American vs European

When Early Exercise Matters

American options add choices, but the model still has scope limits.

highlighted = computed this step

More choices

American value is at least European value because the holder has every European choice plus the choice to exercise earlier.

more choices cannot reduce holder value\text{more choices cannot reduce holder value}
American put comparisonThe American put value and exercise decision are recomputed.updowntodayS $100.00V $7.14exerciset1 u0S $90.00V $15.00t1 u1S $120.00V $0.00t2 u0S $81.00V $24.00t2 u1S $108.00V $0.00t2 u2S $144.00V $0.00

Call note

For a call on a non-dividend-paying stock, early exercise is not optimal in this model, so the American call equals the European call.

VAcall=VEcallV_A^{\text{call}}=V_E^{\text{call}}

Scope note

The value depends on constant up and down factors, one stated rate, and frictionless trading. The risk-neutral probability is a pricing device, not a forecast. This is descriptive, not investment advice.

q=1/2prices payoffs, not forecast oddsq=1/2\quad \text{prices payoffs, not forecast odds}