Delta Hedging

Delta hedging is a method of reducing the directional risk of an options position by taking an offsetting position in the underlying asset. The size of the offset is set by the option's delta, the amount by which the option's price moves for a one-unit move in the underlying. When the hedge is in place, a small rise or fall in the underlying produces a gain on one side that cancels the loss on the other.

The technique is the standard way in which option dealers manage risk. A market maker who sells an option to a customer does not usually want a bet on the direction of the market, and hedges the delta at once. Delta hedging is also the practical counterpart of the theory behind the Black-Scholes model, which values an option as the cost of hedging it continuously.

The number of units of the underlying needed to hedge an options position is:

• $\Delta$ is the delta of one option.
• $n$ is the number of contracts, positive if bought and negative if sold.
• $m$ is the contract multiplier, the number of units of the underlying per contract.

A positive result means buying the underlying and a negative result means selling it short. A dealer sells 10 call contracts on a share trading at $100. Each contract covers 100 shares and each call has a delta of 0.57. The hedge is −0.57 × (−10) × 100 = +570, so the dealer buys 570 shares. If the share rises by $1, the calls sold lose about $570 and the shares gain $570.

The hedge does not stay correct. Suppose the share rises to $102 and the delta of each call becomes 0.67. The required holding is now 670 shares, and the dealer buys another 100. If the share then falls back to $100, the delta returns to 0.57 and the dealer sells those 100 shares at $100, having bought them at $102. This repeated adjustment is called dynamic hedging or rebalancing.

The example shows a general rule. A trader who is short options has negative gamma and must buy the underlying after it rises and sell after it falls. Each round trip loses money. A trader who is long options has positive gamma and does the reverse, selling after rises and buying after falls, which earns money on each round trip. Deliberately harvesting those gains is called gamma scalping.

Offsetting this is time decay. Over a short interval, the profit or loss of a delta-hedged option position is approximately:

Here $\Gamma$ is the position's gamma and $\Delta S$ is the change in the underlying price, while $\Theta$ is the position's theta, its change in value per unit of time, and $\Delta t$ is the time elapsed.

For a long option position, the first term is a gain from movement and the second is a loss from the passing of time. The two balance when the underlying moves exactly as much as the option's price assumed. A delta-hedged position therefore makes money if realized volatility exceeds the implied volatility paid, and loses if it falls short. Delta hedging converts a bet on direction into a bet on volatility, which is the basis of volatility trading.

In theory the hedge is adjusted continuously. In practice it is adjusted at intervals, and the trader must choose how often. Frequent rebalancing tracks delta closely but runs up transaction costs. Infrequent rebalancing saves cost but leaves the position exposed between adjustments. Common rules rebalance at fixed times, such as once a day, or whenever the net delta drifts beyond a set band.

The hedging instrument need not be the underlying itself. Index options are usually hedged with index futures, and options on illiquid shares may be hedged with a related, more liquid instrument at the cost of some mismatch. Other options can also be used, which allows gamma and volatility exposure to be reduced at the same time as delta.

• Gamma risk. The hedge is exact only for small moves. A sudden jump in price, such as an overnight gap, moves the option's value by more or less than the hedge covers.
• Volatility risk. A change in implied volatility changes the option's value with no move in the underlying. This exposure is measured by vega.
• Model risk. Delta comes from a model. If the volatility input is wrong, the hedge ratio is wrong.
• Liquidity and financing. The hedge requires the ability to trade the underlying at will, to borrow shares for short selling, and to fund the position.

Because dealers hedge mechanically, their trades can influence the underlying. When dealers are net short options, their hedging buys into rising prices and sells into falling ones, which can amplify moves. Heavy buying of short-dated calls by investors has at times forced dealers to buy shares as prices rose, an effect known as a gamma squeeze. When dealers are net long options, their hedging leans against price moves and tends to dampen them. Dynamic hedging by portfolio insurers, who sold index futures as prices fell, is widely regarded as having contributed to the severity of the October 1987 crash. These effects are part of options risk management at both the firm and the market level.