Option Values at a Glance

So far, our discussion of option values has assumed that investors hold the option until matu­rity. That is certainly the case with European options that cannot be exercised before maturity but may not be the case with American options that can be exercised at any time. Also, when we valued the Amazon call, we could ignore dividends, because Amazon did not pay any. Can the same valuation methods be extended to American options and to stocks that pay dividends?

Another question concerns dilution. When investors buy and then exercise traded options, there is no effect on the number of shares issued by the company. But sometimes the company itself may give options to key employees or sell them to investors. When these options are exercised, the number of outstanding shares does increase, and therefore the stake of existing stockholders is diluted. Option valuation models need to be able to cope with the effect of dilution. The Beyond the Page feature shows how to do this.

In this section, we look at how the possibility of early exercise and dividends affect option value.

American Calls—No Dividends Unlike European options, American options can be exer­cised any time. However, we know that in the absence of dividends, the value of a call option increases with time to maturity. So, if you exercised an American call option early, you would needlessly reduce its value. Because an American call should not be exercised before matu­rity, its value is the same as that of a European call, and the Black-Scholes model applies to both options.

European Puts—No Dividends If we wish to value a European put, we can use the put-call parity formula from Chapter 20:

Value of put = value of call – value of stock + PV(exercise price)

American Puts—No Dividends It can sometimes pay to exercise an American put before maturity in order to reinvest the exercise price. For example, suppose that immediately after you buy an American put, the stock price falls to zero. In this case, there is no advantage to holding onto the option because it cannot become more valuable. It is better to exercise the put and invest the exercise money. Thus, an American put is always more valuable than a European put. In our extreme example, the difference is equal to the pres­ent value of the interest that you could earn on the exercise price. In all other cases, the difference is less.

Because the Black-Scholes formula does not allow for early exercise, it cannot be used to value an American put exactly. But you can use the step-by-step binomial method as long as you check at each point whether the option is worth more dead than alive and then use the higher of the two values.

European Calls and Puts on Dividend-Paying Stocks Part of the share value comprises the present value of dividends. The option holder is not entitled to dividends. Therefore, when using the Black-Scholes model to value a European option on a dividend-paying stock, you should reduce the price of the stock by the present value of the dividends to be paid before the option’s maturity.

Dividends don’t always come with a big label attached, so look out for instances where the asset holder gets a benefit and the option holder does not. For example, when you buy foreign currency, you can invest it to earn interest; but if you own an option to buy foreign currency, you miss out on this income. Therefore, when valuing an option to buy foreign cur­rency, you need to deduct the present value of this foreign interest from the current price of the currency.[1]

American Calls on Dividend-Paying Stocks We have seen that when the stock does not pay dividends, an American call option is always worth more alive than dead. By holding on to the option, you not only keep your option open, but also earn interest on the exercise money. Even when there are dividends, you should never exercise early if the dividend you gain is less than the interest you lose by having to pay the exercise price early. However, if the dividend is sufficiently large, you might want to capture it by exercising the option just before the ex-dividend date.

The only general method for valuing an American call on a dividend-paying stock is to use the step-by-step binomial method. In this case, you must check at each stage to see whether the option is more valuable if exercised just before the ex-dividend date than if held for at least one more period.

Source:  Brealey Richard A., Myers Stewart C., Allen Franklin (2020), Principles of Corporate Finance, McGraw-Hill Education; 13th edition.

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