## American option using BSM model and related all equation

missvidlalsare
Good Student
Posts: 14
Joined: Mon Jan 30, 2017 4:05 am

### American option using BSM model and related all equation

Not able to understand BSM model and black approximation for American option given on page 86 and 87 schweser notes VAR may Nov 17
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edupristine
Finance Junkie
Posts: 946
Joined: Wed Apr 09, 2014 6:28 am

### Re: American option using BSM model and related all equation

Hi,

A logical way of answering this question is proof by contradiction. First note that if it is not optimal to exercise an american option prior to expiry, then the option should have the same value as the European option.

So assume that it is not optimal to exercise the option prior to expiry, i.e. assume that the american option has the same value as the European option.

Determine the value of the option using the Black-Scholes formula, for a range of initial spot prices, with all other parameters fixed.

Plot a graph of the option values against the spot prices, and on the same chart plot the payoff against the spot prices.

You observe that for a call option with no dividends, for any given spot price, the option value always exceeds the corresponding payoff value. This means that it will always be profitable to sell on, or indeed hold the option, rather than to exercise - i.e. it is never optimal to exercise the option prior to expiry.

On the other hand, where there is a dividend and the spot price is large enough, the payoff is larger than option value - i.e. it is optimal to exercise the option. This is where the initial assumption that it is not optimal to exercise the option prior to expiry is contradicted, in which case we can't hold on to the argument that the European and American call options are of the same value.

missvidlalsare
Good Student
Posts: 14
Joined: Mon Jan 30, 2017 4:05 am

### Re: American option using BSM model and related all equation

Thank you so much sir