Question

A stock that does not pay dividend is trading at $20. A European call option with strike price of $15 and maturing in one year is trading at $6. An American call option with strike price of $15 and maturing in one year is trading at $8. You can borrow or lend money at any time at risk-free rate of 5% per annum with continuous compounding. Devise an arbitrage strategy.

Answer 1

As it is never feasible to exercise an American call option on a non dividend paying stock before Maturity, the American call option on a non-dividend paying stock is as good as European option.

Here , as the American call option is trading at a higher price it has to be sold and the European call option purchased

The Arbitrage strategy works as follows :

1. Today, buy the European Call option at $6 by selling the American Call option at $8. Invest the remaining $2 at risk free rate for one year.

2. If at any point of time , the American call option is exercised before maturity , say after 3 months, when the stock price is $25, the European call option would be trading at a higher price than the intrinsic value i.e higher than.( $25-$15 = $10) as it also has some time value before maturity. In any case, the Euopean call option can be easily sold for $10. Get the $15 from the American call buyer and buy the stock from the market at $25 and deliver. In this case , one can make at least $2 as arbitrage profit. (in present value terms).

One can check that even if the American call option is exercised immediately, one can sell the the European call option at $6, get $15 as the Strike price from American call buyer and buy the stock from market at $20, thereby making $1 more than the $2 arbitrage profit as stated above.

If the American call option is exercised at maturity, exercise the European option, get the stock by paying $15 and deliver the stock against $15 from the American Call Buyer. Again, one can make, $2 as arbitrage profit (in present value terms)