Question

A. What would happen in the options market if the price of an American call were
less than the value Max (0, S0 − X)? Would your answer differ if the option were
European? Explain.
B. Critique the following statement made by an options investor in American call
option: “My call option is very deep in-the-money. I don’t see how it can go any
higher. I think I should exercise it.”
C. Why do higher interest rates lead to higher call option prices but lower put option
prices?
D. Suppose a European put price exceeds the value predicted by put–call parity. How
could an investor profit? Demonstrate that your strategy is correct by constructing
a payoff table showing the outcomes at expiration.
E. Consider a two-period, two-state world. Let the current stock price be 45 and the
risk-free rate be 5 percent. Each period the stock price can go either up by 10 percent
or down by 10 percent. A call option expiring at the end of the second period has
an exercise price of 40.
1. Find the stock price sequence.
2. Determine the possible prices of the call at expiration.
3. Find the possible prices of the call at the end of the first period.
4. What is the current price of the call?

Answer 1

A

The Intrinsic value of a call option is equal to max(S0-X,0) where S0 is the stock price today and X is the strike price  

If the price of an American call were less than the value Max (0, S0 − X) i.e. less than its Intrinsic value , there can be immediate arbitrage opportunity by purchasing the option and exercising it immediately

Thus, the price of the American Call option can never be less than Max (0, S0 − X)

If the call option were European , it cannot be exercised right away.. However, arbitrage opportunities are still available if short selling is allowed, the stock can be short sold and  the option purchased and remaining amount invested . At maturity, the stock can be bought back at the option striike price. Thus , the payoff at maturity = Max (0, S0 − X) whereas the cost for option purchase at beginning is lesser than the same. Hence there are still arbitrage opportunities available if short selling is allowed

Thus, the price of the European Call option is also never less than Max (0, S0 − X) (provided short selling is allowed)