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.todaydownupS $100.00V $7.14S $90.00V $15.00S $120.00V $0.00S $81.00V $24.00S $108.00V $0.00S $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 option value is a model value under the stated assumptions, not a market price or fair-value claim. It depends on constant up and down factors, one stated rate, early-exercise rules, frictionless trading, and no transaction costs, fees, taxes, credit risk, liquidity limits, or default. 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}