Question

3. Put-call Parity Sad Corp (SC) is a distressed firm that is not expected to pay dividends over the next year. SC stock is currently at $10, and it costs $7 to buy an at-the money call option on SC maturing one year from now. The price of a riskfree zero-coupon bond with a face of $ 100 maturing one year from now is $95. Assume there is no arbitrage. a. (5 points) If the call option described above is a European option, what is the price today of an at-the-money European put option on SC maturing one year from now? b. (5 points) If the call option described above is an American option, what do we know about the price today of the same European put option? c. (5 points) If the call option described above is an American option, what do we know about the price today of an at-the-money American put option on SC maturing one year from now?

Answer 1

Solution:

We know that as per the Put Call Parity

C0 + X*e^-rt = P0 + S0

Where C0 is the call option price

X -> bond price

r -> discount rate /Yield of the bond

t -> time to maturity.

P0 -> price of the put option

S0 -> Spot price of the stock

a) We know that the given is the European call option so we can find the price of the European put option by using the put call parity.

Now we are given C0 = 7$ , S0 = 10$, t= 1 year , X =100 and X*e^-rt = 95 (present value or the price of the bond)

Inputting the values in the formula we get,

7 + 95 = P0 + 10

P0= 102-10 = 92

The price of the put option is 92$.

b) we know that the American options can be exercised anytime before the maturity so in comparison to the European options the price of an American option will always be higher.

The mentioned call option is an American Call which pays no dividend i.e. it will have the same price as an European call options and using the price of the European call we can apply PCP to calculate the price of the EU Put option i.e. 92$ like we calculated in the previous part.

c) The above mentioned theory does not hold true in the case of American put options as the put options are short interest rates and in order for the price of the European Put = Price of the American Put the rates must be zero.

So the price of the American put will be higher than the price of the European Put what we calculated in the previous parts.