Infinance,delta neutral describes a portfolio of related financial securities, in which the portfolio value remains unchanged when small changes occur in the value of the underlying security (having zero delta). Such aportfolio typically containsoptions and their corresponding underlying securities such that positive and negativedelta components offset, resulting in the portfolio's value being relatively insensitive to changes in the value of the underlying security.

A related term,delta hedging, is the process of setting or keeping aportfolio as close to delta-neutral as possible. In practice, maintaining a zero delta is very complex because there are risks associated with re-hedging on large movements in the underlying stock's price, and research indicates portfolios tend to have lower cash flows if re-hedged too frequently.^[1] Delta hedging may be accomplished by trading underlying securities of the portfolio. SeeRational pricing § Delta hedging for details.

Delta measures the sensitivity of the value of an option to changes in the price of the underlying stock assuming all other variables remain unchanged.^[2]

Mathematically, delta is represented aspartial derivative${\displaystyle \frac{\mathrm{\partial }V}{\mathrm{\partial }S}}$ of the option'sfair value with respect to thespot price of theunderlying security.

Delta is a function of S,strike price, andtime to expiry.^[2] Therefore, if a position is delta neutral (or, instantaneously delta-hedged) its instantaneous change in value, for aninfinitesimal change in the value of the underlying security, will be zero; seeHedge (finance). Since Delta measures the exposure of aderivative to changes in the value of the underlying, a portfolio that is delta neutral is effectivelyhedged, in the sense that its overall value will not change for small changes in the price of its underlying instrument.

Optionsmarket makers, or others, may form a delta neutral portfolio using related options instead of the underlying. The portfolio's delta (assuming the same underlier) is then the sum of all the individual options' deltas. This method can also be used when the underlier is difficult to trade, for instance when an underlyingstock is hard to borrow and therefore cannot besold short.

For example, in the portfolio${\displaystyle \mathrm{\Pi }=-V+kS}$, an option has the valueV, and the stock has a valueS. If we assumeV islinear, then we can assume${\displaystyle S\frac{\delta V}{\delta S}\approx V}$, therefore letting${\displaystyle k=\frac{\delta V}{\delta S}}$ means that the value of${\displaystyle \mathrm{\Pi }}$ is approximately0.

The existence of a delta neutral portfolio was shown as part of the original proof of theBlack–Scholes model, the first comprehensive model to produce correct prices for some classes of options. SeeBlack-Scholes: Derivation.

From theTaylor expansion of the value of an option, we get the change in the value of an option,${\displaystyle C(s)}$, for a change in the value of the underlier${\displaystyle (ϵ)}$:

${\displaystyle C(s+ϵ)=C(s)+ϵC^{\prime }(s)+1/2{ϵ}^2{C}^{″}(s)+...}$

where${\displaystyle C^{\prime }(s)=\mathrm{\Delta }}$(delta) and${\displaystyle C^{″}(s)=\mathrm{\Gamma }}$(gamma); seeGreeks (finance).

For any small change in the underlier, we can ignore thesecond-order term and use the quantity${\displaystyle \mathrm{\Delta }}$ 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,${\displaystyle \mathrm{\Gamma }}$, cannot be ignored: seeConvexity (finance).

In practice, maintaining a delta neutral portfolio requires continuous recalculation of the position'sGreeks and rebalancing of the underlier's position. Typically, this rebalancing is performed daily or weekly.^{[citation needed]}

• Delta Hedging, investopedia.com
• Theory & Application for Delta Hedging

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• Derivatives (finance)
• Mathematical finance

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