Question

ABC Company's current stock price is $100.

The risk free rate is 4%

a) The maximum value of a European put option with an exercise price of $50 and 6 months remaining till expiration is closest to?

b)The minimum value of an American call option with an exercise price of $80 and 6 months till expiration is closest to?

Answer 1

a) A put option gives the right to sell one share of stock at a certain price. Under no circumstance can the put be worth more than the sale or strike price. If it were, everyone would sell the option and invest the proceeds at the risk-free rate over the life of the option. Maximum value of put = $50

b) Consider the following two portfolios:
• Portfolio P1: one Americanan call, c, with exercise price X plus a zero-coupon risk-free bond that pays X at T.
• Portfolio P2: one share o f the underlying stock, S.
At expiration, T, Portfolio P1 will always be the greater of X (when the option expires outof- the-money) or ST (when the option expires in-the-money). Portfolio P2, on the other hand, will always be worth ST . Therefore, P1 is always worth at least as much as P2 at expiration. If we know that at T, P1 > P2, then it always has to be true because if it were not, arbitrage would be possible. Therefore, we can state that
c + Xe^(-rT) > S0
Since the value of a call option cannot be negative (if the option expires out-of-the-money, its value will be zero), the lower bound for a European call on a nondividend-paying stock is:
c > max (S0 - Xe^(-rT), 0)

in the given question , (S0 - Xe^(-rT) = 100 - 80e^(-0.04*0.5) =100- 79.60 = 20.4

so minimum value of call option = $20.4