Hedging a Call Option with a Two-State Binomial Model

Summary

This explanation derives a call option’s no-arbitrage value by constructing a hedge in a one-period, two-state stock model. The stock is assumed to be worth 100 today and either 110 or 90 at expiration, while a call with a strike of 100 pays 10 in the up state and nothing in the down state. Holding half a share for each short call makes the hedged portfolio worth the same amount in either outcome.

With the risk eliminated and the interest rate assumed to be zero, the portfolio must have the same value today as a risk-free investment paying that amount at expiration. Equating the current stock-and-option portfolio value to this amount gives a call value of 5. The example illustrates replication and arbitrage-free pricing without needing the probability of either state. It is deliberately simplified: it assumes only two possible outcomes, zero interest, and a hedge that can be established and maintained without discussing trading costs or other market frictions.

Key ideas

• A call’s payoff in each possible state determines the hedge required to offset its risk.
• In the example, holding half a share against each short call equalizes the portfolio value across both states.
• A riskless hedged portfolio must be priced consistently with an equivalent risk-free investment.
• The resulting call value does not require specifying the probability of an up or down move.
• The derivation assumes zero interest and a simplified two-state market.

Tags

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.