Delta neutral

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In finance, a portfolio containing options is delta neutral when it consists of positions with offsetting positive and negative deltas (exposure to changes in the value of the underlying instrument), and these balance out to bring the net delta of the portfolio to zero.

A related term, delta hedging is the process of setting or keeping the delta of a portfolio as close to zero as possible.

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[edit] Mathematical interpretation

Main article: greeks (finance)

Mathematically, delta - \frac{\partial V}{\partial S} - is the partial derivative of the instrument or portfolio's fair value with respect to the price of the underlying security, and indicates sensitivity to the heteroscedasticity of the underlying parameters of the impartial asset.

Therefore, if a position is delta neutral (or, instantaneously delta-hedged) its instantaneous change in value, for an infinitesimal change in the value of the underlying, will be zero; see Hedge (finance). Since delta measures the exposure of a derivative to changes in the value of the underlying, a portfolio that is delta neutral is effectively hedged. That is, its overall value will not change for small changes in the price of its underlying instrument.

[edit] Creating the position

Delta hedging - i.e. establishing the required hedge - may be accomplished by buying or selling an amount of the underlier that corresponds to the delta of the portfolio. By adjusting the amount bought or sold on new positions, the portfolio delta can be made to sum to zero, and the portfolio is then delta neutral.

Options market makers, or others, may form a delta neutral portfolio using related options instead of the underlier. 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 underlying stock is hard to borrow and therefore cannot be sold short.

[edit] 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.

From the Taylor expansion of the value of an option, we get the change in the value of an option, C(s) \,, for a change in the value of the underlier (\Delta\,):

C(s + \Delta\,) = C(s) + \Delta\,C'(s) + {1/2}\,\Delta^2\, C''(s) + ...

where C'(s) = \delta\,(delta) and C''(s) = \Gamma\,(gamma). (see The Greeks)

For any small change in the underlier, we can ignore the second-order term and use the quantity \delta\, to determine how much of the underlier to buy or sell to create a hedged portfolio.

When the change in the value of the underlier is not small, the second-order term, \Gamma\,, cannot be ignored. In practice, maintaining a delta neutral portfolio requires continual recalculation of the position's greeks and rebalancing of the underlier's position. Typically, this rebalancing is performed daily or weekly.

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