Delta-hedging
Quantitative Finance glossary · Derivatives Pricing & Volatility Surfaces
Definition
Continuous rebalancing of the underlying position to keep the portfolio delta-neutral: hold -Δ_t shares against a long option. In the BSM idealisation rebalancing is continuous and frictionless, perfectly replicating the payoff at cost equal to the BSM price. In practice, rebalancing is discrete (incurring gamma slippage proportional to 1/2Gamma,(Δ S)^2 per step) and pays transaction costs, so dealers hedge at intervals that trade off these two; the canonical analysis is Leland (1985), who derives a vol-adjusted BSM with proportional costs.
Delta-hedging in plain English
Delta-hedging is how an options dealer cancels out the risk of the stock moving: since delta tells you the option behaves like holding some number of shares, you take the opposite share position to stay 'neutral' — flat no matter which way the stock ticks. In the idealized textbook world you rebalance this offset constantly and for free, and it perfectly recreates the option's payoff at exactly its fair price. In real life you can only adjust every so often and each trade costs money, so a little error creeps in each time the stock jumps (roughly 1/2Gamma(Δ S)^2 per step) — like trying to keep a canoe pointed straight by paddling only occasionally instead of continuously. Dealers pick a rebalancing rhythm that balances those two annoyances; the classic study of this trade-off is Leland (1985).
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