Due to the fact that an American option gives more rights than a European option, the value of an American option can never be lower than a European option. This does not mean that an American option should cost more.
There is a rule. Early exercise of an American option on a share that does not pay dividends is never the best solution. Consequently, the price of an American option must be equal to the price of an equivalent European option on a similar share.
If an American call option that is in-the-money is exercised in advance, the optionholder will only be able to exercise the intrinsic value (difference between the spot price of the underlying and the strike price). However, the same call can be sold on exchange at the market price, which consists of the intrinsic and the time value.
Early execution will simply result in the loss of the time component. And because the stock does not pay dividends, the investor will not lose anything from the option exercise in order to receive the dividend. Since it makes no practical sense to exercise an American option before the expiration date, the price of the American option should be equal to that of a European option.
It should be noted that if a stock does pay a dividend, the discounted value of the dividend payout may exceed the loss in time value of the option. In this case, executing the option before the ex-dividend date will generate more profit than selling it.
On the other hand, sometimes the exercise of in-the-money American put option before the exercise date makes sense even if the stock does not pay dividends. For example, a trader holds a put option with an expiration date in one year and a strike of $110. The price of the underlying asset is almost zero and its volatility is very low. Since the price of the asset cannot be negative, the profit potential is almost exhausted and the funds received from the early exercise of the option can be reinvested at the risk-free interest rate right now. Thus, the interest earned can sometimes exceed the potential profit from a share falling to zero. In such cases, the option may have a negative time value.
Put-call parity and American options
Put-call parity only applies to European options. (Read Put-call parity: formula, examples, graphs). For American options on a similar underlying asset with a similar strike and expiry date, the following equation can be used:
C – P ≤ S – Xe-rt
Call – Put ≤ Asset price - Discounted strike price
We use the inequality mark because the value of the American put option may exceed the value of the European put (in case the put option is deeply in-the-money).