# 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)????

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:
$$\begin{array}[ccc] & & & F_{0,T}^Pu^2 \\ & F_{0,T}^Pu & \\ S = F_{0,T}^P + De^{-rt} & & F_{0,T}^Pud \\ & F_{0,T}^Pd & \\ & & F_{0,T}^Pd^2 \\ \end{array}$$