Lesson 1: 2.0: Delta and Gamma

Delta is the most used Greek, and it is genuinely three tools at once. Each interpretation is useful in a different situation.

• Rate of change: How much the option premium moves per $1.00 move in the stock.

• Share equivalence: How many shares of stock the option position behaves like.

• Probability estimate: Roughly the chance the option finishes in-the-money.

Delta ranges from 0 to +1.00 for calls and 0 to −1.00 for puts. Long calls and short puts carry positive delta (they profit when the stock rises). Long puts and short calls carry negative delta.

★ Core Fact: These three interpretations are not competing definitions. They are three consequences of the same underlying mathematics, and working traders switch between them constantly.

Worked example: You hold a call with a delta of 0.60, currently priced at $4.00.

• Stock rises $1.00 → option gains roughly $0.60, to about $4.60

• Stock rises $2.00 → option gains roughly $1.20, to about $5.20

• Stock falls $1.50 → option loses roughly $0.90, to about $3.10

In dollar terms, since one contract covers 100 shares: a $2.00 stock move on a 0.60 delta call produces roughly $120 of profit per contract.

The word 'roughly' is doing real work here. Delta itself changes as the stock moves — that is gamma, covered in Section 4. For small moves the approximation is close. For large moves it drifts.

This interpretation is the most practically valuable for managing risk, because it converts an options position into something directly comparable to stock.

★ Share equivalent = Delta × 100 × Number of contracts

Worked examples:

• Long 1 call at 0.60 delta = 60 share equivalents. Behaves like owning 60 shares.

• Long 5 calls at 0.30 delta = 5 × 0.30 × 100 = 150 share equivalents.

• Short 3 puts at 0.25 delta = +75 share equivalents (short puts are positive delta).

• Long 2 puts at 0.40 delta = −80 share equivalents.

Why this matters: a trader who owns 10 contracts of a 0.70 delta call may feel they hold a modest position. In reality they carry 700 share equivalents. On a $100 stock, that is $70,000 of directional exposure. Delta reveals what your account is genuinely exposed to.

⚠️ Risk Warning: Traders routinely underestimate their exposure because contracts feel small. Always convert to share equivalents before deciding whether a position is appropriately sized.

Gamma measures how much delta itself changes per $1.00 move in the stock. If delta is speed, gamma is acceleration.

Worked example: A call has delta 0.50 and gamma 0.04.

• Stock rises $1.00 → delta becomes roughly 0.54

• Stock rises another $1.00 → delta becomes roughly 0.58

• Stock falls $1.00 from the start → delta becomes roughly 0.46

This is why option gains accelerate in your favour and decelerate against you when you are long. As a winning long call moves ITM, its delta climbs toward 1.00 and each subsequent dollar of stock movement delivers more profit. That convexity is the structural appeal of owning options.

For sellers the same mechanic works in reverse and is dangerous. A short option moving against you gains delta, so losses accelerate.

Gamma is highest for at-the-money options with little time remaining. This single fact explains a great deal of options behaviour.

Approximate gamma for an ATM option on a $100 stock:

• 60 days to expiration: low gamma — delta shifts slowly

• 14 days: moderate gamma

• 1 day: extremely high gamma — delta can swing from 0.20 to 0.80 on a small move

This is why short-dated ATM options behave so violently. On expiration day, a contract can go from nearly worthless to deeply profitable on a 1% stock move, then reverse entirely within minutes.

⚠️ Risk Warning: Selling short-dated at-the-money options exposes you to maximum gamma risk. The premium looks attractive because decay is fast, but a modest adverse move can produce losses far larger than the premium collected. This is the primary reason this course recommends 30 to 45 day expirations for premium selling.

Open the Options page in Signal Desk, load a chain for a stock you follow and pick a monthly expiration about 45 days out. Find the strike whose Delta reads closest to 0.30, the one nearest 0.50, and the one nearest 0.16.

Note where each sits relative to the current stock price. The 0.50 will be essentially at-the-money; the 0.16 will be noticeably further out. Click a put near 0.30 and read the "Strike vs. price" figure: it tells you in percent how far below today's price that strike sits. You have just located the strikes premium-selling strategies target, and you found them by probability rather than by guessing.

Then switch to the nearest weekly expiration and observe how much closer to the money the 0.16 delta strike sits. Less time means a smaller expected move, and the chain is showing you that directly.