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)????


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)).


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.


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