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 1/20 minus 9/10 over 6/5 minus 9/10 gives numerator 3/20 and denominator 3/10, so q is 1/2.

q=(1+r)−du−d=1+1/20−9/106/5−9/10=3/203/10=1/2,1−q=1/2q=\frac{(1+r)-d}{u-d}=\frac{1+1/20 - 9/10}{6/5 - 9/10}=\frac{3/20}{3/10}=1/2,\quad 1-q=1/2
Backward induction treeThe call value is recomputed at every tree node.todaydownupS $100.00V $10.20S $90.00V $1.43S $120.00V $20.00S $81.00V $0.00S $108.00V $3.00S $144.00V $39.00

Up node value

The up node is worth $20.00, found from the all-up and middle terminal payoffs.

Vu=1/2⋅$39.00+1/2⋅$3.001+r=$20.00V_u=\frac{1/2\cdot \$39.00+1/2\cdot \$3.00}{1+r}=\$20.00

Down node value

The down node is worth $1.43, found from the middle and all-down terminal payoffs.

Vd=1/2⋅$3.00+1/2⋅$0.001+r=$1.43V_d=\frac{1/2\cdot \$3.00+1/2\cdot \$0.00}{1+r}=\$1.43

Model note

These node values are model prices under the stated binomial assumptions, not market prices. The calculation uses frictionless one-period no-arbitrage with one stated rate and no transaction costs, fees, taxes, credit risk, or liquidity limits. This is descriptive, not investment advice.

node values are model prices\text{node values are model prices}