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# value of the put

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#### Solution Preview

Please see the attached file.

Part a
First of all, notice that since the bond prices are all one, this implies that the interest rate is 0%. This will simplify matters a lot.

We can now solve the binomial model as usual. First, we compute the possible terminal values for the call option.

If the price ends at UU, the price of the stock will be \$2, and the call option will entitle its bearer to buy it for \$1.50. Therefore, the value of the call option is thus \$0.50:

Now, both at DU and DD, the value of the call will be zero. The call gives you the right to buy the stock for \$1.50, but the price in these nodes is never greater than \$1.50 (it's \$1.50 in DU and \$0.10 in DD). Therefore, the call is worthless at these points. So:

Now let's go back one period, to either U or D.

At U, we're certain that in the next period, the call will be worth \$0.50. Since the interest rate is zero, we conclude that the value of the call at U (that is, in the 2nd period, when t=1) must be \$0.50:

The value of the call at D is also easy to compute, since we know for sure that the value of the call in the next period will be zero, no matter what happens with the stock price. Therefore,

Now, we go to the 1st period (t=0), to calculate the value of the call at state zero. This is done through arbitrage arguments in the following way:

Consider a portfolio in which you buy h shares of the stocks and sell 1 call at t=0. If the stock goes to U (rises to \$2), then the value of this portfolio will be:

(The value of the call is subtracted because you sold it)

If the stock goes to D (falls to \$1), then the value of the portfolio will be:

Now, we can ...

#### Solution Summary

This solution finds value of the put.

\$2.19