A static hedge is built to reproduce the cash flows of the instrument being hedged, approximately or, in theory, exactly. The combination is built from vanilla options, for example a bought call together with a sold put of a different strike. Unlike a dynamic hedge, which adjusts the position in the underlying asset as the market moves, it needs no continuous trading123.
In the barrier option example, the payoff when maturity is reached without touching the barrier is replicated by a vanilla call, and the quantities of the other options are chosen so that the value of the combination is zero when the barrier is touched3.
Existing methods that aim at perfect replication rely on strong assumptions that options of particular strike prices and maturities can be traded and on the distribution of the underlying asset4. A combination built on the assumption of constant volatility has a large hedging error when volatility changes3.
Notes
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Dominique Y. Dupont, "Hedging Barrier Options: Current Methods and Alternatives" (University of Twente, February 2002), page 8. It gives an example of hedging a down-and-out call with strike K and barrier H (H < K) by buying a call with strike K and selling puts with strike H²/K. ↩
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Dupont, cited above, page 4. It states that with market frictions dynamic hedging becomes too costly, and practitioners rebalance less often and adopt static hedging. ↩
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Yuki Shinbara, "確率局所ボラティリティ・モデルのもとでのヘッジ戦略" (Hedging strategies under stochastic local volatility models), Kinyu Kenkyu, Institute for Monetary and Economic Studies, Bank of Japan (October 2011, in Japanese), Section 1 "はじめに" (Introduction) (page 126) and Section 2 (1) ロ (page 132). ↩ ↩2 ↩3
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Dupont, cited above, page 1 (abstract). It states that current static hedging methods rely on strong assumptions on the availability of traded options with certain strikes or maturities, or on the distribution of the underlying asset. ↩