Pricing Options with a Known Discrete Dividend in a Binomial Model
Summary
The document explains how a known cash dividend affects a binomial stock-price tree used to price a European put. The question concerns a dividend paid between the two steps: the stock price drops to its ex-dividend level on the payment date, then follows up and down movements over the remaining period. The answers recommend treating the known dividend separately from uncertain price movement, rather than counting the mechanical dividend drop as volatility.
One approach builds the tree from the prepaid forward value and adds the present value of the dividend to nodes before payment; after distribution, the dividend is no longer included. Another answer advises adjusting volatility to account for the difference between the stock price and the stock value net of the dividend. These comments outline the modeling intuition but do not work through the requested self-financing replication or provide a complete numerical tree. The exact construction depends on the dividend timing and tree conventions.
Key ideas
- A known cash dividend causes a predictable price adjustment that should be separated from uncertain movements.
- A binomial tree can be built using the prepaid forward value of the stock.
- Before the dividend is paid, nodes include its present value; subsequent nodes exclude it.
- The response suggests adjusting volatility when modeling the stock net of a known dividend.
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Full text
# How to price an option on a dividend-paying stock using the binomial model?
# How to price an option on a dividend-paying stock using the binomial model?
This is actually an exercise from a course. But I don't completely understand the wording of the question.
- A stock is now trading at 100 dollars.
- Its price over the next 6 months evolves as a two step binomial process.
- Over each 3 month period, the price can go up by a factor $u$, or down $d=\frac{1}{u}$.
- The annual risk free rate is 5% (cont.).
- We consider an European put with strike price $K=93$ dollars and expiring in 6 months.
Part a) and b) are about pricing the put using risk-neutral pricing approach.
But part c) states:
> Now suppose that in 3 months, the stock pays a dividend of 10 dollars. On the payment date, the stock price immediately adjusts to its ex-dividend level and then either goes up by a factor $u=1.1$ or down $d=1/u$ over the subsequent 3 months. Construct a dynamic self-financing strategy that replicates the payoff of the put.
Alright, so my question is. I don't know what happens when people know the stock is going to pay out 10 dollars of dividend in 3 months.
Is it during the next period, there are 2 states:(100*1.1-10=100, 100/1.1-10=80.90)????
## Answer by QuantK (score 1)
https://quant.stackexchange.com/a/16274
You could solve this by constructing a binomial tree with the stock price ex-dividend. Also keep in mind that you have to adjust your volatility by muliplying with S/(S-PV(D)).
## Answer by nomen (score 1)
https://quant.stackexchange.com/a/17451
So, the deal is that since the dividend is known in advance, the stock price change it causes should not count as volatility. So, instead of starting a binomial tree with $S$, you want to start with the prepaid forward price of $S$, scale up and down with $u$ and $d$, and add the then-present value of the dividend to the stock price to the nodes where the dividend hasn't yet been distributed. So your tree will look something like:
$$ \begin{array}[ccc] & & & F_{0,T}^Pu^2 \\ & F_{0,T}^Pu & \\ S = F_{0,T}^P + De^{-rt} & & F_{0,T}^Pud \\ & F_{0,T}^Pd & \\ & & F_{0,T}^Pd^2 \\ \end{array} $$
Notice that we "removed" (i.e., did not include) the dividend in the second or third columns.
As QuantK said, you need to adjust the volatility. The idea is the same as above: the dividend is known, so stock price volatility is "due" to the changes in the forward price.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.