Replication & Price

The No-Arbitrage Price

A derivative price is pinned by the cost of its replicating portfolio.

highlighted = computed this step

Portfolio cost

The replicating portfolio costs $50.00 for the stock part minus the $45.00 borrowed.

1/2⋅$100.00−$45.00=$5.001/2\cdot \$100.00 - \$45.00=\$5.00
Replicating call valueThe derivative value is recomputed from the one-period tree.updownTodayS0 $100.00V0 $5.00Up stateSu $120.00Vu $15.00Down stateSd $90.00Vd $0.00

No-arbitrage price

The call must cost $5.00, the same as the replicating portfolio. Any other price creates a riskless arbitrage.

V0=$50.00−$45.00=$5.00V_0=\$50.00 - \$45.00=\$5.00

Model note

This is one price by no-arbitrage inside the stated model. Mispricing means a riskless profit in the model; this is descriptive, not advice.

same payoff⇒same price\text{same payoff}\Rightarrow \text{same price}