2.4.2 Gamma hedging

Delta hedging must be dynamically executed because delta changes with underlying spot price. Gamma is the rate of change of delta with changes in spot. So then the aggregate gamma exposure of a portfolio indicates how sensitive a delta hedge is to price changes in the underlying. Importantly, gamma is always positive for bought options and negative for sold options. If a portfolio has negative net gamma, the delta hedger must buy when spot price goes up, and sell when spot price goes down, thus amplifying the underlying price movement. Conversely, if a portfolio has positive gamma, the delta hedger must sell into rising prices and buy into sell offs, thus muting price moves and stabilizing the market.

Under naïve assumptions, all puts are bought and all calls are sold, so a market maker will typically have positive gamma when calls are near the money, and negative gamma when puts are near the money. Thus, when then net gamma of the portfolio is positive, typically for steady rising spot prices, the gamma hedge prevents large excursions in price, and “pins” the price to its trajectory. When net gamma is negative, the hedge exacerbates volatility, increasing it. Understanding how the gamma environment affects underlying price action can provide insights into expectations for future realized volatility. The table below provides a summary of this behavior from the perspective of the market maker.

Let’s return to our example in the delta hedging section above for AAPL, where the market maker is short an AAPL call with a delta of 0.80, which they have hedged by buying 80 shares of AAPL underlying stock. If the gamma for the contract is 0.05, then they have a gamma exposure of -0.05 since they have sold the option. If AAPL stock price moves up $1, the delta exposure becomes -0.85, and so they must buy another 5 shares of AAPL to remain delta hedged. If instead the market maker is long that call (someone sold the call to the MM), then their delta exposure is +0.80, and gamma exposure is +0.05, so they must short 80 shares and then short 5 more shares on a move up of $1. Thus, when someone buys an option from a market maker, the negative gamma in the market maker’s portfolio requires them to buy into rising prices and sell into dropping prices, amplifying moves in the market. Conversely, any options sold to a market maker shifts their gamma positive, meaning they must sell into rising prices and buy into falling prices, dampening moves in the market.