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A European option gives us the right to buy or sell an asset at a fixed price, but only on a particular expiry date. Surprisingly for the case of vanilla options, despite the apparent extra utility of American options, it turns out that the price of American and European options is almost always the same! In general, American options are MUCH harder to price than European options, since they depend in detail on the path that the underlying takes on its way to the expiry date, unlike Europeans which just depend on the terminal value, and no closed form solution exists.
So we can always take the European price to be a lower bound on American prices. Also note that Put-Call Parity no longer holds for Americans, and becomes instead an inequality. How can we go any further? This is the volatility-dependent part of the price, since we are shielded by the optionality from price swings in the wrong direction, but are still exposed to upside from swings in our favour.
Consider the graph above, which shows the BS value of a simple European call under typical parameters. Time value is maximal at-the-money, since this is the point where the implicit insurance that the option provides is most useful to us far in- or out-of-the-money, the option is only useful if there are large price swings, which are unlikely.
What is the extra value that we should assign to an American call relative to a European call due to the extra optionality it gives us? In the case of an American option, at any point before expiry we can exercise and take the intrinsic value there and then. This means that we can sell the option on the market for more than the price that would be received by exercising an American option before expiry — so a rational investor should never do this, and the price of a European and American vanilla call should be identical.
It seems initially as though the same should be true for put options, but actually this turns out not quite to be right. Consider the graph below, showing the same values for a European vanilla put option, under the same parameters. Notice that here, unlike before, when the put is far in-the-money the option value becomes smaller than the intrinsic value — the time value of the option is negative!
What is it that causes this effect for in-the-money puts? It turns out that it comes down to interest rates. Roughly what is happening is this — if we exercise an in-the-money American put to receive the intrinsic value, we receive cash straight away. But if we left the option until expiry, our expected payoff is roughly , where is the forward value. For vanilla options, this is given by.
The plot below shows Theta for the two options shown in the graphs above, and sure enough where the time value of the European put goes negative, Theta becomes positive — the true option value is increasing with time instead of decreasing as usual, as the true value converges to the intrinsic value from below.
In between European and American options lie Bermudan options, a class of options that can be exercised early but only at one of a specific set of times. Since we have an analytical price, we can also calculate an expression for the GREEKS of this option by differentiating by the various parameters that appear in the price. Moreover, differentiating equation  above shows that the greeks of a digital put are simply the negative of the greeks of a digital call with the same strike.
Graphs of these are shown for a typical binary option in the following graphs. One final point on pricing, note that the payoff of a digital call is the negative of the derivative of a vanilla call payoff wrt. This means that any binary greek can be calculated from the corresponding vanilla greek as follows.
Price and first-order greeks for a digital call option. Second-order greeks for a digital call option. Greeks for digital puts are simply the negative of these values One final point on pricing, note that the payoff of a digital call is the negative of the derivative of a vanilla call payoff wrt.