When American Options Can Be Valued Like European Options

Summary

The document explains when early exercise changes the value of American options on a non-dividend-paying underlying. Under the no-arbitrage assumption and a nonnegative risk-free rate, early exercise of a call is not optimal because it gives up remaining time value. With a positive rate, early exercise of a sufficiently deep-in-the-money put can be worthwhile because the exercise proceeds can earn interest.

For the stated case of zero dividends and a zero risk-free rate, these arguments support treating American calls and puts like European options for pricing and Greeks, subject to the assumptions. The answer points to a standard options reference but offers no numerical example or independent empirical evidence. It also notes an exception: negative rates can make early exercise of calls optimal, though the answer describes rates sufficiently negative for this to occur as rare. The conclusion depends on the stated market assumptions and should not be generalized to dividend-paying underlyings or other rate environments.

Key ideas

• With no dividends and nonnegative rates, early exercise of a call sacrifices time value without a compensating benefit.
• A positive risk-free rate can make early exercise of a sufficiently in-the-money put attractive.
• At a zero risk-free rate, the stated reasoning supports valuing American options like European options under the assumptions.
• Sufficiently negative rates can change the early-exercise conclusion for calls.

Tags

# Can American options with no dividends and zero risk-free rate be treated as European?


# Can American options with no dividends and zero risk-free rate be treated as European?












Let's say you've got American options on a future of a stock index. There are no dividends, and no risk-free rate either (assume $r=0$). Can these options then be treated as European from the perspective of using Black-Scholes to price them and calculate the Greeks?

## Answer by Bob Jansen (score 7, accepted)

https://quant.stackexchange.com/a/3441

All this assumes the absence of arbitrage:

As you probably know without dividends it's is never optimal to early exercise a call option on a non dividend paying stock because then the time value is lost, if $r$ is non-negative.

Early exercise of an American put option can be optimal if the option is sufficiently far in the money and $r > 0$. Then you can earn interest on the money gained.

So yes, see also Hull, 7nd edition, chapter 9 'Properties of Stock Options', section 9.6 in particular.

In the rare case that $r < 0$ it can be optimal to exercise call options. However $r$ rarely, if ever, goes sufficiently negative on its own.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.