Problem $16-03$

Assuming that a one$-$year call option with an exercise price of $$30$ is available for the stock of the DEW Corp., consider the following is a two$-$period price tree for DEW stock over the next year:

Now S$1$ One Year

$$42\mathrm{.}32$

$$36\mathrm{.}80$

$$32\mathrm{.}00$ $$31\mathrm{.}28$

$$27\mathrm{.}20$

$$23\mathrm{.}12$

You also know that the risk$-$free rate is RFR $=5\%$ per subperiod $($i$.$e$.,10\mathrm{.}25\%$ annualized$).$

If the sequence of stock prices that DEW stock follows over the year is $$32\mathrm{.}00,$ $$36\mathrm{.}80,$ and $$31\mathrm{.}28,$ describe the composition of the initial riskless portfolio of stock and options you would form and all the subsequent adjustments you would have to make to keep this portfolio riskless. Do not round intermediate calculations. Round your answers to two decimal places. Use a minus sign to enter negative values, if any.

The riskless portfolio will initially contain $1$ share of stock and

$-$Select$-$

call options. After an initial up move to $$36\mathrm{.}80$ the hedge ratio is

$.$

Given the initial DEW price of $$32\mathrm{.}00,$ what are the probabilities of observing each of the three terminal stock prices in one year? $($Hint: In arriving at your answer, it will be useful to consider $(1)$ the number of different ways that a particular terminal price could be achieved and $(2)$ the probability of an up or down movement.$)$ Do not round intermediate calculations. Round your answers to three decimal places.

Ending Price Total Probability

$$42\mathrm{.}32$

$$31\mathrm{.}28$

$$23\mathrm{.}12$

Use the binomial option model to calculate the present value of this call option. Do not round intermediate calculations. Round your answer to the nearest cent.

$

Calculate the value of a one$-$year put option on DEW stock having an exercise price of $$30$; be sure your answer is consistent with the correct response to Part c$.$ Do not round intermediate calculations. Round your answer to the nearest cent.

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