Gamma

What gamma measures

Delta tells you how much an option moves with the underlying; gamma tells you how fast delta itself changes. A gamma of 0.0000278 per $1 means delta grows by about 0.0278 for every $1,000 the price rises (and shrinks by the same when it falls).

Gamma is the same for a call and a put at the same strike and expiry, and it is always positive for an option you own. Long options have positive gamma; short options have negative gamma.

Convexity: why gamma helps the buyer

Positive gamma means your delta grows in the direction the market is moving: on a rally your call behaves more and more like the underlying, on a sell-off less and less. Gains accelerate and losses slow down. In the delta example, a $1,000 rise added $13 more than delta alone predicted; a $1,000 fall would also have lost about $13 less. That curvature is called convexity.

A seller has the opposite: negative gamma, where every move pushes delta against them. Short options lose faster the further the price runs, which is why a short straddle can be comfortable on a quiet day and painful on a volatile one.

Gamma near expiry

For a $100,000 at-the-money option with BTC at $100,000 and 50% implied volatility:

• with 30 days left, gamma is about 0.0000278: delta moves 0.03 per $1,000;
• with 7 days left, about 0.0000576: twice as fast;
• with 1 day left, about 0.000152: delta swings 0.15 for every $1,000.

As expiry approaches, an at-the-money option's delta has to decide between 0 and 1 over an ever smaller price range, so gamma concentrates there. Far from the strike, gamma falls toward zero, because delta is already settled near 0 or ±1.

Gamma and theta are the two sides of one trade

Gamma isn't free. The options with the most gamma (at the money, short-dated) are also the ones losing the most value each day to theta. Buying gamma means paying theta every day and needing the price to move enough to make up for it; selling gamma means collecting theta and hoping it doesn't. Whether that trade pays depends on how much the underlying actually moves compared with what implied volatility priced in.