In case of dividend payments at discrete points of time the tree
of stock prices changes. By changing the price tree we have to
distinguish two different cases. In the first case, dividends are
paid as a percentage of the stock price. In the second case,
dividends are paid as a fixed amount of money. We confine
ourselves to the case that dividends are paid only once during the
time to maturity, say, at time
Dividends
paid at several points of time can be dealt with analogously. We
assume that the underlying is a stock.
Using arbitrage arguments it can be shown that the stock price jumps down by the amount
of the dividend at the time the dividend is paid. Let's consider the following argument
to visualize this. At time
which is immediately before the dividend is paid, we
buy the stock, cash in the dividend, and sell the stock at time
By doing this,
we make a gain of
, which has to be zero if arbitrage is
excluded. Therefore, for
it is sufficient if
jumps down by
at time
Suppose that
is contained, say, in the
th time interval,
i.e.
Let the dividend paid at time
be a percentage
of the stock price, that is the
dividend amount that is paid is equal to
It follows
that the stock price at time
is smaller by the dividend
amount than the stock price without the dividend payment.
Accordingly, all stock prices in the tree after time
change
in the same way: all prices
are multiplied by
the factor
Following this correction the option
values can be determined recursively as in the no dividend case.
Example 8.3
We consider a call option on a stock paying a dividend of

1% of the stock price at time

All other parameters of
this example are those already given in Table
7.1.
The results are shown in Table
7.4. First we
ignore the dividend and compute the stock price tree as shown in
Table
7.1. Following, all stock prices from the
dividend date on, i.e. from time

on (note that we
divided the time period into 5 time steps

),
are multiplied by the factor

In Table
7.4 the values in parentheses correspond to the
stock prices that are decreased by the dividend amount,
i.e.

respectively

Thus, the option prices at maturity change due to equation
(
7.4), for example

Having determined the option
values at maturity the preceding option values are again computed
by recursively applying equation (
7.2). Note,

corresponds to the stock price

rather
than to

for

, i.e.

However,
the current time

is not concerned,
i.e.

is still the option price corresponding to
the current stock price

Table 7.4:
Evolution of option prices (dividends as a percentage of
the stock price)
Current stock price |
230.00 |
Exercise price |
210.00 |
Time to maturity |
0.50 |
Volatility |
0.25 |
Risk free interest rate |
0.04545 |
Discrete dividend  |
0.01 |
Dividend date |
0.15 |
Time steps |
5 |
Option type |
European call |
Stock prices |
Option prices |
341.50558 |
|
|
|
|
|
128.091 |
315.54682 |
|
|
|
|
103.341 |
(338.09) |
291.56126 |
|
|
|
80.542 |
(312.39) |
78.646 |
269.39890 |
|
|
59.543 |
(288.65) |
57.655 |
(288.65) |
248.92117 |
|
41.942 |
(266.70) |
38.329 |
(266.70) |
36.432 |
230.00000 |
28.384 |
(248.92) |
24.087 |
(246.43) |
18.651 |
(246.43) |
212.51708 |
(230.00) |
14.592 |
(227.70) |
9.547 |
(227.70) |
0.392 |
196.36309 |
|
(212.52) |
4.886 |
(210.39) |
0.199 |
(210.39) |
181.43700 |
|
|
(194.40) |
0.101 |
(194.40) |
0.000 |
167.64549 |
|
|
|
(179.62) |
0.000 |
(179.62) |
154.90230 |
|
|
|
|
(165.97) |
0.000 |
|
|
|
|
|
|
(153.35) |
Time |
0.00 |
0.10 |
0.20 |
0.30 |
0.40 |
0.50 |
Dividend |
1.00 |
1.00 |
0.99 |
0.99 |
0.99 |
0.99 |
SFEBiTree.xpl
|
We assume now that at an ex ante fixed point in time
a fixed
amount of money (for example
EUR) is paid. Now, the
stock price jumps down by an amount which is independent of the
stock price. It follows that the tree is not totally recombining
anymore. The stock price tree splits up which can be visualized in
a simple example. Suppose at time
a
fixed dividend of
is paid. Figure 7.1 shows the
stock price tree for this example. Before the dividend payment at
time
the nodes correspond to stock prices of the kind
and
After the dividend payment, however, stock prices at
time
are given by
and
Proceeding from these
prices the tree consists of
possible
prices in time
at time
it consists of
and so on.
The stock price tree gets very vast the more time steps are
considered, and is less useful for practical computations. To
overcome this problem, we use the fact that the dividend is
independent of the stock price and therefore not random anymore.
We decompose the stock price
in a random and a deterministic
component:
with
being the current present value of the dividend payment, i.e. before dividend
payment, it is the time
discounted value of
afterwards it is
 |
(8.5) |
In particular, at maturity it holds
and
Fig.:
Evolution of the stock price tree (dividends as a fixed amount of money
 |
In order to compute the option price we first construct a stock price tree of the random
stock price component
beginning in
Starting at
maturity
we obtain:
The other option prices are given as in the no dividend case by:
The original option prices then correspond to
given above. However, they
do not correspond to the stock price
, rather than to the actual stock
price
Example 8.4
In this example, there are two dividend payments at time

and

Both dividends are

1.00 EUR. The parameters and results are given in
Table
7.5. First, we compute the time

present value of all dividends with equation (
7.5):

for

,

for

and

for
Table 7.5:
Evolution of option prices (discrete dividends as a fixed money amount)
Current stock price |
100.00 |
Exercise price |
100.00 |
Time to maturity |
1.00 |
Volatility |
0.30 |
Risk free interest rate |
0.10 |
Discrete dividend  |
1.00 |
Payment date |
0.25 |
Discrete dividend  |
1.00 |
payment date |
0.75 |
Time steps |
6 |
Option type |
European Put |
Prices |
Option prices |
204.55 |
|
|
|
|
|
|
0.000 |
180.97 |
|
|
|
|
|
0.000 |
(204.55) |
160.12 |
|
|
|
|
0.000 |
(180.97) |
0.000 |
141.65 |
|
|
|
0.179 |
(161.10) |
0.000 |
(160.11) |
125.32 |
|
|
1.373 |
(142.63) |
0.394 |
(141.65) |
0.000 |
110.88 |
|
3.906 |
(126.28) |
2.810 |
(126.32) |
0.866 |
(125.32) |
98.10 |
7.631 |
(112.81) |
6.990 |
(111.85) |
5.720 |
(110.88) |
1.903 |
86.79 |
(100) |
12.236 |
(99.06) |
12.100 |
(99.09) |
11.567 |
(98.10) |
76.78 |
|
(88.72) |
18.775 |
(87.76) |
19.953 |
(86.79) |
23.215 |
67.93 |
|
|
(77.74) |
27.211 |
(77.78) |
30.421 |
(76.78) |
60.10 |
|
|
|
(68.91) |
36.631 |
(67.93) |
39.897 |
53.17 |
|
|
|
|
(61.09) |
45.178 |
(60.10) |
47.05 |
|
|
|
|
|
(53.17) |
52.955 |
|
|
|
|
|
|
|
(47.05) |
Zeit |
0.00 |
0.17 |
0.33 |
0.50 |
0.67 |
0.83 |
1.00 |
Div.  |
1.903 |
1.935 |
0.960 |
0.975 |
0.992 |
0.00 |
0.00 |
SFEBiTree4.xpl
|
In particular, it holds

for

Following, we
construct the stock price tree as in Table
7.1, but
this time we start in

rather than
in

Proceeding from the boundary values

we compute once again recursively the put prices at earlier points
in time by means of equation (
7.3). We have to take
into account that for example the option price

belongs to the stock price

and not to

, which accounts for
the dividend. It follows that the put option price at a current
stock price

is equal to
