Delta Neutral

Delta neutral is the description given to a position, or a whole portfolio, in which the positive and negative deltas cancel so that the total is zero. Delta measures how much a position gains or loses when the underlying asset moves by one unit. With a total delta of zero, a small rise or fall in the underlying leaves the value of the position roughly unchanged.

Traders build delta-neutral positions when they want exposure to something other than the direction of the market: the level of volatility, the passing of time, or a price difference between related instruments. Option market makers keep their books close to delta neutral as a matter of routine. The term describes a state at one moment. Because deltas change as prices move, a position that is neutral now will not remain so without adjustment.

A portfolio is delta neutral when:

• $n_i$ is the number of units held of instrument $i$, counted in units of the underlying and negative for short positions.
• $\Delta_i$ is the delta of that instrument. The underlying itself has a delta of 1.

A trader buys 20 put contracts on a share, each covering 100 shares, with a delta of −0.40 per option. The options contribute 20 × 100 × (−0.40) = −800. To neutralize this, the trader buys 800 shares, which contribute +800. The total is zero. If the share rises by $1, the shares gain $800 and the puts lose about $800.

The process of adding the offsetting position is delta hedging. Where a portfolio holds options on several different shares, traders often convert each delta into money terms or weight it by the share's beta to an index, so that exposures can be added together and offset with index futures.

• Options against the underlying. A long or short option position is offset with shares or futures, as in the example.
• Options against options. A long straddle buys an at-the-money call and put on the same strike. Their deltas are close to +0.5 and −0.5 and roughly cancel. A strangle does the same with out-of-the-money options, and an iron condor is a delta-neutral position with limited risk.
• Ratio positions. Different numbers of options at different strikes are combined so that the deltas offset, for example one option with a delta of 0.60 against two with a delta of 0.30.
• Related securities. In convertible arbitrage, a fund buys a convertible bond and sells short enough of the issuer's shares to cancel the bond's equity delta.

Removing delta does not remove risk. It isolates the other sensitivities, described by the remaining option Greeks.

Gamma measures how fast delta changes. A delta-neutral position that is long options has positive gamma: after a large move in either direction it becomes profitable, because the option that gained delta outweighs the hedge. A position that is short options has negative gamma and loses on large moves. Theta works the other way. Long options lose value each day, and short options earn it. Vega measures the effect of a change in implied volatility: a long option position gains when it rises.

A delta-neutral long option position is therefore a bet that the underlying will move more than option prices imply, and a delta-neutral short option position is a bet that it will move less. The outcome depends on realized volatility compared with the implied level at which the options were traded.

Dealers engaged in market making earn the bid-ask spread on options and do not want their results to depend on market direction, so they hedge delta continuously. Volatility traders use delta-neutral positions to express views on volatility itself, as in volatility arbitrage, or on the relationship between the volatility of an index and of its members, as in dispersion trading. Traders who hold delta-neutral long option positions and trade the underlying against each price swing are practicing gamma scalping.

Delta shifts with the underlying price, with time and with volatility, so a neutral position drifts. In the example above, if the share falls, the puts' delta might move from −0.40 to −0.55. The options then contribute −1,100 against +800 from the shares, and the trader must buy another 300 shares to restore balance. Traders set a tolerance band and rebalance when net delta moves outside it, weighing the risk of an unhedged move against the cost of trading.

Some go further and make the position gamma neutral as well, by adding other options, so that delta stays near zero over a wider range of prices. Each added constraint requires another instrument and adds cost.

The neutrality is local and model-based. It holds for small moves over short periods and depends on deltas calculated from a pricing model with an assumed volatility. A gap in price can produce a large gain or loss before any adjustment is possible. Frequent rebalancing incurs commissions and bid-ask costs that can exceed the edge being pursued. A short-option position that looks balanced can lose many times its expected profit in a sharp move, so delta neutral should not be read as low risk.