Delta hedging is the partial or complete offset of an asset’s price risk through the holding of options on the asset. Options confer the right, but not the obligation, to buy (with call options) or sell (with put options) an underlying asset for a particular price (the strike price) by a particular date (the expiration date). The purchase price of an option is the premium. Options in which the underlying asset is trading near the strike price are called at-the-money (ATM); there are similar terms for in-the-money (ITM) and out-of the-money (OTM) options.

## Delta

Delta measures the change in an option’s price due to a price change in the underlying asset. Long (purchased) puts positions have a range of deltas, from zero to minus one, since they gain in value when the underlying loses value. Long call prices vary directly with underlying prices, so they have a delta range from zero to one. An ATM put with a delta of minus 0.50 would increase in price by $0.50 for every $1 dollar decline in the underlying. As the expiration date approaches, an option’s time value declines and delta tends to increase. Delta will also change if the underlying asset has a change in volatility. Note that the sign of a short (sold) option position’s delta is opposite that of a long position.

## Hedge Ratio

Delta hedging involves calculating the number of options necessary to offset price movement in an underlying position — the hedge ratio. For instance, a trader would require three puts, each with a delta of minus 0.33, to completely offset the downward price risk of an underlying asset; the hedging ratio in this case would be three-to-one. Traders can adopt a bullish (positive) or bearish (negative) stance by changing their actual hedge ratios away from neutral. To completely offset the downward price risk of an underlying asset; the hedging ratio in this case would be three-to-one. Traders can adopt a bullish (positive) or bearish (negative) stance by moving their actual hedge ratios away from neutral.

## Hedge Dynamics

The rate of change of delta is called gamma and it's not linear. Take the example of a call purchased OTM on an appreciating asset with an original delta of 0.3. At first, the call has no intrinsic value (the asset price is less than the strike price), only time value due to the possibility that the call will achieve intrinsic value before expiration. As the asset appreciates, not only does the call gain value, its delta also increases--at the gamma rate. A deep ITM call reaches a delta of 1.0, meaning the asset and call move in tandem. In this situation, a trader will have to short an escalating amount of underlying asset to rebalance to a neutral hedge ratio. Gamma describes precisely how much more asset must be shorted to keep a position neutral.

## Delta Positive Strategies

A bullish trader may undertake a delta positive strategy to bet that the underlying asset will appreciate. The trader may choose to purchase additional calls since their value increases with that of the underlying. Alternatively, the trader could short puts — she is obligated to purchase the underlying asset if the asset value decreases and falls below the strike price. This is actually a bullish strategy, because a short put only costs the short position money if prices decline. By selling puts, the short trader collects premiums that insulate her to some extent from a price decline.

## Delta Negative Strategies

Bearish traders may convert a neutral delta ratio to bearish by purchasing puts, since puts profit from a depreciating underlying. A less obvious alternative is to short calls -- the trader collects premium but doesn’t have to take assignment of the underlying asset unless the asset’s price goes above the strike price.