Early Exercise

The American Option

An American option adds the right to exercise before expiry.

highlighted = computed this step

Exercise timing

An American put can be exercised any time up to expiry. A European option waits until expiry. Here the strike is $105.00 on a two-step tree that starts at $100.00, moves up by 6/5 or down by 9/10, and uses a 5% risk-free rate.

K=$105.00,S0=$100.00,r=5%K=\$105.00,\quad S_0=\$100.00,\quad r=5\%
American put treeThe 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

Tree values

The diagram shows the recomputed American put values. The orange badge marks a node where immediate exercise beats holding.

American value is recomputed at each node\text{American value is recomputed at each node}

Model note

This is a binomial model with constant up and down factors, one stated rate, frictionless trading, and no default or trading costs.

d<1+r<ud<1+r<u