The Hedge Ratio

The Delta-Hedged Portfolio

A delta-hedged position is riskless over the step, which fixes the price.

highlighted = computed this step

The delta-hedged portfolio

Hold 13/21 share of the underlying against one short option. Because delta is chosen so the position moves the same amount in the up and down states, the hedged portfolio is riskless over the step.

Δ(Su−Sd)=Vu−Vd\Delta(S_u-S_d)=V_u-V_d

The price is the cost of the hedge

A riskless one-step position must earn the stated rate, so no-arbitrage fixes today's option value. Rolling this hedge back through the tree gives the model price $10.20.

ΔS−V is riskless⇒V=$10.20\Delta S-V\text{ is riskless}\Rightarrow V=\$10.20

Model note

The delta hedge is riskless only under the stated model, and holding it requires the frictionless assumptions above. This is descriptive, not investment advice.

riskless under the stated model\text{riskless under the stated model}