BOPM FOR PUTS AND THE DIVIDEND-ADJUSTED BOPM. Chapter 6. Binomial Stock Movement. Assumption. Assume there is a European put option on the stock that expires at the end of the period. Example: X = $100. Binomial Put Movement. Replicating portfolio.
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The equilibrium price of the put is governed by arbitrage.
If the market price of the put is above (below) the equilibrium price, then an arbitrage can be exploited by going short (long) in the put and long (short) in the replicating portfolio. For an example, see JG: 185-186.
If a dividend is paid and the ex-dividend date occurs at the end of any of the periods, then the price of the stock will fall. The price decrease will cause the call price to fall and the put price to increase.
The dividend may also make the early exercise of a call profitable, making an American call more valuable than a European.
For American call options on dividend-paying stocks, early exercise can be incorporated into the BOPM by constraining each possible call value to be the maximum of either its European value as determine by the BOPM or the IV just prior to the ex-dividend date.
In our previous case, the $1 dividend in Period 1 would not make early exercise profitable if the call were American.
As shown in the next Figure, if the dividend in Period 2 were $2 instead of $1, then there is an early exercise advantage at the top node in Period 1. The price of an American call in this case is 3.38, compared to a European value of $3.02.
The dividend adjustments required for European puts are similar to those for European calls.
When applicable, the dividend is subtracted from the stock price to determine the ex-dividend put price. This price is then used to determine H, I, and Po.
The dividend adjustment for an American put does differ from the adjustment for the American call. For the put, the price at each node is the maximum of its European value as determined by the BOPM or its IV on the ex-dividend date (see JG:210):