What is delta hedging and how does it work?

Setting up the hedge

Suppose a dealer sells 20 call contracts on a 100-dollar stock, each with a delta of 0.50. Each contract covers 100 shares, so the calls behave like 1,000 shares, and the dealer, being short them, has a delta of minus 1,000. To neutralize that, the dealer buys 1,000 shares. Net delta is now zero.

If the stock moves up ten cents, the calls gain about ten cents times 1,000, or 100 dollars, which is a 100-dollar loss to the dealer. The 1,000 shares gain 100 dollars. The two cancel. For small moves, the position is insensitive to direction, which is the point.

The hedge can also be done with futures, other options, or a correlated instrument, but the principle is the same: offset the delta so the remaining exposure is to things other than direction.

Why the hedge drifts

Delta is not fixed. If the stock rises from 100 to 102, the calls move further into the money and their delta rises, say to 0.60. The short calls now behave like minus 1,200 shares, but the dealer only owns 1,000. The position is short 200 shares of exposure, and to rebalance the dealer must buy 200 more shares, at the higher price.

If the stock then falls back to 100, delta returns to 0.50, the dealer is long 200 more shares than it needs, and it sells them, at the lower price. Buying high and selling low on every round trip is the cost of being short gamma. The dealer collected premium for the calls up front, and the hedging losses are what that premium was paying for.

The reverse holds for a long-gamma position. Someone long the calls and short stock as a hedge sells shares as the stock rises and buys them back as it falls, buying low and selling high. That is the core of gamma scalping.

How often to rebalance

In theory, a delta hedge is rebalanced continuously. In practice, every trade has costs, so hedgers choose a schedule. Some rebalance at fixed times, such as every hour or at the close. Others rebalance when delta drifts outside a band, for example when net delta exceeds a set number of shares.

Rebalancing more often keeps the position closer to neutral but costs more in spreads and commissions. Rebalancing less often saves costs but lets the position drift, adding risk. Large dealers with low costs rebalance frequently; individual traders usually less so.

Time and volatility move delta too. As expiry approaches, out-of-the-money deltas decay toward zero; as implied volatility changes, deltas shift. A hedge can need adjusting on a day when the stock does not move.

Why this matters to the market

Market makers are on the other side of a large share of options trades, and they delta hedge continuously. Their hedging is mechanical. When they are short gamma in aggregate, their rebalancing buys into rallies and sells into declines, adding to moves. When they are long gamma, their rebalancing sells rallies and buys declines, damping moves.

That is the link between options positioning and stock prices. A gamma exposure map is, in effect, an estimate of how much delta hedging the market's dealers will be forced to do at each price, and in which direction.

The scale is what makes it matter. One trader rebalancing a few hundred shares has no effect on price. Dealers rebalancing hedges against the open interest of an entire index options market are trading in size that can be a meaningful share of the underlying's volume, especially near heavily traded strikes close to expiry.

What delta hedging does not remove

A delta-hedged position is still exposed to gamma, vega and theta. It gains or loses depending on how much the stock actually moves compared with the implied volatility that was priced into the options. A short-option hedger loses if the stock moves more than implied and gains if it moves less.

It is also exposed to gaps. A hedge rebalanced at the close cannot protect against an overnight jump, and the loss from a large gap on a short-gamma position can be many times the daily premium collected.