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.

1 As a hedged position
- 1.1 Example
2 Creating the position
3 Theory
4 Larger Movements
5 See also

[edit] As a hedged position

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] Example

If a call option on 100 shares of IBM stock, strike at $90 and expiring in 3 months has a delta of 45 shares (or 0.45 when delta is considered the likelihood of expiring in the money), the delta neutral portfolio would consist of:

Long 1 call (a contract represents 100 shares of stock)
Short 45 shares of IBM

Assuming that the price of IBM stock rises by $0.10, then the price of the call will (according to theory) rise by $0.045. Therefore, the total value of the portfolio remains unchanged because the increase in the call's value is matched by the decrease in the value of the short stock position.

[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 a 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.

[edit] Larger Movements

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.