Backward Induction

One Step Back

Backward induction starts from expiry and prices the previous nodes.

highlighted = computed this step

One-period rule

The risk-neutral probability comes from the previous one-period no-arbitrage model. Plugging in 1 plus 0 minus 9/10 over 6/5 minus 9/10 gives numerator 1/10 and denominator 3/10, so q is 1/3.

q=(1+r)−du−d=1+0−9/106/5−9/10=1/103/10=1/3,1−q=2/3q=\frac{(1+r)-d}{u-d}=\frac{1+0 - 9/10}{6/5 - 9/10}=\frac{1/10}{3/10}=1/3,\quad 1-q=2/3
Backward induction treeThe call value is recomputed at every tree node.updowntodayS $100.00V $5.67t1 u0S $90.00V $1.00t1 u1S $120.00V $15.00t2 u0S $81.00V $0.00t2 u1S $108.00V $3.00t2 u2S $144.00V $39.00

Up node value

The up node is worth $15.00, found from the all-up and middle terminal payoffs.

Vu=1/3⋅$39.00+2/3⋅$3.001+r=$15.00V_u=\frac{1/3\cdot \$39.00+2/3\cdot \$3.00}{1+r}=\$15.00

Down node value

The down node is worth $1.00, found from the middle and all-down terminal payoffs.

Vd=1/3⋅$3.00+2/3⋅$0.001+r=$1.00V_d=\frac{1/3\cdot \$3.00+2/3\cdot \$0.00}{1+r}=\$1.00