2 Dividends in the Binomial Model I (20 points) Let's add some dividends to the binomial model. We will have three times t=0,1,2. Time periods are of length t=1, the stock starts at S0=$100, and risk-free zero rates are always r=0.02. At each time, the stock goes up by a factor of u=1.1 or down by a factor of d=1.11, with probability 65% of it going up and 35% of it going down. At t=1 we will have a dividend payout of $5. We will be pricing a call option, so we will want to allow the investor to exercise before the dividend is paid out, which is the only time they would ever consider early exercise in reality. We will model this with the following timing at t=1. If the investor exercises their option at t=1, then exercise happens before the stock goes ex-dividend. For example, in the up state, they would exercise while the stock is still worth uS0=$110. The stock goes ex-dividend right after the time when investors can do early exercise, and it is the ex-dividend prices that are multiplied by u and d to get the time t=2 prices. For example, the price of the stock in the up-up state at t=2 is Suu=u($110$5)=$115.50. This is the ex-dividend price of $110$5 multiplied by u. (a) (5 points) Draw the binomial tree corresponding to this setup. Include the stock, savings, and probabilities. At t=1, write down the stock price both before and after it goes ex-dividend. Note: it will not be a recombinant tree, even though u and d are the same each period. If you did things correctly, you will get 4 different values of the stock at t=2. (b) ( 3 points) As a warmup, price a European call option that expires at t=2 with strike price K= $100. (c) (7 points) Now, price an American call option that expires at t=2 with strike price K=$100. (d) (5 points) Briefly (a few sentences at most!) explain why you obtained a higher price in (c) than (b)