1.  Definition

American options give the right (but not the obligation) to the holder to buy or sell the underlying asset at a predetermined price, on or before the maturity date.

2.  American v European options

The difference between European and American options lies within their optionality features: A European contract can be exercised only on the maturity date, while an American contract can be exercised at any point on or before maturity. Thus, an American option will have an early exercise premium: American option prices will be at least as high as their European counterparts.

3.  Pricing of American options

This early exercise option makes pricing American calls and puts a more complex task, compared to a European option. Together with the price, an early exercise boundary is also retrieved. If the underlying asset crosses this boundary, exercising the option becomes the optimal strategy.

It turns out that early exercise is never optimal for a call option, on an asset that pays no dividends. Intuitively, buying the asset at a fixed price earlier cannot be a better strategy than waiting on maturity before making that decision: After buying the asset the price might drop, and losses will be realized; waiting is better in that case. One would be indifferent towards price increases, since the asset can be acquired at the exercise price on the maturity date anyway.

Apparently, high dividend payments can make early exercise beneficial, considering that holding the asset entitles one to these dividends. Also, American puts can be exercised early, if the spot price is very low.

The price of American option together with the exercise boundaries can be retrieved by

• Appriximations, such as the Barone-Adesi formula
• Numerically solving the corresponding Black-Scholes partial differential equation under a free boundary condition
• Using a lattice, e.g. binomial, trinomial, etc.

4.  Examples

In the examples below, the red area represents the region where early exercise is optimal. If the price is in the blue region, it is optimal to hold the option and not exercise. In all graphs we assume: strike price $100, interest rate of 5% and volatility of 20%. One axis gives the different maturities, up to one year; the other axis gives the possible spot prices, up to $150.

The graph below gives the price of an American put, together with the region of optimal early exercise (in red). When the stock-maturity combination moves into this area, we should exercise immediately.

The American call is always worth more than its European counterpart (shown in green). This is due to an extra premium for the early exercise.

An American call (on an non-dividend paying asset) should never be exercised early. The whole region below is blue.

On the other hand, if the stock pays dividends (here the yield is 20%), early exercise can become optimal even for a call.

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