# Black-Scholes Model for American Options

There is no close-form solution for American-style option up to now. For applying Black-Schloes-Merton model to American options, let us consider non-dividend paying American call and put options, and dividend paying American call and put options separately.

Non-Dividend Paying American Call Option
Analysis shows in case of non-dividend paying American call option it is always optimal to exercise the option at expiry. So, in case of non-dividend paying American call option the value of American call option is equivalent to non-dividend paying European call option.

Hence, when there are no dividends the value of American call option can be calculated by using the Black-Scholes-Merton formula

Where

Same as the European call option because in case of non-dividend paying American call option it is always optimal to exercise the option at expiry.

Non-Dividend Paying American Put Option
We can approximate the value of the non-dividend paying American put option using put-call parity for non-dividend paying American option

This inequality gives the value of non-dividend paying American put option with lower and upper bounds, for example

The value of the put is between these two values, inclusive.

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