**Black Scholes Doesn't Account for Early Exercise**

The first option pricing model, the Black Scholes model, was designed to evaluate European-style options, which don't permit early exercise. So Black and Scholes never addressed the problem of when to exercise an option early and how much the right of early exercise is worth. Being able to exercise an option at any time should theoretically make an American-style option more valuable than a similar European-style option, although in practice there is little difference in how they are traded.

Different models were developed to accurately price American-style options. Most of these are refined versions of the Black Scholes model, adjusted to take into account dividends and the possibility of early exercise. To appreciate the difference these adjustments can make, you first need to understand when an option should be exercised early.

And how will you know this? In a nutshell, an option should be exercised early when the option's theoretical value is at parity and its delta is exactly 100. That may sound complicated, but as we discuss the effects interest rates and dividends have on option prices, I will also bring in a specific example to show when this occurs. First, let's look at the effects interest rates have on option prices, and how they can determine if you should exercise a put option early.

**The Effects of Interest Rates**

An increase in interest rates will drive up call premiums and cause put premiums to decrease. To understand why, you need to think about the effect of interest rates when comparing an option position to simply owning the stock. Since it is much cheaper to buy a call option than 100 shares of the stock, the call buyer is willing to pay more for the option when rates are relatively high, since he or she can invest the difference in the capital required between the two positions.

When interest rates are steadily falling to a point where the Fed Funds' target is down to around 1.0% and short-term interest rates available to individuals are around 0.75% to 2.0% (like in late 2003), interest rates have a minimal effect on option prices. All the best option analysis models include interest rates in their calculations using a risk-free interest rate such as U.S. Treasury rates.

Interest rates are the critical factor in determining whether to exercise a put option early. A stock put option becomes an early exercise candidate anytime the interest that could be earned on the proceeds from the sale of the stock at the strike price is large enough. Determining exactly when this happens is difficult, since each individual has different opportunity costs, but it does mean that early exercise for a stock put option can be optimal at any time provided the interest earned becomes sufficiently great.

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**The Effects of Dividends**

It's easier to pinpoint how dividends affect early exercise. Cash dividends affect option prices through their effect on the underlying stock price. Because the stock price is expected to drop by the amount of the dividend on the ex-dividend date, high cash dividends imply lower call premiums and higher put premiums.

While the stock price itself usually undergoes a single adjustment by the amount of the dividend, option prices anticipate dividends that will be paid in the weeks and months before they are announced. The dividends paid should be taken into account when calculating the theoretical price of an option and projecting your probable gain and loss when graphing a position. This applies to stock indices as well. The dividends paid by all stocks in that index (adjusted for each stock's weight in the index) should be taken into account when calculating the fair value of an index option.

Because dividends are critical to determining when it is optimal to exercise a stock call option early, both buyers and sellers of call options should consider the impact of dividends. Whoever owns the stock as of the ex-dividend date receives the cash dividend, so owners of call options may exercise in-the-money options early to capture the cash dividend. That means early exercise makes sense for a call option only if the stock is expected to pay a dividend prior to expiration date.

Say you own a call option with a strike price of 90 that expires in two weeks. The stock is currently trading at $100 and is expected to pay a $2 dividend tomorrow. The call option is deep in-the-money, and should have a fair value of 10 and a delta of 100. So the option has essentially the same characteristics as the stock. You have three possible courses of action:

- Do nothing (hold the option).
- Exercise the option early.
- Sell the option and buy 100 shares of stock.

If you exercise the option early and pay the strike price of 90 for the stock, you throw away the 10-point value of the option and effectively purchase the stock at $100. When the stock goes ex-dividend, you lose $2 per share when it opens two points lower, but also receive the $2 dividend since you now own the stock.

Since the $2 loss from the stock price is offset by the $2 dividend received, you are better off exercising the option than holding it. That is not because of any additional profit, but because you avoid a two-point loss. You must exercise the option early just to ensure you break even.

What about the third choice - selling the option and buying stock? This seems very similar to early exercise, since in both cases you are replacing the option with the stock. Your decision will depend on the price of the option. In this example, we said the option is trading at parity (10) so there would be no difference between exercising the option early or selling the option and buying the stock.

But options rarely trade exactly at parity. Suppose your 90 call option is trading for more than parity, say $11. Now if you sell the option and purchase the stock you still receive the $2 dividend and own a stock worth $98, but you end up with an additional $1 you would not have collected had you exercised the call.

Alternately, if the option is trading below parity, say $9, you want to exercise the option early, effectively getting the stock for $99 plus the $2 dividend. So the only time it makes sense to exercise a call option early is if the option is trading at or below parity, and the stock goes ex-dividend tomorrow.

**Conclusion**

Although interest rates and dividends are not the primary factors affecting an option's price, the option trader should still be aware of their effects. In fact, the primary drawback I have seen in many of the option analysis tools available is that they use a simple Black Scholes model and ignore interest rates and dividends. The impact of not adjusting for early exercise can be great, since it can cause an option to seem undervalued by as much as 15%.

Remember, when you are competing in the options market against other investors and professional market makers, it makes sense to use the most accurate tools available.

To read more on this subject, see

*Dividend Facts You May Not Know*.