American Compared With European Option Exercise

A deep in the money American put earns its holder cash sooner than the identical European put, and the gap is priced by interest rates.

An American option can be exercised on any day up to and including expiry, while a European option can only be exercised on the expiry date itself, and that single difference in timing is worth real money whenever a position is deep in the money before expiry arrives. Listed single stock options in the United States are American style by default, while index products such as the S&P 500 options trade European style, and a candidate who cannot explain why the two conventions coexist has not fully understood the mechanics behind either one.

The right to exercise early is only valuable if there is a reason to want the cash or the stock before expiry, and the main reason is interest. A deep in the money put holder who is entitled to receive the strike price gives up interest on that cash for every day the exercise is deferred, so the American version of the same put should trade at or above the European version, never below it. A deep in the money call holder faces the opposite calculation, since exercising early means paying the strike out early rather than later, and giving up cash sooner is a cost rather than a benefit, which is why American and European calls on a stock that pays no dividend end up worth the same in practice.

Take a stock trading at $40 with a put struck at $60 and 30 days to expiry, and assume a 5 percent annualized risk free rate. Exercising an American put immediately locks in the full $20 of intrinsic value today, since the holder delivers stock and receives the strike in cash. A European put on the same terms cannot be exercised until expiry, so its value is bounded below by the discounted strike minus the stock price, which is $60 times 0.9959 minus $40, or $19.75. Priced through a standard options model at a reasonable volatility assumption for a stock this deep in the money, the European put comes out almost exactly at that $19.75 lower bound, since there is almost no chance the stock…

The American put therefore trades close to its $20.00 intrinsic value while the European put trades close to $19.75, a 25 cent gap. Part of that gap is pure interest: giving up the chance to invest $20 for 30 days at 5 percent costs roughly 8 cents, and the rest reflects that a European holder who wants the cash today has no choice but to sell the option in the market rather than exercise it, and the market price already reflects that the cash is not available until expiry.

For a call on a stock that pays no dividend, holding the option is always at least as good as exercising it early, because exercising throws away the remaining time value in exchange for intrinsic value the holder could instead capture by simply selling the call in the market. This result, and the narrower conditions under which it breaks down once dividends are introduced, is covered in full with a worked dividend threshold in the companion piece on early exercise of American calls.