Then
D
(
t
) is constant in
t
between consecutive payment dates,
and jumps up by
a
j
S
(
t
j

) at time
t
j
.
By (5.27) the sum
S
(
t
) +
D
(
t
) is continuous in
t
. Therefore
S
(
t
)
jumps down by
a
j
S
(
t
j

) at time
t
j
. Also
dS
(
t
) =
S
(
t
)
α
(
t
)
dt
+
S
(
t
)
σ
(
t
)
dW
(
t
) =
S
(
t
)
R
(
t
)
dt
+
S
(
t
)
σ
(
t
)
d
f
W
(
t
)
for
t
∈
(
t
j
,
t
j
+1
) between consecutive payment dates. Thus
S
(
t
) =
S
(
t
j
)
e
R
t
t
j
σ
(
u
)
d
f
W
(
u
)+
R
t
t
j
(
R
(
u
)

1
2
σ
(
u
)
2
)
du
if
t
∈
(
t
j
,
t
j
+1
)
,
S
(
t
j
+1
) = (1

a
j
+1
)
S
(
t
j
)
e
R
t
j
+1
t
j
σ
(
u
)
d
f
W
(
u
)+
R
t
j
+1
t
j
(
R
(
u
)

1
2
σ
(
u
)
2
)
du
.
By recursion it follows for all
t
∈
[0
,
T
] that
S
(
t
) =
S
0
Y
t
j
≤
t
(1

a
j
)
·
e
R
t
0
σ
(
u
)
d
f
W
(
u
)+
R
t
0
(
R
(
u
)

1
2
σ
(
u
)
2
)
du
(5.32)
For constant coefficients
R
(
t
) =
r
,
σ
(
t
) =
σ
, and
a
j
,
S
(
t
) =
S
0
Y
t
j
≤
t
(1

a
j
)
·
e
σ
f
W
(
t
)+
(
r

1
2
σ
2
)
t
.
The riskneutral pricing formula (5.30) then yields for the price
V
(
t
)
of the European call with strike
K
and maturity
T
V
(
t
) =
¯
S
(
t
)Φ
d
+
(
t
,
¯
S
(
t
)
)

e

r
(
T

t
)
K
Φ
d

(
t
,
¯
S
(
t
)
)
where
¯
S
(
t
) =
S
(
t
)
Y
t
<
t
j
≤
T
(1

a
j
).
Remark.
We can compute the value
X
(
t
) of a portfolio strategy
which starts with one share of stock and instantaneously reinvests
all dividends in the stock. Let Δ(
t
) be the number of shares in the
portfolio at time
t
. This is an increasing process with Δ(0) = 1
and
d
Δ(
t
) = Δ(
t

)
dD
(
t
)
S
(
t
)
.
Case 1)
D
(
t
) =
R
t
0
A
(
u
)
S
(
u
)
du
. Then we find Δ(
t
) =
e
R
t
0
A
(
u
)
du
.
Case 2)
D
(
t
) =
∑
t
j
≤
t
a
j
S
(
t
j

). Then Δ(
t
) is constant between
consecutive payment dates and at payment date
t
j
we have
Δ(
t
j
)

Δ(
t
j

) = Δ(
t
j

)
a
j
S
(
t
j

)
S
(
t
j
)
= Δ(
t
j

)
a
j
1

a
j
.
Thus Δ(
t
j
) = Δ(
t
j

)
1
1

a
j
= Δ(
t
j

1
)
1
1

a
j
and so Δ(
t
) =
Q
t
j
≤
t
1
1

a
j
.
Now since
X
(
t
) = Δ(
t
)
S
(
t
), using either (5.31) or (5.32) we
obtain in each case that
X
(
t
) =
S
0
exp
R
t
0
σ
(
u
)
d
f
W
(
u
) +
R
t
0
(
R
(
u
)

1
2
σ
(
u
)
2
)
du
.
In conclusion, if the dividend paying stock model is given by
(5.27), then the associated value process
X
(
t
) from continuous
dividend reinvestment follows the same stochastic process as the
nondividend paying stock
S
(
t
) in (5.14).
In particular, the discounted value process
X
(
t
)
B
(
t
)
=
S
0
exp
R
t
0
σ
(
u
)
d
f
W
(
u
)

R
t
0
1
2
σ
(
u
)
2
du
is again a martingale under the riskneutral measure
e
P
.
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 Fall '09
 J.WISSEL
 Probability theory