected
risk
free
rate
of
return,
r*,
is
6%.
If
ST
>
K,
then
CT
=
K
If
ST
≤
K,
then
CT
=
0
d N ~
C e E C e K
[ ] ( )
* *
r T
− ⋅ − ⋅
= ⋅ = ⋅ ⋅
T
r T
.38892 -­‐ N ~
e $45
( )
.06 1
− ⋅
= ⋅ ⋅
.06
2
e $45 .34867 $14.78
0
−
= ⋅ ⋅ =
[ ] [ ]
E C = K ⋅ Pr S >
K
T T
d N ~
K
( ) 2
( )
= ⋅
⎞
0 * 2
r .5 σ T
⋅ ⋅ − + ⎟⎠
σ ⋅
T
( )
.38892
2
.06 .5 .2 1
⎞
⋅ ⋅ − + ⎟⎠
.2 1
ln S
K
⎛
⎜⎝
ln 40
45
d
2
= −
⋅
⎛
⎜⎝
=
=
The
fair
value
of
this
security
known
as
a
“cash
or
nothing
call
op>on”
is
$14.78
[ ]
( ) 0
Pr S ≤K = [ ]
T
d N ~
Pr S >K =
T
d N ~
( ) 2
Confidence
Intervals
28
What
are
the
upper
and
lower
bounds
on
a
future
stock
price
for
which
one
is
95%
(=1-­‐α)
confident?
St+
and
St-­‐
are
the
upper
and
lower
bounds
at
>me
T
=
0.5
years
S S e
*
μ T 1.95996 σ T
0
⋅ + ⋅ ⋅
⋅
$40.00 e
$57.17
+
T
=
=
=
S S e
.16 0.5 1.95996 0.2 0.5
*
⋅ + ⋅ ⋅
⋅
μ T 1.95996 σ T
0
⋅ − ⋅ ⋅
⋅
$40.00 e
$32.84
.16 0.5 1.95996 0.2 0.5
−
T
=
=
=
⋅ − ⋅ ⋅
⋅
Confidence
Level
(1-­‐α)
α α/2 -­‐Z +Z
90% 10% 5.00% -­‐1.64485 1.64485
95% 5% 2.50% -­‐1.95996 1.95996
99% 1% 0.50% -­‐2.57583 2.57583
( −
1 . 95996 ~
) N Dynamic
Equity
Models
7

8. 8/28/14
Value
at
Risk
(VaR)
29
What
is
the
maximum
loss
that
an
investor
would
expect
over
some
>me
period
t
?
For
example,
what
is
the
maximum
loss
expected
with
95%
confidence
from
owning
an
equity
over
a
10
day
period?
The
equity
has
μ*=
16%,
σ
=
20%,
and
S0
=
$40.00.
Unlike
the
confidence
interval,
which
uses
a
two
tailed
confidence
,
VaR
is
a
one-­‐tail
interval.
Confidence
Level
(1-­‐α)
α -­‐Z
90% 10% -­‐1.28155
95% 5% -­‐1.64485
99% 1% -­‐2.32635
S S e
*
μ T 1.64485 σ T
− ⋅
0
=
⋅ − ⋅ ⋅
$40.00 e
$37.70
1.64485 0.2 10
252
.16 10
⋅ − ⋅ ⋅
252
T
=
=
⋅
( −
1 . 64485 ~
) N Value
at
Risk
(VaR)
30
The
minimum
95%
confident
price
is
$37.67,
thus
the
95%
maximum
expected
loss
is
$3.63
or
value
at
risk,
VaR
And
commonly
approximated
for
short
>me
periods
as
follows
VaR = $40.00 − $34.34 = $5.66
VaR
is
computed
directly
as
follows
( )
VaR S 1 e
*
μ T z σ T
= ⋅ −
0
⎛
⋅ + ⋅ ⋅
$40.00 1 e
= ⋅ −
$2.30
1.64485 0.2 10
252
.16 10
⋅ − ⋅ ⋅
252
=
⎞
⎟⎟
⎠
⎜⎜
⎝
VaR = S ⋅ 1 −
e
0
μ* T z σ T
⎛
⎜⎝⎛
⋅ + ⋅ ⋅
$40.00 1 e
= ⋅ −
$2.54
⎟⎠⎞
1.64485 0.2 10
252
=
⎞
⎟⎟
⎠
⎜⎜
⎝
− ⋅ ⋅
Expected
Value
Exceeding
Cri>cal
Value
31
~
The
same
N problem
as
last
slide,
but
now
-­‐
what
is
the
expected
value
of
the
equity
posi>on
given
that
the
cri>cal
value,
K,
has
been
exceeded?
[ ] [ ] ( z )
1
z N ~
~( )
N ~
N 2
( z )
( z ) 2
*
> = ⋅
μ T 1
E S |S K E S
T T T
S e
⋅
= ⋅ ⋅
0
The
deriva>on
details
are
not
included
in
this
course.
( )
⎞
0 * 2
μ .5 σ T
⋅ ⋅ + + ⎟⎠
σ ⋅
T
( )
σ T
⎞
0 * 2
μ .5 σ T
ln S
K
⎜⎝⎛
ln S
K
z
z
1
2
⋅ ⋅ − + ⎟⎠
⋅
⎛
⎜⎝
=
=
1.10581
0.65860
.18 5.0
ln $4,000
$6,000
⎞
⋅ + ⎟⎠
.2 5.0
.14 5.0
⋅
⎛
⎜⎝
ln $4,000
$6,000
⎞
⋅ + ⎟⎠
.2 5.0
z
z
1
2
=
=
⋅
⎛
⎜⎝
=
=
E S |S $6,000 $4,000 e.16 5 .086560
[ ]
.074492
> = ⋅ ⋅
$8902.16 .086560
= ⋅
$10,344
.074492
T T
=
Example:
Price
Distribu>on
at
>me
T
(5Yrs)
0.014
0.012
0.01
0.008
0.006
0.004
0.002
0
N [ ln(S ) μ T
,
σ T
]
N[4.388879
E[ST|ST>K]=$103.44
S ~ e 0 + ⋅ ⋅
,
.447214]
T
~ e
Mode[ST]=$65.95
K=$60
Median[ST]=$80.55
E[ST]=$89.02
S0
$0 $20 $40 $60 $80 $100 $120 $140 $160 $180 $200
32
Dynamic
Equity
Models
8

9. 8/28/14
Another
Simple
Binary
Op>on
33
A
security,
C,
is
offered
as
follows:
If
an
equity
currently
priced
at
$40,
S0,
exceeds
$45,
K,
aZer
exactly
one
year
(T=1.0),
then
the
buyer
of
this
security
will
receive
the
price
of
the
equity,
ST,
if
the
equity,
S,
is
less
than
or
equal
to
K,
then
the
buyer
will
receive
nothing.
If
ST
>
K,
then
CT
=
ST
If
ST
≤
K,
then
CT
=
0
[ ] [ ] [ ]
E C = Pr S > K ⋅ E S |S >
K
T T T T
d N ~
( ) [ ] ( )
1
d N ~
= ⋅ ⋅
( )
( ) 1
S E d N ~
2 T
S e
~
N r *
T
d ( )
⎞
0 * 2
r .5 σ T
⋅ ⋅ + + ⎟⎠
σ ⋅
T
( )
.18892
2
.06 .5 .2 1
⎞
⋅ ⋅ + + ⎟⎠
.2 1
ln S
K
⎛
⎜⎝
ln 40
45
d
1
= −
⋅
⎛
⎜⎝
=
=
⋅ The
fair
value
of
this
security
known
as
a
“asset
= ⋅ ⋅
0
2
[ ]
( ) ( )
*
r T
C e E C
T
= ⋅ − ⋅
.18892 -­‐ N ~
40 $
d N ~
S
= ⋅ = ⋅
0 1
$40 .42509 $17.00
0
= ⋅ =
or
nothing
call
op>on”
is
$14.78
[ ]
( ) 0
Pr S ≤K = [ ]
T
d N ~
Pr S >K =
T
d N ~
( ) 2
Comparing
the
Two
Binary
Op>ons
¨ cash
or
nothing
call
op>on ¨ asset
or
nothing
call
op>on
0.05
0.04
0.03
0.02
0.01
0
S K S K T > T ≤
E[S |S K] T T >
K
$10 $20 $30 $40 $50 $60 $70 $80 $90
34
[ ] [ ] [ ]
E C = Pr S > K ⋅ E S |S >
K
T T T T
d N ~
( ) [ ] ( )
1
d N ~
= ⋅ ⋅
( )
( )
S E d N ~
2 T
S e
~
N r *
⋅
T
d = ⋅ ⋅
0
− r
* ⋅
T
[ ]
T
( ) 0 1
C
e E C
0
1
2
= ⋅
d N ~
S
= ⋅
[ ] [ ]
E C = K ⋅ Pr S >
K
T T
d N ~
K
( )
2
= ⋅
[ ]
K = E K|S >
K
d N ~
C e K
( ) 2
*
r T
0
T
− ⋅
= ⋅ ⋅
≤ = [ ]
[ ]
( ) ( ) 0 2
Pr S K
T
~
N d =
-­‐ ~
d N Pr S >K =
T
d N ~
( ) 2
Essen>al
Concepts
35
Appendix:
Probability
and
Expecta>on
Summary
36
[ ]
( ) ( ) ( )
Pr S K
T
> =
z = ~
N ~
-­‐ N ~
z = 1 N −
z 2 0 0
z N ~
[ ] [ ] ( )
1
z N ~
( )
2
E S |S K E S
> = ⋅
T T T
Risk
Neutral
d N ~
Pr S K
N [ ] ( )
[ ] [ ] ( d )
1
~
d N ~
( ) 2
> =
T 2
E S |S K E S
> = ⋅
T T T
[ ]
( ) ( )
Pr S K
T
≤ =
~
N z =
-­‐ ~
z N 0 2
z N ~
1
( )
2
= −
Risk
Neutral
[ ]
( ) 2
Pr S K
T
d -­‐ N ~
≤ =
[ ]
E[
]
Pr
Risk
neutral
probability
Risk
neutral
expecta>on
Dynamic
Equity
Models
9

10. 8/28/14
Appendix:
1
Tail
Confidence
37
90%
95%
99%
Confidence
95%
confident
that
return
rate
lies
above
the
shaded
area
Appendix:
2
Tail
Confidence
38
90%
95%
99%
Confidence
95%
confident
that
return
rate
lies
between
the
shaded
areas
Dynamic
Equity
Models
10