108156.pdf

9. Financial Engineering 211

is random and depends on the stock price after 20 days,

KC=

(S(20/365)−60)+−C

C

.

To compute the expected return on an option we can find the parameters

of the binomial model consistent with the option price and the expected

return on stock, assuming that the market probability of up and down

price movements is 1/2. First we find the risk-free return over a single day,

r= (1 + 12%)

3601

− 1 ∼= 0 .0315%.

Then we write down a condition on the up and down daily stock returns

such that the expected annual return is 31%,

u+d

2

= (1 + 31%)^3601 − 1 ∼= 0 .075%.

The call price gives another condition foruandd, and we finally arrive at

the following values:^3

u∼= 1 .85%,d∼=− 1 .70%.

Now we can compute the standard deviation for the period in question

(using the actual market probabilitiespk=

( 20

k

)

0. 520 ,k=0, 1 ,...,20),

σS∼= 8 .0962%.

Finally, we compute the expected return and risk of the investment in op-

tions,

μC∼= 14 .1268%,σC∼= 153 .006%.

The return is impressive, but so is the risk. Observe that with probability

0 .4119 the investor can lose all his or her money.

3.Forward Contracts.The forward price is approximately $60.38. Suppose

that entering into a forward contract requires a 20% deposit of the initial

stock price, that is $12 per share. The investor can afford to enter into 1, 250

forward contracts. The expected return and risk in the binomial model are

μF∼= 4 .3993%,σF∼= 40 .4811%.

Note that if the stock price falls below $48. 38 ,the investor will lose the

deposit and suffer an additional loss, resulting in a return below−100%.

(^3) Take any value ofu, compute the corresponding 21 stock price values after 20 days,
the option payoffs, risk-neutral probability, and finally the option price. Using the
Goal Seek facility in a spreadsheet application (or trial and error), find the value
ofusuch that the option price is as given.