Copyright c
±
2008 by Karl Sigman
1 Geometric Brownian motion
Note that since BM can take on negative values, using it directly for modeling stock prices is
questionable. There are other reasons too why BM is not appropriate for modeling stock prices.
Instead, we introduce here a nonnegative variation of BM called
geometric Brownian motion
,
S
(
t
), which is deﬁned by
S
(
t
) =
S
0
e
X
(
t
)
,
(1)
where
X
(
t
) =
σB
(
t
) +
μt
is BM with drift and
S
(0) =
S
0
>
0 is the intial value. We view
S
(
t
)
as the price per share at time
t
of a risky asset such as stock.
Taking logarithms yields back the BM;
X
(
t
) = ln(
S
(
t
)
/S
0
) = ln(
S
(
t
))

ln(
S
0
). ln(
S
(
t
)) =
ln(
S
0
) +
X
(
t
) is normal with mean
μt
+ ln(
S
0
), and variance
σ
2
t
; thus, for each
t
,
S
(
t
) has a
lognormal
distribution.
As we will see in Section 1.4: letting
r
=
μ
+
σ
2
2
,
E
(
S
(
t
)) =
e
rt
S
0
(2)
the expected price grows like a ﬁxedincome security with continuously compounded
interest rate
r
.
In practice,
r >> r
, the real ﬁxedincome interest rate, that is why one invests in stocks. But
unlike a ﬁxedincome investment, the stock price has variability due to the randomness of the
underlying Brownian motion and could drop in value causing you to lose money; there is risk
involved here.
1.1 Lognormal distributions
If
Y
∼
N
(
μ,σ
2
), then
X
=
e
Y
is a nonnegative r.v. having the
lognormal distribution
; called
so because its natural logarithm
Y
= ln(
X
) yields a normal r.v.
X
has density
f
(
x
) =
(
1
xσ
√
2
π
e

(ln(
x
)

μ
)
2
2
σ
2
,
if
x
≥
0;
0
,
if
x <
0.
This is derived via computing
d
dx
F
(
x
) for
F
(
x
) =
P
(
X
≤
x
) =
P
(
Y
≤
ln(
x
)) = Φ((ln(
x
)

μ
)
/σ
)
,
where Φ(
x
) =
P
(
Z
≤
x
) denotes the c.d.f. of
N
(0
,
1) rv
Z
.
Observing that
E
(
X
) =
E
(
e
Y
) and
E
(
X
2
) =
E
(
e
2
Y
) are simply the moment generating
function (MGF)
M
Y
(
s
) =
E
(
e
sY
) of
Y
∼
N
(
μ,σ
2
) evaluated at
s
= 1 and
s
= 2 respectively
yields
E
(
X
) =
e
μ
+
σ
2
2
E
(
X
2
) =
e
2
μ
+2
σ
2
V ar
(
X
) =
e
2
μ
+
σ
2
(
e
σ
2

1)
.
1
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View Full DocumentAs with the normal distribution, the c.d.f.
F
(
x
) =
P
(
X
≤
x
) = Φ((ln(
x
)

μ
)
/σ
) does not
have a closed form, but it can be computed from the unit normal cdf Φ(
x
). Thus computations
for
F
(
x
) are reduced to dealing with Φ(
x
).
We denote a lognormal
μ
,
σ
2
r.v. by
X
∼
lognorm
(
μ,σ
2
)
.
1.2 Back to our study of geometric BM,
S
(
t
) =
S
0
e
X
(
t
)
For 0 =
t
0
< t
1
<
···
< t
n
=
t
, the ratios
L
i
def
=
S
(
t
i
)
/S
(
t
i

1
)
,
1
≤
i
≤
n,
are independent
lognormal r.v.s. which reﬂects the fact that it is the percentage of changes of the stock price
that are independent, not the actual changes
S
(
t
i
)

S
(
t
i

1
). For example
L
1
def
=
S
(
t
1
)
S
(
t
0
)
=
e
X
(
t
1
)
,
L
2
def
=
S
(
t
2
)
S
(
t
1
)
=
e
X
(
t
2
)

X
(
t
1
)
,
are independent and lognormal due to the normal independent increments property of BM;
X
(
t
1
) and
X
(
t
2
)

X
(
t
1
) are independent and normally distributed. Note how therefore we can
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 Summer '10
 KarlSigma
 Normal Distribution, geometric bm

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