The Greeks — Options Tutorial

Options Don’t Just Move with the Stock

A common beginner frustration: AAPL went up $2, but my call option lost money. How?

Because stock price is only one of several forces acting on an option’s price. Time is passing. Implied volatility is shifting. These forces push and pull simultaneously. The Greeks quantify each one — they tell you how much the option price changes when a single factor moves by one unit.

Four core Greeks:

Greek	What It Measures	One-Liner
Delta	Effect of stock price movement	Stock moves $1, option moves how much?
Gamma	Rate of change of Delta	Delta isn’t fixed — it accelerates
Theta	Effect of time passing	How much do you lose per day?
Vega	Effect of implied volatility change	IV moves 1%, option moves how much?

Delta: Direction Sensitivity

Delta measures how much the option price changes when the stock moves $1.

Call Delta ranges from 0 to 1. Put Delta ranges from -1 to 0.

Example: AAPL trades at $180. You hold a $180 call with Delta = 0.50.

AAPL rises $1 -> option price rises about $0.50
AAPL drops $1 -> option price drops about $0.50

Delta depends on how far in or out of the money the option is:

Moneyness	Call Delta	Put Delta
Deep ITM	Near 1.00	Near -1.00
ATM	Around 0.50	Around -0.50
Deep OTM	Near 0.00	Near 0.00

Delta as a probability proxy. The absolute value of Delta roughly approximates the probability the option expires in the money. A call with Delta 0.30 has roughly a 30% chance of finishing ITM. Not a precise probability, but useful for quick judgment.

Delta-neutral. If you hold 10 contracts of a Delta 0.50 call (total Delta = 500 shares equivalent), you can short 500 shares of stock to hedge. Small moves in either direction cancel out. This is delta-neutral positioning — the bread and butter of market makers and institutional traders.

Gamma: How Fast Delta Changes

Delta isn’t constant. As the stock moves, Delta moves with it. Gamma measures that rate of change.

If the stock moves $1, how much does Delta change? That’s Gamma.

Say your call has Delta = 0.50 and Gamma = 0.05:

AAPL rises $1 -> Delta goes from 0.50 to about 0.55
AAPL rises another $1 -> Delta goes from 0.55 to about 0.60

Key characteristic: Gamma is highest at the money. Deep ITM and deep OTM options have low Gamma — their Delta barely budges. ATM options are most sensitive.

Gamma explodes near expiration. This is where traders get burned. In the last few days before expiry, ATM Gamma becomes enormous. Delta can swing from 0.30 to 0.80 and back to 0.20 within minutes.

For buyers, high Gamma is a gift — if the stock makes a sudden large move, Delta ramps up fast, accelerating profits. For sellers, high Gamma is a nightmare — small stock movements cause wild Delta swings, making hedging nearly impossible.

Theta: Time Decay

Options expire. Every day that passes, time value erodes. Theta measures the rate of that erosion.

Theta is typically negative for option holders. Theta = -$0.05 means, all else equal, the option loses $0.05 per day.

Concrete feel: you buy an AAPL $180 call for $3.00 with 30 days to expiration. Theta = -$0.08.

After 10 days, time decay alone costs $0.80. Option’s time value drops to about $2.20.
If AAPL hasn’t moved much, you’re already down 27%.

Theta accelerates. Time decay is not linear — it speeds up as expiration approaches. The last 30 days decay faster than the prior 30. The final week is a cliff.

Days to Expiration	Daily Theta (Illustrative)
60	-$0.04
30	-$0.08
7	-$0.18
1	-$0.45

This is why “buying lottery ticket options expiring Friday” is so dangerous — even if you get the direction right, time decay might eat your profits faster than the stock moves.

Buyer’s perspective: Theta is your enemy. You’re paying rent every day. Seller’s perspective: Theta is your friend. You’re collecting rent every day.

Vega: Volatility Sensitivity

Vega measures how much the option price changes when implied volatility (IV) moves by 1 percentage point.

Say Vega = $0.12, current IV = 30%:

IV rises from 30% to 31% -> option price rises $0.12
IV drops from 30% to 29% -> option price drops $0.12

IV Crush: post-event volatility collapse. The classic scenario is earnings. Before an earnings announcement, the market expects a big move, so IV gets bid up. After the announcement, uncertainty disappears, and IV drops sharply.

Say a stock’s IV is 65% before earnings and drops to 35% after — a 30-point decline. If your option’s Vega is $0.12, that IV crush alone costs $3.60 per share. Even if the stock moved in your direction, you could still lose money.

Vega is critical for event trades. Before earnings, Fed decisions, or economic data releases, check how inflated IV already is. If it’s elevated, the “insurance premium” is already expensive — you need the stock to move more than the market expects just to break even.

Buyers have positive Vega — rising volatility helps. Sellers have negative Vega — falling volatility helps.

How the Greeks Interact: A Practical Example

Setup: you buy 1 contract of the AAPL $180 call, 15 days to expiration.

Parameter	Value
AAPL price	$180.00
Strike	$180.00
Option price	$2.50
Delta	0.50
Gamma	0.08
Theta	-$0.12
Vega	$0.10

Scenario 1: AAPL rises $2, IV unchanged, 1 day passes

Delta contribution: +$2 x 0.50 = +$1.00
Gamma adjustment: Delta rises to about 0.66, but we use the initial Delta for this approximation
Theta cost: -$0.12
Vega contribution: $0 (IV unchanged)
New option price: roughly $2.50 + $1.00 - $0.12 = $3.38

A decent day. Direction was right, time cost was manageable.

Scenario 2: AAPL rises $2, but IV drops 5 points, 1 day passes

Delta contribution: +$1.00
Theta cost: -$0.12
Vega cost: -5 x $0.10 = -$0.50
New option price: roughly $2.50 + $1.00 - $0.12 - $0.50 = $2.88

You were right on direction, the stock moved $2, and your option is up only $0.38. More than half the directional gain was eaten by IV crush and time decay. This is the post-earnings trap — direction right, still barely profitable.

Quick Reference

Greek	What It Measures	Buyer	Seller	When It Matters Most
Delta	Stock +$1, option changes by…	Want high Delta (bigger directional gain)	Want low Delta or hedge it away	Trending markets
Gamma	Stock +$1, Delta changes by…	High Gamma helps (accelerates profits)	High Gamma hurts (hard to hedge)	Near expiration + ATM
Theta	Per day, option loses…	Enemy (bleeding daily)	Friend (earning daily)	Last 30 days, accelerating
Vega	IV +1%, option changes by…	Positive Vega (want IV to rise)	Negative Vega (want IV to fall)	Before/after major events

Exercise

You hold 1 contract of the NVDA $900 call with these parameters:

NVDA price: $900.00
Strike: $900.00 (ATM)
Option price: $18.00
Delta: 0.50
Gamma: 0.03
Theta: -$0.35
Vega: $0.45
Days to expiration: 12
Current IV: 55%

Question: Tomorrow, NVDA rises $5 and IV drops from 55% to 50%. What is the approximate new option price? Was this a good day for the option holder?