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Write both expiry equations
For original claim amounts 30 at underlying 130 and zero at 80, let Δ be underlying units and B current cash. The two equations are 130Δ+1.05B=30 and 80Δ+1.05B=0. The same B appears in both states.
Subtract to isolate the underlying holding
The equations differ by 50Δ=30, so Δ=0.6. Substituting into the lower state gives current B=−48/1.05≈−45.714286. The negative cash position is borrowing within this model.
Add the current costs
Underlying cost is 0.6×100=60. Add current cash −45.714286 for replication cost 14.285714. The cost is measured at time zero; using future repayment −48 in that sum mixes dates.
| State | Underlying leg | Expiry cash leg | Total |
|---|---|---|---|
| Higher | 78 | −48 | 30 |
| Lower | 48 | −48 | 0 |
Keep the replication conditional
The holdings match the two supplied claim amounts. This is a mathematical model construction, not an assertion that a real investor can borrow, trade fractional quantities or settle at these prices without costs.