🎯 In 30 seconds: The Greeks are sensitivity numbers. Delta: price change per $1 move in the stock. Gamma: how fast delta changes. Theta: value lost per day. Vega: price change per 1 point of implied volatility. Rho: price change per 1 point of interest rates. Multiply by 100 for one contract. They are estimates from a pricing model, good for small changes, and they shift as price and time change.
Why options need their own dashboard
A share of stock moves $1 when the stock moves $1. Simple. An option's price depends on several things at once: the stock price, time left, implied volatility and interest rates. The Greeks break that down, one input at a time. Each tells you how much the option price should change if one input changes a little and everything else stays the same.
The Options Industry Council calls them a theoretical guidepost, not a guarantee. That is the right attitude. They come from a pricing model, usually Black-Scholes or a close relative, explained in How options are priced.
One option, all five Greeks
Everything in this chapter uses one hypothetical option, priced with the Black-Scholes model so the numbers agree with each other:
| Company X stock | $50.00 |
| Option | $50 call, 30 days to expiration |
| Implied volatility | 30% |
| Interest rate used | 4% |
| Model price | $1.80 ($180 per contract) |
| Greek | Value (per share) | Per contract (x 100) | Plain meaning |
|---|---|---|---|
| Delta | 0.53 | $53 per $1 stock move | like owning 53 shares |
| Gamma | 0.092 | delta rises by 0.092 per $1 up | how fast delta changes |
| Theta | -0.031 | -$3.10 per day | cost of one day passing |
| Vega | 0.057 | $5.70 per 1 point of IV | sensitivity to volatility |
| Rho | 0.020 | $2.00 per 1 point of rates | sensitivity to interest rates |
Now each one, with a check against the model.
Delta: the price sensitivity
Delta is the expected change in the option price for a $1 change in the stock.
- Calls have delta between 0 and 1. Puts have delta between 0 and -1.
- At the money, a call's delta is near 0.50. Deep in the money it approaches 1, and the option moves almost like 100 shares. Far out of the money it approaches 0.
- The matching $50 put has delta -0.47. Call delta minus put delta at the same strike is about 1: 0.53 - (-0.47) = 1.00.
Check. If Company X rises to $51, delta alone predicts $1.80 + $0.53 = $2.33. The model's actual new price is $2.37. Delta gets you close. Gamma explains the rest.
Position delta adds up across contracts. Five of these calls: 5 x 0.53 x 100 = about 265 "share equivalents". For small moves, your position behaves like owning 265 shares, a useful way to see how much exposure you really have. A position of 5 calls costing $900 has roughly the same short-term exposure as $13,250 of stock.
Delta as a rough probability
Traders often read delta as the chance an option finishes in the money. A 0.20 delta call: roughly a 20% chance. It is a handy shortcut, and it is only approximate.
For our call, the model's own probability of finishing in the money is about 50%, a little below the 0.53 delta. For a $55 call with the same 30 days, delta is 0.15 and the model probability is about 13%. Both say the same thing: probably not.
Two cautions. First, the "probability" comes from the model's assumptions, including the implied volatility, so it is not a forecast. Second, finishing in the money is not the same as making a profit. You also have to recover the premium.
Gamma: how fast delta changes
Gamma is the change in delta for a $1 move in the stock. Our call has gamma 0.092. If the stock rises $1, delta goes from 0.53 to about 0.62. If it falls $1, delta drops to about 0.44.
Check, with gamma. A better estimate for a $1 rise is delta + half of gamma: $1.80 + $0.53 + 0.5 x 0.092 = $2.38. The model says $2.37. For a $1 fall: $1.80 - $0.53 + 0.046 = $1.32, model $1.31. Close both ways.
Gamma is highest for at-the-money options and grows as expiration nears.
Look at the red curves. With 2 days left, the same $50 call has gamma 0.36, four times the 30-day figure. A $1 rise takes the 2-day call from $0.45 to $1.12, a gain of about 150%. A $1 fall would wipe out most of its value. That is the raw mechanics behind 0DTE and expiration-week options: huge gamma, both ways.
Option buyers are long gamma: big moves help them. Option sellers are short gamma: big moves hurt them, and the hurt speeds up.
Theta: the daily cost of time
Theta is the change in option price from one day passing, everything else unchanged. Our call has theta -0.031, so it loses about $3.10 per contract per day.
Check. Tomorrow, with the stock still at $50 and 29 days left, the model price is $1.764, down $0.031. Exactly as theta said.
Theta grows as expiration nears for at-the-money options. With 60 days left this call's theta is about -0.023 a day. With 2 days left it is about -0.11 a day. The full decay curve is in What an option is worth.
Theta and gamma are two sides of one deal. The buyer pays theta every day in exchange for gamma, the chance of a big move. The seller collects theta and accepts the gamma risk.
Vega: sensitivity to volatility
Vega is the change in option price for a 1 percentage point change in implied volatility. Our call has vega 0.057.
Check. If IV rises from 30% to 31% with nothing else changing, the model price goes from $1.796 to $1.853, up $0.057.
Longer-dated options have more vega: with 60 days left this call's vega is 0.080, with 2 days left only 0.015. Vega is why options can lose money even when the stock moves your way, if implied volatility drops at the same time. That happens almost every earnings season. See Implied volatility for a worked case of IV crush.
Rho: sensitivity to interest rates
Rho is the change in option price for a 1 percentage point change in interest rates. Our call has rho 0.020: if rates went from 4% to 5%, the model price would rise about 2 cents, to $1.816. Puts have negative rho.
For short-dated options rho is small next to the other Greeks. It matters more for long-dated options such as LEAPS, where a year or more of interest is involved.
Putting it together: a one-day estimate
You can combine the Greeks for a quick estimate of tomorrow's price. Hypothetical: tomorrow Company X is up $0.80 and IV slips from 30% to 29%.
| Delta: 0.53 x 0.80 | +$0.424 |
| Gamma: 0.5 x 0.092 x 0.80 x 0.80 | +$0.029 |
| Theta: one day | -$0.031 |
| Vega: 0.057 x (-1) | -$0.057 |
| Estimated change per share | +$0.365, about +$36.50 per contract |
Estimated new price: $1.796 + $0.365 = about $2.16. The full model, repriced with all the changes at once, also gives about $2.16. The Greeks work well for moves like this. For big jumps, such as an earnings gap, reprice the whole option instead.
Try it
Open the options chain for a stock on your Watchlist and turn on the Greeks columns. For the at-the-money call in the nearest weekly or monthly expiration and in one about 90 days out, record delta, gamma, theta and vega. Predict tomorrow's price for each using the one-day estimate above and the stock's actual move. Compare with the real midpoint the next day. Do this for five days in your trading journal and note when the estimate was off and why.
- A call has a delta of 0.53. About how much should it gain if the stock rises $1? Answer: About $0.53 per share, or $53 per contract.
- You hold 5 calls, each with a delta of 0.53. About how many shares does the position act like for small moves? Answer: About 265 shares (5 x 0.53 x 100).
- A call has a theta of -0.031. What does one day passing cost per contract, everything else unchanged? Answer: About $3.10.
- Who is long gamma and who is short gamma? Answer: Option buyers are long gamma, so big moves help them. Option sellers are short gamma, so big moves hurt them, and the hurt speeds up.