American vs European

The Early-Exercise Premium

The extra exercise right creates an early-exercise premium.

highlighted = computed this step

European comparison

The European put with the same tree cannot exercise early. Its exact value is 800/147 dollars, displayed as $5.44.

VE=800/147≈$5.44V_E=800/147\approx \$5.44
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

Premium

The American put is exact 50/7 dollars, displayed as $7.14. The early-exercise premium is exact 250/147 dollars, displayed as $1.70.

premium=VA−VE=250/147≈$1.70\text{premium}=V_A-V_E=250/147\approx \$1.70

Never negative

The right to exercise early is never worth less than zero, because the holder can always choose not to use it. In this model, American value is at least European value. The premium is a model value under the stated assumptions, not a market price or fair-value claim. This is descriptive, not investment advice.

VA≥VEV_A\ge V_E