### European versus American Options

European options can only be exercised at the expiry date while American options can be exercised ahead of the expiry date. This leads to some circumstances in which the pricing of the two different option types differs.

## American Options

Early exercise if not required, so the right to exercise early embedded in the American option cannot have a negative value. Thus, American options cannot sell for less than European options. Using capital letters to denote American options and lowercase letters for European option, we have:

$$C_0 \geq c_0$$

$$P_0 \geq p_0$$

The minimum values for American options are as follows:

$$C_0 = max(0, S_0 – X)$$

$$P_0 = max(0, X – S_0)$$

The minimum values for European options were previously established as:

$$c_0 \geq max(0, S_0 – \frac{X}{(1+r)^T})$$

$$p_0 \geq max(0, \frac{X}{(1+r)^T} – S_0)$$

### How can we use these Formulas?

By comparing the minimum values of American options with European options, we see that So – X/(1 + r)T is greater than So – X. (For example, $110 –$100/1.05 = $14.96 ≥$110 – \$100 = 10.) Given that the American price cannot be less than the European price, we re-establish the American call price minimum as equivalent to the European call price minimum.

Comparing X – S0 with (X/(1 + r)T) – So shows that the American put price is never less than the European put price meaning the American minimum put price is Max(0, X – S0).

The somewhat counterintuitive finding is that an American call is always worth more in the market than exercised which means an American call option should basically never be exercised.

## Exceptions

### Call Options

However, a possible factor that can affect early exercise is when the underlying asset is a dividend-paying stock. When a stock goes ex-dividend, its price falls by the amount of the dividend and an investor holding a call option may find it worthwhile to exercise the call just before the stock goes ex-dividend. If the underlying is a bond paying a coupon, a similar argument may hold. However, if there are significant carrying costs associated with the underlying (such as in the case of an option written on a commodity), then the motivation for early exercise is weakened.

### Put Options

As the minimum value of the American put exceeds the minimum value of the European put, the motivation for early exercise is stronger. Dividends and coupons would discourage the early exercise of a put option but carrying costs would encourage the early exercise.

## Question

In considering an American call option, which statement is least accurate?

A. An American call option will be exercised early if the underlying price is trading at less than the exercise price

B. An American call option will be exercised early if the underlying asset carries significant holding costs

C. An American call option will be exercised early if the underlying stock is about to go ex-dividend

Solution

An American call option is likely to be exercised early just prior to the underlying going ex-dividend.

Option A is incorrect. As the option holder cannot be forced to exercise early, they would never choose to do if the exercise price was higher than the price of the underlying.

Option C is incorrect. Significant holding costs make owning the option preferable to owning the underlying asset.

Explain under which circumstances the values of European and American options differ

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