Gamma (Options)

Gamma, in options trading, is the rate at which an option's delta changes when the price of the underlying asset changes. Delta says how much the option's price moves for a $1 move in the underlying. Gamma says how much delta itself moves. An option with a delta of 0.50 and a gamma of 0.04 will have a delta of about 0.54 after the underlying rises by $1.

Gamma is one of the option Greeks and measures the curvature of an option's value. It is the reason an option's payoff is not a straight line, and it determines how often a hedged position must be adjusted and whether those adjustments make or lose money. Traders who manage option books watch gamma as closely as delta, particularly as expiry approaches.

Gamma is the second derivative of the option's value with respect to the underlying price:

Under the Black-Scholes model, for an option on an asset that pays no dividends:

• $V$ is the option's value and $S$ is the price of the underlying.
• $\sigma$ is the volatility and $T$ is the time to expiry in years.
• $N'(d_1)$ is the standard normal density evaluated at $d_1$, the same quantity used in the formula for delta.

The formula is the same for a call option and a put option with the same strike and expiry. Gamma is always positive for an option that is held and negative for one that has been sold.

A share trades at $100. An option with a $100 strike has three months to expiry, with volatility of 20 percent and an interest rate of 5 percent. For these inputs $d_1 = 0.175$ and $N'(0.175) = 0.393$. The denominator is 100 × 0.20 × √0.25 = 10. Gamma is 0.393 ÷ 10 = 0.039.

The call's delta is 0.57. If the share rises to $101, the delta becomes about 0.57 + 0.04 = 0.61. Gamma also improves the estimate of the price change for a larger move:

For a $5 rise, delta alone gives 0.57 × 5 = $2.85. The gamma term adds 0.5 × 0.039 × 25 = $0.49, for an estimated gain of $3.34. For a $5 fall, the estimate is −2.85 + 0.49 = −$2.36. The holder gains more on the way up than is lost on the way down, which is the benefit of positive gamma.

Gamma is highest for an at-the-money option and falls away as the option moves deep in or out of the money, where delta is already close to 1 or 0 and has little room to change. Time has a strong effect. For an at-the-money option, gamma rises as expiry nears, because a small price move then decides whether the option finishes with value. In the last days before option expiry, at-the-money gamma becomes very large. For options far from the strike, gamma shrinks toward zero as time runs out.

Lower volatility also raises at-the-money gamma, since the option's value is then concentrated in a narrower band of prices.

A position with positive total gamma is called long gamma. It arises from buying options, as in a long straddle. Its delta rises when the price rises and falls when the price falls, so the position gets longer in a rally and shorter in a decline. When it is delta hedged, the hedger sells into rises and buys into falls, locking in small gains on each swing. This is the mechanism of gamma scalping.

A position with negative total gamma is short gamma. It arises from selling options, as in a short straddle. The hedger must buy after rises and sell after falls, losing a little each time, and a large sudden move produces a loss that grows faster than the move.

Gamma is paid for through time decay. A long gamma position has negative theta and loses value each day the market stands still. A short gamma position collects that decay. The two roughly offset when the underlying moves as much as the option's implied volatility predicted. A long gamma position profits when realized volatility is higher, and a short gamma position profits when it is lower.

For a dealer, gamma shows how quickly a hedge goes out of date. A book with large gamma needs frequent delta hedging and is exposed to gaps in price, when no adjustment is possible. Risk limits are therefore set on gamma as well as on delta, and stress tests revalue the book for large moves instead of relying on the Greeks alone.

Short-dated at-the-money options create a particular problem at expiry. If the underlying sits close to a strike where a trader holds a large position, delta swings between nearly 0 and nearly 1 with each small price change, and the trader cannot know until the close whether the options will be exercised. This is known as pin risk.

Gamma also links option positions to the behavior of the underlying market. When dealers as a group are short gamma, their hedging adds to price moves. When they are long gamma, it dampens them. Heavy buying of short-dated calls can force dealers to buy the underlying as it rises, an effect popularly called a gamma squeeze.