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Excerpted from Marek Musiela and Marek Rutkowski's Martingale Methods in Financial Modelling, Second Edition.enter image description here

I think I understand formula 3.71: paying cash dividend $\kappa_j$ at time $T_j$ will cause the stock price to drop by the same amount immediately, as is reflected in the subtraction from the firm capital value $G_t$ by $D_t$ in the stock price. (I understand the capital value as the stock price that would have been if no past cash dividends had been issued.)

But what does the second formula (3.72) mean? To me, $\tilde D_t$ means the present value (at $t$) of all future dividends to be paid after $t$. But what does the sum $S_t=G_t+\tilde D_t$ mean? How can the stock price exceed the capital value of the firm? Also, as opposed to $G_0=S_0$ implied by the first formula, the second formula implies that $G_T=S_T$, how is it reasonable? It looks as if by the expiry, the dividends paid will have had no impact on the stock price at all.

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    $\begingroup$ The first dividend model basically assumes that $S_T$ follows a shifted lognormal distribution. This model is best understood in situations where there is a single dividend at expiry: think of it as diffusing the spot price from $S_0$ to $S_T$ as usual and then subtract $D$ (=> shifted lognormal). The second dividend model - best known as escrowed model - on the other hand assumes that $S_T$ is lognormal. The model is best applied in situations where there is a single dividend close to inception: think of diffusing the spot price from $S_0* = S_0 - D$ and then acting as usual.. $\endgroup$ – Quantuple Apr 28 '17 at 9:18
  • $\begingroup$ These are merely models. The second one is often met in the literature because it allows to use the BS formula. I would be careful though as there exist better modelling approaches. $\endgroup$ – Quantuple Apr 28 '17 at 9:19
  • $\begingroup$ @Quantuple but for the second, the stock price $S_t$ is greater than the price that would have been without past cash dividends, which I think is rather ridiculous: doesn't the dividend cause the stock price to drop? $\endgroup$ – Vim Apr 28 '17 at 12:24
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    $\begingroup$ Simply put, the second model views the current stock price as related to a company's current equity $G_t$ + the discounted value of all the future dividends that this stock will pay $\tilde{D}_t$. This is similar to viewing the stock as a bond distributing coupons in the form of dividends. The price of the bond with coupons is higher than the zero coupon bond. The first model takes the opposite view: The stock's value is related the current company equity, minus all the dividends that have been paid in the past $D_t$. $\endgroup$ – Quantuple Apr 28 '17 at 12:57
  • $\begingroup$ @Quantuple thanks. These seem to be two distinct ways of determining the stock price. I'm not a finance student, but as far as I know, stock price should be the equity the company owns divided by the total number of shares, which is proportional to the equity. So does such a discrepancy actually come from two different measures of the equity? (Equity is capital minus debt, so even more fundamentally, does this come from two different measures of debt?) $\endgroup$ – Vim Apr 28 '17 at 13:10
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Let $G_t$ represents the price of the stock as if it paid no dividends.

The models you discuss above correspond to two different ways of viewing the price of a dividend-paying stock

  • Case 1: $G_t$ minus the (capitalised) value of all past cash distributions that you were entitled to by holding to the stock. $$ S_t = G_t - D_t $$ in that case you start from $S_0 = G_0$ and finish at $S_T = G_T - D_T$. The advantage with that model is that it still allows the BS formula to be used, provided one modifies (shifts) the strike price.

  • Case 2: $G_T$ plus the (discounted) value of all future cash distributions that you are entitled to by holding the stock. $$ S_t = G_t + \tilde{D}_t $$ in that case you start from $S_0 = G_0 - \tilde{D}_0$ and finish at $S_T = G_T$. This is known as the escrowed model. The advantage with that model is that it still allows the BS formula to be used, provided one modifies the spot price, see this related question.

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    $\begingroup$ Actually the first case also allows the use of a modified BS: this time raising the strike by the value of all future cash dividends compounded to the expiry, as opposed to reducing the stock price by that discounted to time $0$ as in the second one. $\endgroup$ – Vim Apr 28 '17 at 14:09
  • $\begingroup$ Yes you are absolutely right. One is shifted lognormal the other is just lognormal. $\endgroup$ – Quantuple Apr 28 '17 at 14:11

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