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A delta hedge cancels the option position’s immediate sensitivity to a small change in the underlying price. It does not freeze that sensitivity: gamma measures how delta changes as the underlying moves, leaving a curved residual even after the starting delta is zero. Implied volatility is a separate concept: the volatility input that makes an option model reproduce a market price.
What delta neutral means
Delta measures the option value’s first response to a small move in the underlying, holding other inputs fixed. A hedge adds an offsetting exposure to make the combined starting delta zero. CFA Institute’s contingent claims reading treats delta as a local approximation and gamma as the remaining curvature. This is why “neutral” describes a point, not every possible future underlying price.
For a written call, the option position has negative delta and negative gamma. Holding the corresponding amount of the underlying can cancel its initial delta, but the underlying itself has no option curvature. A larger move therefore exposes the mismatch. The CFA programme page places this type of calculation in its study setting.
A small worked hedge
All figures here are invented and illustrative. Consider one call on one unit of an underlying, with delta 0.60 and gamma 0.04 per £1 change in that underlying. A portfolio that is short this call and holds 0.60 units of the underlying starts with zero delta. Now let the underlying rise by £5 while other pricing inputs are held fixed.
The call’s delta component gains £3.00: 0.60 times £5. Its gamma component adds £0.50: half of 0.04 times £5 squared. The estimated call gain is £3.50. The 0.60 units of underlying gain £3.00, leaving an estimated £0.50 loss on the written-call hedge. The calculation is an approximation, not an exact repricing of an option.
Gamma also indicates why the hedge ratio changes. Holding gamma fixed for this illustration, the call delta rises by about 0.20 to 0.80 after the move. The starting 0.60-unit hedge is then short 0.20 units of delta. In an exam calculation, keep the signs of the short option and the long underlying separate before adding their exposures.
Where implied volatility fits
The CFA Institute guide to option Greeks distinguishes delta, gamma and vega, the sensitivity to volatility. Implied volatility runs the valuation problem in reverse: start with an observed option price and find the volatility input that makes the model match it. It is model dependent and does not make a delta hedge immune to changes in underlying price, time or volatility.
The common exam error is to call a delta-neutral portfolio risk-free. Instead, state the movement being hedged, the point at which delta is measured, and which inputs are held constant. The study methodology can help organise that sequence when revising option risk.
Sources
- Valuation of Contingent Claims | CFA Institute · cfainstitute.org · Retrieved
- Why you don’t need to learn Greek to understand investment risk | CFA Institute · cfainstitute.org · Retrieved
