The Greeks, Part 1: Delta and Gamma

7.1 How much will my option move?

Mint owned a $100,000 BTC call. Bitcoin rose $1,000. Mint expected the option to gain $1,000 too, but it only gained about $544.

"Why only half?" asked Mint.

"Because your option isn't bitcoin," said Hash. "It's a probability-weighted bet on bitcoin. Options traders measure exactly how sensitive each option is to different factors. Those measurements are called the Greeks, because most are named after Greek letters. The first two tell you how your option responds to price."

7.2 The Greeks in one table

Greek	Measures sensitivity to	Covered in
Delta	The underlying price	This chapter
Gamma	How fast delta changes as price moves	This chapter
Theta	The passage of time	Chapter 8
Vega	Implied volatility	Chapter 8

7.3 Delta

Delta is roughly how much an option's price changes when the underlying moves by $1.

Calls have positive delta, between 0 and 1: they gain when the price rises. Puts have negative delta, between 0 and −1: they gain when the price falls.

Here are deltas for 30-day BTC options, with BTC at $100,000 and 55% IV:

Option	Delta	Approximate change if BTC rises $1,000
$90,000 put (OTM)	−0.23	−$230
$100,000 put (ATM)	−0.47	−$470
$100,000 call (ATM)	+0.53	+$530
$110,000 call (OTM)	+0.30	+$300
$120,000 call (far OTM)	+0.14	+$140

Mint's at-the-money call had a delta of about 0.53, so a $1,000 rise added roughly $530 to $544 in value. (The small difference comes from gamma, below.)

Moneyness	Call delta	Put delta
Deep in the money	Close to +1	Close to −1
At the money	About +0.5	About −0.5
Far out of the money	Close to 0	Close to 0

7.4 Three ways to read delta

1. Price sensitivity. As above: how much the option moves per $1 move in the underlying.

2. Equivalent position. Delta shows how much of the underlying an option position behaves like. A call with delta 0.53 on 1 BTC behaves, for small moves, like holding 0.53 BTC. Holding 10 such calls behaves like holding about 5.3 BTC.

3. Rough probability. Delta is often used as a rough estimate of the chance the option expires in the money. A 0.30 delta call is loosely thought of as having about a 30% chance of finishing in the money. This is an approximation, not an exact probability.

"So a $120,000 call with 0.14 delta is roughly a one-in-seven shot," said Mint.

"Roughly," said Hash. "Which is why it's cheap."

7.5 Delta-neutral and hedging

As Module 7, Chapter 12 explained, a position's total delta shows its overall directional exposure. Options traders use delta to build and manage hedges.

Position	Delta
Long 10 ATM BTC calls (delta 0.53 each)	+5.3 BTC
Short 5.3 BTC perps	−5.3 BTC
Total	About 0

The combined position is roughly delta-neutral: small moves in BTC barely change its value. Professional market makers who sell options to the public typically hedge their delta this way, so they earn from the premium rather than betting on direction. But delta changes as the price moves, which is where gamma comes in.

7.6 Gamma

Gamma measures how much delta changes when the underlying moves. It's the "acceleration" of an option's value.

For Mint's 30-day at-the-money call, gamma means delta rises by about 0.025 for every $1,000 rise in BTC.

BTC price	Call delta (approximate)
$100,000	0.53
$101,000	0.56
$110,000	About 0.75

As BTC rises, Mint's call behaves more and more like owning bitcoin. As BTC falls, it behaves less and less like it. That's what makes long options attractive: gains accelerate when you're right and losses slow down when you're wrong.

Chapter 7 · Delta and gamma across prices

Two stacked charts share a BTC price axis from $70,000 to $130,000 for a 30-day $100,000 call at 55% implied volatility. The top chart shows delta as an S-curve rising from near 0 to near 1: about 0.53 at the money and about 0.75 at $110,000. The bottom chart shows gamma as a bell curve peaking at the money at about 0.025 per $1,000 move and falling toward zero on both sides. A vertical line marks the $100,000 strike across both charts.

7.7 Where gamma is highest

Gamma is highest for at-the-money options close to expiry.

ATM call (BTC at $100,000)	Delta change per $1,000 move
30 days to expiry	~0.025
7 days to expiry	~0.052

Near expiry, an at-the-money option's delta can swing rapidly from near 0 to near 1, depending on which side of the strike the price ends up. This makes short-dated at-the-money options very sensitive to small moves.

7.8 Long gamma and short gamma

	Long gamma (option buyers)	Short gamma (option sellers)
When the price moves a lot	Gains accelerate	Losses accelerate
When the price is calm	Slowly lose to time decay (Chapter 8)	Slowly earn time decay
Behaviour	Benefits from volatility	Hurt by volatility

Option sellers are short gamma. If the market moves sharply against them, their losses grow faster and faster, like the insurer facing a disaster. This is why selling options can be so dangerous during crashes.

7.9 Gamma and the market

Large concentrations of options can affect the underlying market. Market makers who are short gamma may need to buy as prices rise and sell as prices fall to keep their hedges balanced, which can amplify moves. When they're long gamma, their hedging can dampen moves, buying dips and selling rallies.

Some traders track estimates of market makers' gamma exposure around large strikes, especially near major expiries. It's an advanced topic, and estimates are uncertain, but it explains why prices sometimes seem "pinned" near large strikes before expiry, or move sharply after.

7.10 Try it yourself

On an options chain that shows Greeks, find the delta and gamma of BTC calls at five strikes for the nearest monthly expiry. Plot delta against strike: you should see it fall from near 1 to near 0. Then compare the gamma of the at-the-money option for a weekly expiry and a quarterly expiry.

Key takeaways from this chapter

1. The Greeks measure an option's sensitivity to price, time, and volatility.
2. Delta is how much an option's price changes per $1 move in the underlying: about 0.5 at the money.
3. Delta also shows the equivalent position size and roughly approximates the chance of expiring in the money.
4. Positions can be made delta-neutral by hedging with the underlying or perps.
5. Gamma measures how quickly delta changes, and is highest for at-the-money options near expiry.
6. Option buyers are long gamma and benefit from big moves; sellers are short gamma and are hurt by them.
7. Market makers' gamma hedging can amplify or dampen moves around large strikes.