Question

1. A European call option and put option on a stock both have a strike price of $20 and an expiration date in three months. Both sell for $3. The risk-free interest rate is 10% per annum, the current stock price is $19, and a $1 dividend is expected in one month. Is there an arbitrage opportunity? If there is an arbitrage opportunity, clearly state what condition must be satisfied to eliminate the arbitrage opportunity. What is the strategy followed to make a profit from the arbitrage opportunity? What is the profit expressed as a present value?

Answer 1

The call is worth $3, put-call parity shows that the put should be worth :

3+20e^-0.10*3/12 + e^-0.1*1/12 - 19 = 4.50

This is greater than $3. The put is therefore undervalued relative to the call. The correct arbitrage strategy is to buy the put, buy the stock, and short the call. This costs $19. If the stock price in three months is greater than $20, the call is exercised. If it is less than $20, the put is exercised. In either case the arbitrageur sells the stock for $20 and collects the $1 dividend in one month.

The present value of the gain to the arbitrageur is

-3-19+3+20e^-.10*3/12+e^{-0.10*1/12 = 1.50}