Delta neutral

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Delta neutral refers to a portfolio containing options, that has been hedged so that its overall value will not change for small changes in the underlier's price. The hedge is ordinarily accomplished by buying or selling an amount of the underlier that corresponds to the delta of the portfolio that is to be made delta neutral. It is also possible to hedge using related options instead of the underlier. This second tactic is common when the underlier is difficult to trade, for instance when a underlying stock is hard to borrow and therefore cannot be sold short.

Contents

1 Theory
2 Example
3 Larger Movements
4 See also

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) + \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.

Example

If a call option on 100 shares of IBM stock, strike at $90 and expiring in 3 months has a delta of 45, the delta neutral portfolio would consist of:

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

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 remain unchanged because the increase in the call's value is matched by the decrease in the value of the short stock position.

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.

Delta neutral

Theory

Example

Larger Movements

See also