Two Dividend Models for Pricing Stocks and Their Options
Summary
The document explains two ways to represent a stock’s price when it pays cash dividends. In the first, the price equals a hypothetical no-dividend stock value minus the capitalized value of dividends already paid. This formulation starts with the observed stock price equal to the no-dividend value and ends with the accumulated past distributions reflected in the difference.
The second, called the escrowed model, adds the discounted value of future dividends to the hypothetical no-dividend value. It begins with the observed price below that hypothetical value by the present value of expected future distributions, and ends when no future dividends remain. The answer notes that both formulations can preserve use of the Black–Scholes formula with an adjusted strike or spot, respectively. The excerpt gives a conceptual distinction rather than a derivation, and its application depends on the chosen model and treatment of dividend cash flows.
Key ideas
- A past-dividend formulation subtracts the capitalized value of prior distributions from a hypothetical no-dividend stock price.
- An escrowed formulation adds the discounted value of future dividends to the hypothetical no-dividend value.
- In the escrowed model, the observed stock price and hypothetical value coincide at the horizon when future dividends are exhausted.
- Both formulations can be paired with Black–Scholes by adjusting the strike or spot input.
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# What is the other type of impact of dividends on the stock price in this formula?
# What is the other type of impact of dividends on the stock price in this formula?
Excerpted from Marek Musiela and Marek Rutkowski's Martingale Methods in Financial Modelling, Second Edition.
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.
## Answer by Quantuple (score 1, accepted)
https://quant.stackexchange.com/a/33944
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.