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Finance Option Price Movement Implications

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C1. Value a one-year call option on a stock (Alpha PLC, Ltd.) whose current price is $35.
In one year the stock will be worth either $42 or $29.75, and the riskless interest
rate for the next year is (what else?) 10% The option's exercise price is $40.
What is the option's payoff in the stock's "up" state? Why? What is the option's
payoff in the stock's "down" state? Why? What is the options' value today?

C2. Value another one-year european call option. This time the stock (Omega Corp.) is
currently selling for $50 and the interest rate on one-year treasury bills is 8%.
The stock's terminal value will be either up 40% from its present value or down
35%. The option's strike price is $40.

C3. Now let's see how the riskless interest rate affects the value of the call option on
Omega Corp. Re-do problem #C2 with the riskless rate set to 12% instead of 8%.
What happens to the call option's value? Why do you think this occurs?

C4. Return again to the case of Alpha PLC, Ltd. in Problem C1. What happens to the
call option's value if Alpha's stock price becomes more variable? In particular,
value the call option if the terminal stock price could be either $49 or $26.25.
Why is the option worth more when the Alpha's stock value is more variable?

C5. Return one more time to the initial conditions in problem C1. What is the value of a
call option on Alpha PLC when the option's strike price is $37.50 (instead of
$40.00)? Explain the change in value.

P1. Value a one-year put option on a stock (Alpha PLC, Ltd.) whose current price is $35.
In one year the stock will be worth either $42 or $29.75, and the riskless interest
rate for the next year is (naturally!) 10% The option's exercise price is $30.
What will the option pay off in the stock's "up" state? Why? What will the option
pay off in the stock's "down" state? Why? What is the option worth today?

P2. What is the effect on the option value you computed in P1 if the riskless rate rises?
Why?

P3. Starting from the conditions in Problem P1, what happens to the option's value if
Alpha's stock could be worth either $25 or $45 on the option's maturity date?
Why does the option value rise?

P4. Put options are often compared to insurance. Like insurance, you can choose to
protect yourself to a greater or lesser extent. With an insurance policy, you do
this by varying the policy's deductible. With an option, you vary your protection
by varying the option's strike price. Return again to the conditions in Problem P1.
What is the put option worth today if price changes to $35? What if the exercise
price is $40?
What do you think these option price movements imply about the cost of buying
an insurance policy with a higher vs. a lower deductible?

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The option price movement implications are given.

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C1) To be able to solve this problem, and the subsequent ones, we must first set up the problem with the data we are given. We have the following situation:

The binomial tree shown above can be used to calculate the price of the option today. We will use the following terminology:

C = Value of the call option today
X = Exercise price = $40
T = Time period for which the option is active = 1 year
R = Risk-free interest rate = 10%
S = Price of the stock today = $35
Su = Price of the stock after one year in the up state = $42
Sd = Price of the stock after one year in the down state = $29.75
Cu = Value of the option after one year in the up state
Cd = Value of the option after one year in the down state
u = Stock return in the up state
d = Stock return in the down state
Pr = Risk-neutral probability of a stock movement in the up state
(1 - Pr) = Risk-neutral probability of a stock movement in the down state

We will use the following formulas using binomial risk-neutral option pricing:

(1)
(2)

First we calculate the stock returns in both states:

Now we can calculate the risk-neutral probabilities from formula (1):

Now we calculate the option values (pay-offs) in both states:

Cu = Su - X = 42 - 40 = 2.00 (since the option will be exercised)
Cd = Sd - X = 29.75 - 40 = 0 (since the option will not be exercised in this case and there will be zero pay-off)

We can now use formula (2) to obtain the value of the option ...

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