8.2 Discrete Dividends

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 $ t^*, 0 < t^* \leq T.$ 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 $ t^*-dt,$ which is immediately before the dividend is paid, we buy the stock, cash in the dividend, and sell the stock at time $ t^*+dt.$ By doing this, we make a gain of $ D + S_{t^*+dt} - S_{t^*-dt}$, which has to be zero if arbitrage is excluded. Therefore, for $ dt
\rightarrow 0$ it is sufficient if $ S_t$ jumps down by $ D$ at time $ t^*.$

8.2.1 Dividends as a Percentage of the Stock Price

Suppose that $ t^*$ is contained, say, in the $ i$th time interval, i.e.  $ t_{i-1} < t^* \leq t_i.$ Let the dividend paid at time $ t_i$ be a percentage $ \delta$ of the stock price, that is the dividend amount that is paid is equal to $ \delta S_i.$ It follows that the stock price at time $ t_i$ is smaller by the dividend amount than the stock price without the dividend payment. Accordingly, all stock prices in the tree after time $ t_i$ change in the same way: all prices $ S_j^k,\ j \geq i,$ are multiplied by the factor $ ( 1-\delta ).$ 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 $ \delta
=$ 1% of the stock price at time $ 0.15.$ 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 $ t_2=0.2$ on (note that we divided the time period into 5 time steps $ 0 \leq t \leq 0.5$), are multiplied by the factor $ ( 1-\delta ).$ In Table 7.4 the values in parentheses correspond to the stock prices that are decreased by the dividend amount, i.e.  $ S_j^k,\ j < i = 2$ respectively $ 0.99 \cdot S_j^k,\ j \geq
i = 2.$ Thus, the option prices at maturity change due to equation (7.4), for example $ V_5^4 = V(0.99 \cdot S_5^4,t_5)
= 0.99 \cdot 291.56 - K = 78.646.$ Having determined the option values at maturity the preceding option values are again computed by recursively applying equation (7.2). Note, $ V_j^k$ corresponds to the stock price $ 0.99 \cdot S_j^k$ rather than to $ S_j^k,$ for $ j \geq 2$, i.e.  $ t_j \geq t^*.$ However, the current time $ t_0=0 < t^* = 0.15$ is not concerned, i.e.  $ V_0^0 = 28.384$ is still the option price corresponding to the current stock price $ S_0 = 230.$


Table 7.4: Evolution of option prices (dividends as a percentage of the stock price)
Current stock price $ S_t$ 230.00
Exercise price $ K$ 210.00
Time to maturity $ \tau$ 0.50
Volatility $ \sigma$ 0.25
Risk free interest rate $ r$ 0.04545
Discrete dividend $ \delta$ 0.01
Dividend date $ t^*$ 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
10871 SFEBiTree.xpl


8.2.2 Dividends as a Fixed Money Amount

We assume now that at an ex ante fixed point in time $ t^*$ a fixed amount of money (for example $ 5.00$ 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 $ t^*,\ t_1<t^* \leq t_2 < T,$ a fixed dividend of $ D$ is paid. Figure 7.1 shows the stock price tree for this example. Before the dividend payment at time $ t_1$ the nodes correspond to stock prices of the kind $ uS_0$ and $ S_0/u.$ After the dividend payment, however, stock prices at time $ t_2$ are given by $ u^2S_0 -D, S_0-D$ and $ S_0/u^2-D.$ Proceeding from these $ 3$ prices the tree consists of $ 6$ possible prices in time $ t_3,$ at time $ t_4$ it consists of $ 9$ 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 $ S_j$ in a random and a deterministic component:

$\displaystyle S_j = \tilde{S}_j +D_j,$

with $ D_j$ being the current present value of the dividend payment, i.e. before dividend payment, it is the time $ t_j \leq t^*$ discounted value of $ D,$ afterwards it is $ 0:$

$\displaystyle D_j=\left\{ \begin{array}{ll} De^{-r(t^*-t_j)}&, \text{\rm for}\ \; t_j \leq t^*,\\ 0& , \text{\rm for}\ \; t^*<t_j. \end{array} \right.$ (8.5)

In particular, at maturity it holds $ D_n=0$ and $ S_n=\tilde{S}_n.$

Fig.: Evolution of the stock price tree (dividends as a fixed amount of money
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In order to compute the option price we first construct a stock price tree of the random stock price component $ \tilde{S}_n$ beginning in $ \tilde{S}_0 =S_0-D_0.$ Starting at maturity $ T=t_n$ we obtain:

$\displaystyle \tilde{V}_{n-1}=e^{-r\Delta t}{\mathop{\text{\rm\sf E}}}[\max (0,\tilde{S}_n-K) \vert \tilde{S}_{n-1}]$

The other option prices are given as in the no dividend case by:

$\displaystyle \tilde{V}_{j-1}=e^{-r\Delta t}{\mathop{\text{\rm\sf E}}}[\tilde{V}_j \vert \tilde{S}_{j-1}] .$

The original option prices then correspond to $ \tilde{V}_j^k$ given above. However, they do not correspond to the stock price $ \tilde{S}_j^k$, rather than to the actual stock price

$\displaystyle S_j^k=\tilde{S}_j^k+D_j. $

Example 8.4  
In this example, there are two dividend payments at time $ t^*_1=0.25$ and $ t^*_2=0.75.$ Both dividends are $ D^{(1)} =
D^{(2)} =$ 1.00 EUR. The parameters and results are given in Table 7.5. First, we compute the time $ t_j$ present value of all dividends with equation (7.5): $ D_j = D^{(1)}e^{-r(t_1^*-t_j)} + D^{(2)}e^{-r(t_2^*-t_j)}$ for $ t_j \leq t_1^*$, $ D_j = D^{(2)}e^{-r(t_2^*-t_j)}$ for $ t_1^* <
t_j \leq t_2^*$ and $ D_j = 0$ for $ t_2^*<t_j.$

Table 7.5: Evolution of option prices (discrete dividends as a fixed money amount)
Current stock price$ S_t$ 100.00
Exercise price $ K$ 100.00
Time to maturity $ \tau$ 1.00
Volatility $ \sigma$ 0.30
Risk free interest rate $ r$ 0.10
Discrete dividend $ D^{(1)}$ 1.00
Payment date $ t_1^*$ 0.25
Discrete dividend $ D^{(2)}$ 1.00
payment date $ t_2^*$ 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. $ D_j$ 1.903 1.935 0.960 0.975 0.992 0.00 0.00
11118 SFEBiTree4.xpl


In particular, it holds $ D_j = 0$ for $ t_j > t^*_2.$ Following, we construct the stock price tree as in Table 7.1, but this time we start in $ \tilde{S}_0 =S_0-D_0 = 98.10$ rather than in $ S_0 = 100.$ Proceeding from the boundary values $ \tilde{V}_6^k
= K - \tilde{S}_n^k, k=0, ..., 3, \tilde{V}_6^k = 0, k=4, ..., 6$ 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 $ \tilde{V}_3^2 =
2.810$ belongs to the stock price $ S_3^2 = \tilde{S}_3^2 + D_3 =
111.85$ and not to $ \tilde{S}_3^2 = 110.88$, which accounts for the dividend. It follows that the put option price at a current stock price $ S_0 = 100 $ is equal to $ \tilde{V}_0^0 = 7.631.$