Is there a pricing relationship between knock-in and knock-out barrier options?

Question

AcadiFi · Accepted Answer

Yes, there is an elegant parity relationship. For identical strike, maturity, and barrier level, the sum of a knock-in and a knock-out option (both calls or both puts) equals the value of an otherwise identical vanilla option, assuming no rebate.

Intuition: at any moment before expiry, the underlying either has or has not touched the barrier. If it has, the knock-in is alive and the knock-out is dead. If it has not, the knock-out is alive and the knock-in is dormant. At expiry, exactly one of the two pays off the vanilla payoff, and together they replicate the vanilla.

Example from analyst Helena Bjornsdottir at Northfjord Asset Management: a vanilla call on Torvald Mining Corp struck at 55 is worth 4.80. A down-and-out call with barrier 45 trades at 3.20. By parity, the down-and-in call with the same parameters should be worth approximately 1.60.

This parity is powerful for arbitrage checks and for constructing hedges. If a dealer has sold a down-and-in call, they can partially hedge by buying a vanilla call and shorting a down-and-out call.

Note that the parity breaks if rebates are paid on knock-out, or if the two options have different monitoring frequencies. Always verify the contract specifications before applying the relationship.

Is there a pricing relationship between knock-in and knock-out barrier options?

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