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Chapter 4: Q4.1-14E (page 217)

Consider the situation of pricing a stock option as in

Example 4.1.14.We want to prove that a price other than \(20.19 for the option to buy one share in one year for \)200 would be unfair in some way.

a.Suppose that an investor (who has several shares of

the stock already) makes the following transactions.

She buys three more shares of the stock at \(200 per

share and sells four options for \)20.19 each. The investor

must borrow the extra \(519.24 necessary to

make these transactions at 4% for the year. At the

end of the year, our investor might have to sell four

shares for \)200 each to the person who bought the

options. In any event, she sells enough stock to pay

back the amount borrowed plus the 4 percent interest.

Prove that the investor has the same net worth

(within rounding error) at the end of the year as she

would have had without making these transactions,

no matter what happens to the stock price. (Acombination

of stocks and options that produces no change

in net worth is called a risk-free portfolio.)

b.Consider the same transactions as in part (a), but

this time suppose that the option price is \(xwhere

x <20.19. Prove that our investor loses|4.16x鈭�84触

dollars of net worth no matter what happens to the

stock price.

c.Consider the same transactions as in part (a), but

this time suppose that the option price is \)xwhere

x >20.19. Prove that our investor gains 4.16x鈭�84

dollars of net worth no matter what happens to the

stock price.

Short Answer

Expert verified
  1. Here gain\(20.19\)and\(181.734\).
  2. Hence the investor losses\(\left| {4.16x - 84} \right|\) dollars
  3. Hence the investor losses \(\left( {4.16x - 84} \right)\) dollars

Step by step solution

01

Given Information

Let company \(A\) has currently priced \(200\). Also \(x\) the price of stock one year from now discrete random variable that only take two values \(260\) and \(180\).

02

Find the stock price that no change

Let\(p\)be the probability that\(x = 260\), one could by the stock for\(200\). Then want to calculate value of these stock option. The possibility of selling them or because want to company\(A\)offer to what other company鈥檚 offer. Let \(y\)be the value of option for one share when it expires in one year. Since nobody pay\(200\)for the stock of the price. If\(x\)is less than\(200\),the value of the stock option is\(0\)if \(x = 180\). If\(x = 260\)one could by the stock for\(200\)per share and then immediately sell it for\(260\). Then if\(y = h\left( x \right)\)where\(h\left( x \right) = \left\{ \begin{array}{l}0,x = 180\\60,x = 260\end{array} \right.\)

Assume that an investor could earn\(4\% \)risk free on any money invested for this same year. If no other investment option were available then let take present value of one year is take\(E\left( y \right)\). Let the equals value\(c\)such that\(E\left( y \right) = 1.04c\).

Then calculating\(E\left( y \right)\)is to get

\(\begin{array}{c}E\left( y \right) = 0 \times \left( {1 - p} \right) + 60 \times p\\ = 60p\end{array}\)

So the fair price of an option to buy one share would be

\(\begin{array}{c}c = \frac{{60p}}{{1.04}}\\ = 57.69p\end{array}\)

Then assume that\(E\left( x \right) = 200 \times 1.04\) and since\(E\left( x \right) = 260p + 180\left( {1 - p} \right)\).

Therefore

\(\begin{array}{l}200 \times 1.04 = 260p + 180\left( {1 - p} \right)\\260p + 180 - 180p = 208\\p = 0.35\end{array}\)

Obtained\(p = 0.35\). Then one share for\(200\)in

one year would be\(57.69 \times 0.35 = 20.19\) and also \(519.24 \times 0.35 = 181.734\) .

03

Prove that investor losses dollars

Here given that\(x\)is option price. Where\(x < 20.19\). Investor losses that\(E\left( x \right) = 4.16x\) and\(84p + 84\left( {1 - p} \right) = 84\).

Then

\(\begin{array}{l}4.16x = 84\\4.16x - 84 = 0\end{array}\)

Hence the investor losses \(\left| {4.16x - 84} \right|\) dollars.

04

Show that investor gain dollars

Similarly for this method\(x\)is option price. Where\(x > 20.19\). Then by use sme process which is used in previously. Then the following method given by\(E\left( x \right) = 4.16x\)and\(84p + 84\left( {1 - p} \right) = 84\).

Then

\(\begin{array}{l}4.16x = 84\\4.16x - 84 = 0\end{array}\)

Finally prove that the investor gain \(\left( {4.16x - 84} \right)\) dollars.

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