Theory
The existence of a delta neutral portfolio was shown as part of the original proof of the Black–Scholes model, the first comprehensive model to produce correct prices for some classes of options. See Black-Scholes: Derivation.
From the Taylor expansion of the value of an option, we get the change in the value of an option, for a change in the value of the underlier :
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- where (delta) and (gamma); see The Greeks.
For any small change in the underlier, we can ignore the second-order term and use the quantity to determine how much of the underlier to buy or sell to create a hedged portfolio. However, when the change in the value of the underlier is not small, the second-order term, cannot be ignored: see Convexity (finance).
In practice, maintaining a delta neutral portfolio requires continuous recalculation of the position's Greeks and rebalancing of the underlier's position. Typically, this rebalancing is performed daily or weekly.
Read more about this topic: Delta Neutral
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