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 1700/3 cents, which displays as $5.67.

V0=1/3⋅$15.00+2/3⋅$1.001+0=1700/3 cents≈$5.67V_0=\frac{1/3\cdot \$15.00+2/3\cdot \$1.00}{1+0}=1700/3\text{ cents}\approx \$5.67
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

Initial hedge ratio

The initial replication delta is 7/15 share. It is the first-step option value change divided by the first-step stock value change.

Δ0=$15.00−$1.00$120.00−$90.00=7/15\Delta_0=\frac{\$15.00-\$1.00}{\$120.00-\$90.00}=7/15

Model note

Backward induction gives a model price from repeated one-period no-arbitrage. It is descriptive, not investment advice.

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