American Call versus European Call

American Call versus European Call

Many of the results so far only apply to European calls. In many cases, however, American calls behave exactly like European calls. In fact, an American call can be viewed as a European call with the added feature that it can be exercised early. Since exercising an option is never mandatory an American call will be at least as valuable as a European call with the same terms:

                                                  C_a(S₀, T, X) ≥ C_e(S₀, T, X)

We already proved that the minimum value of an American call is Max(0, S₀ - X) while the lower bound of a European call is Max(0, S₀ - X(1+r)^-T). Because S₀ - X(1+r)^-T is greater than S_O - X, the lower bound value of the American call must also be Max(0, S₀ - X(1+r)^-T).

With the lower bound of an American call established, we can now examine whether an American call should ever be exercised early. If the stock price is S_o, exercising the call produces a cash flow of S_O – X. Since the cash flow from exercising, S_O - X, can never exceed the call's lower bound, S₀ - X(1+r)^-T, it will always be better to sell the call in the market.

When the transaction cost of exercising is compared to the transaction cost selling a call, the argument that a call should not be exercised early is strengthened.