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 $42.86 borrowed.

1/2⋅$100.00−$42.86=$7.141/2\cdot \$100.00 - \$42.86=\$7.14
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 $7.14, the same as the replicating portfolio. Any other price creates a riskless arbitrage.

V0=$50.00−$42.86=$7.14V_0=\$50.00 - \$42.86=\$7.14

Model note

This is a model price by no-arbitrage under the stated one-period, two-state assumptions, not a market price. It assumes frictionless trading, one stated rate, no transaction costs, fees, taxes, credit risk, or liquidity limits, and the ability to form the replicating portfolio. Mispricing means a riskless profit in the model. This is descriptive, not investment advice.

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