The Greeks
Every option price moves when something it depends on moves: the stock, volatility, time or interest rates. The Greeks put a number on each of those moves. Here is where they come from, and the whole family on one page.
Where the Greeks come from
The Greeks start with one paper and one exchange. In 1973 Fischer Black and Myron Scholes published "The Pricing of Options and Corporate Liabilities" in the Journal of Political Economy. The same spring, on April 26, the Chicago Board Options Exchange opened as the first exchange for listed options.
At the model's heart is a hedge: buy or sell the stock in just the right amount, keep adjusting it as the stock moves, and the option's risk cancels out. That amount is delta. Theta, vega and rho follow the same idea for time, volatility and interest rates, and gamma measures how fast delta itself changes.
Traders took the model to the market quickly. Within six months of the paper, Texas Instruments produced a handheld calculator programmed to give Black–Scholes prices and hedge ratios. Exchange trading and the model together changed how options were priced and hedged.
Vega is the odd one out: it is not a Greek letter at all. It is usually written with the letter nu (ν), and some academics call it kappa instead.
The second- and third-order Greeks came later, named by the people who trade and model options rather than by the model's authors. Mark Garman gave speed, charm and color their names in 1992. Espen Gaarder Haug's "Know Your Weapon" articles in Wilmott magazine (2003) gathered the family, from vanna and vomma to zomma and ultima.
Today many brokers show delta, gamma, theta and vega beside every strike in the option chain. The higher orders are still mostly the professionals' tools, and they are where the surprises come from.
The family
| Greek | Order | Measures | Connects to | Why it matters |
|---|---|---|---|---|
| ΔDelta | 1st | Price change per $1 move in the stock | Root of the spot family | The direction lever, and the only Greek you can adjust cleanly on its own: shares carry delta and nothing else. |
| νVega | 1st | Price change per 1-point change in implied volatility | Root of the volatility family | Long options carry it; selling premium means being short it. |
| ΘTheta | 1st | Price change per calendar day that passes | Root of the time family | What time takes from an option's price: the buyer pays it to hold the option, the premium seller collects it if nothing else moves. |
| ρRho | 1st | Price change per 1-point move in the risk-free rate | Stands alone | Small on short-dated options, visible on long-dated ones. |
| λLambda | 1st (derived) | Percentage change in the option per 1% move in the stock | Delta scaled by the ratio of stock price to option price | Why cheap out-of-the-money options move so much in percentage terms. |
| ΓGamma | 2nd | Delta change per $1 move in the stock | Child of delta (spot) | Why deltas swing fast at the money near expiry. Long options are long gamma. |
| Vanna | 2nd | Delta change per 1-point change in implied volatility | Bridges delta and vega | After a volatility crush, an out-of-the-money option's delta shrinks and an in-the-money option's delta grows. |
| Charm | 2nd | Delta change per calendar day | Child of delta (time) | Why an out-of-the-money call's delta melts toward zero in the final week, as a covered call's short call often does. |
| Vomma (volga) | 2nd | Vega change per 1-point change in implied volatility | Child of vega (volatility) | Long options are long vomma, most of all away from the money: as volatility rises, their vega grows, so a volatility move counts for more the further it goes. |
| Veta | 2nd | Vega change per calendar day | Child of vega (time) | Why vega shrinks as expiry approaches, even when implied volatility doesn't move. |
| Speed | 3rd | Gamma change per $1 move in the stock | Child of gamma (spot) | Why gamma climbs as the stock approaches the strike and falls away once it passes. |
| Zomma | 3rd | Gamma change per 1-point change in implied volatility | Child of gamma (volatility) | Volatility reshapes the whole gamma profile: higher volatility lowers the at-the-money peak and spreads gamma wider. |
| Color | 3rd | Gamma change per calendar day | Child of gamma (time) | Why an at-the-money short front leg, such as a calendar's, gets twitchier in its last days. |
| Ultima | 3rd | Vomma change per 1-point change in implied volatility | Child of vomma (volatility) | Deep volatility-of-volatility territory, listed for completeness. |
The pattern: the spot ladder runs delta, gamma, speed; the volatility column runs vega, vanna, vomma, zomma, ultima; and the time row runs theta, charm, color, veta.
Units as everywhere on Own the Greeks: per share, time per calendar day as it passes, volatility and rates per point. See Our model.
Education only, not financial advice. Options involve substantial risk. Read the full disclaimer.