The Hedge Ratio

Delta

The Hedge Ratio

Delta is the exact binomial hedge ratio, no calculus needed.

highlighted = computed this step

The hedge ratio

Delta is the replication hedge ratio: the change in the option value over the change in the underlying value across a node's two children. It needs no calculus in the binomial model.

Δ=Vu−VdSu−Sd\Delta=\frac{V_u-V_d}{S_u-S_d}

Delta at the root

At the root the up child is worth $20.00 and the down child $1.43, on underlying $120.00 and $90.00. So the hedge ratio is 13/21 share of stock per option.

Δ0=$20.00−$1.43$120.00−$90.00=13/21\Delta_0=\frac{\$20.00-\$1.43}{\$120.00-\$90.00}=13/21
Binomial treeStock and call values are recomputed at every 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

Model note

The hedge ratio is exact in the stated binomial model with constant move factors, one stated rate, and frictionless trading with no transaction costs, fees, taxes, credit risk, or liquidity limits. It is descriptive, not investment advice.

exact hedge ratio, stated model\text{exact hedge ratio, stated model}