Question

Stock options prices are affected by a number of factors and arbitrage arguments are used to find the fair values of these options.

(1)    Explain the three reasons why the early exercise of an American call option on a non-dividend paying stock is never optimal.

(approx. 100 words)

Answer 1

First of all, this only applies to a call option. The more general formulation is that if you have an option to exchange A for B, and A carries a higher payout than B, you don’t want to exercise early. That applies to the case of a call option to exchange cash (which pays the interest rate) for a non-dividend paying stock (that pays nothing); but not to the case of a put option in which case you’re exchanging a zero payout stock for positive payout cash.

From this formulation you can also see that it might not apply if there are zero or negative interest rates, or non-cash reasons to hold the stock, or costs of holding cash or leaving the position open. Moreover, it’s a theoretical result that holds in perfect markets, there may be tax or administrative or regulatory reasons to exercise early.

Suppose you own a call option to buy one share of XYZ stock in a month for $100. XYZ will pay no dividend over the month. $100 will earn $0.25 of interest over the month.

Suppose you are 100% sure you will exercise in one month. It still makes sense to wait until the end of the month to exercise, because you’ll earn an extra $0.25 keeping the $100 until then. And if you’re not sure you’re going to exercise, that is, if you think XYZ stock might be below $100 in a month’s time, then you certainly don’t want to exercise now, because then you’ll lose money if XYZ falls below $100.

But suppose XYZ is selling today for $200. You could exercise, sell the stock for $200, and invest the $200 to earn $0.50 over the month. But you would be better off shorting XYZ and retaining the option. Now you make $0.75 in interest, on $200 form the short sale proceeds and $100 of exercise price. Plus, if XYZ is selling for under $100 in a month, you can cover the short by purchasing in the market rather than exercising the option.