Backward Induction

Rolling Back to Today

The same one-period rule rolls the tree back to today's price.

highlighted = computed this step

Roll back to today

Apply the same rule once more. Today's exact price is 50000/49 cents, which displays as $10.20.

V0=1/2⋅$20.00+1/2⋅$1.431+1/20=50000/49 cents≈$10.20V_0=\frac{1/2\cdot \$20.00+1/2\cdot \$1.43}{1+1/20}=50000/49\text{ cents}\approx \$10.20
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

Initial hedge ratio

The initial replication delta is 13/21 share. It is the first-step option value change divided by the first-step stock value change.

Δ0=$20.00−$1.43$120.00−$90.00=13/21\Delta_0=\frac{\$20.00-\$1.43}{\$120.00-\$90.00}=13/21

Model note

Backward induction gives a model price under the stated constant-move, one-rate, frictionless assumptions, not a market price. It repeats one-period no-arbitrage with no transaction costs, fees, taxes, credit risk, or liquidity limits. This is descriptive, not investment advice.

same local rule at each node\text{same local rule at each node}