The purpose
An options market maker wants to be paid for pricing volatility, not to be exposed to whether the stock goes up or down. Delta hedging strips out the direction.
Sell a call with delta 0.6 and buy 0.6 shares. Small moves in the stock now leave the combined position roughly unchanged.
Why it goes stale
Delta changes as the stock moves - that is gamma. A hedge that is correct at 100 is wrong at 102.
So delta hedging is a continuous process, not a one-off trade. In theory you rehedge continuously; in practice you rehedge on a schedule or on a delta threshold.
The rehedging trade-off
Rehedge often: small hedge error, high transaction costs.
Rehedge rarely: low costs, large residual risk between adjustments.
Real desks use bands - rehedge when delta drifts beyond a tolerance - with the width set by the ratio of transaction costs to gamma. This is a genuinely interesting optimisation and a good thing to raise unprompted.
What is left after hedging
Delta hedging removes first-order price risk. It leaves:
- Gamma - exposure to the size of moves.
- Vega - exposure to changes in implied volatility.
- Theta - time decay.
That residual is the intended position: a bet on realised volatility versus implied.
The classic interview question
"You sell an option and delta hedge perfectly. Can you still lose money?"
Yes, and the answer distinguishes candidates. You are short gamma, so you rehedge by buying as the market rises and selling as it falls - buying high and selling low, repeatedly. If realised volatility exceeds the implied volatility you sold at, those hedge losses exceed the premium you collected.
Perfect hedging removes directional risk. It does not remove volatility risk, because that is the risk you deliberately took on.
Practise in finance and derivatives.