The Greeks: Delta, Gamma, Theta and Vega

In the lesson on option basics you saw that an option’s price depends on the price of the underlying, the strike, the time to expiry, the expected volatility and interest rates. That is enough to understand options. To work with them you need more: you need to know how strongly the price changes when one of these drivers moves.

That is exactly what the Greeks measure – ratios named after Greek letters. They are the dashboard of every option position: delta is the speedometer, gamma the acceleration, theta the fuel consumption and vega the weather. If you can read them, you understand why an option gains 30% one day and loses the next even though the stock went up.

Delta: how closely does the option follow the stock?#

Delta tells you roughly how much the option price changes when the underlying rises by $1. For our call with a delta of 0.55: if the stock rises from $100 to $101, the call rises from $5.31 to about $5.86.

• A call has a delta between 0 and 1. Far out of the money it is close to 0 (the option barely reacts); deep in the money it is close to 1 (it moves almost like the stock itself).
• A put has a delta between −1 and 0. It gains when the stock falls.
• At the money – when price and strike are equal – a call’s delta is about 0.5, slightly more because of interest rates and the skew of the price distribution.

Delta has two more very practical meanings:

Delta as share equivalent. Holding 10 calls on 100 shares each with a delta of 0.55 behaves, in the short run, like owning 550 shares. Professionals convert their whole book into “delta shares” this way and know how much price risk they really carry.

Delta hedging. Market makers who sell you a call don’t want to bet on direction. So they buy as many shares as delta dictates – 55 shares per 100-share contract at a delta of 0.55. If the stock rises and delta with it, they buy more. If it falls, they sell. These hedging trades noticeably move markets.

Try it: change price, time and volatility and watch how the price and the Greeks shift. The gold line is the tangent to the price curve – its slope is delta.

Gamma: how fast does delta change?#

Delta is not constant. When the stock rises, a call moves deeper into the money and its delta grows. Gamma measures how fast that happens: the change in delta per $1 move in the stock.

Our call has a gamma of 0.032. If the stock rises by $1, delta goes from 0.55 to about 0.58. After a $10 rise it would be well above 0.8. That is why the delta calculation is only accurate for small moves – in big moves the call buyer gains more than delta promises and loses less.

Option buyers are “long gamma”: big moves – in either direction – work in their favour. Sellers are “short gamma”: for them a violent move is dangerous because their losses accelerate.

Theta: the price of time#

An option is a right with an expiry date. The less time is left, the lower the chance of a big move – and the less the time value is worth. Theta measures this loss per day, all else being equal.

With 90 days left our call loses about 3 cents a day. That sounds small, but at a price of $5.31 it is more than 0.5% daily. And decay accelerates: with only 10 days left, the same at-the-money call costs only about $1.70 but loses almost 9 cents a day.

Theta and gamma are two sides of the same coin. The buyer pays theta every day to be long gamma – to profit from big moves. The seller collects theta as a reward for bearing the risk of big moves. Whether an option “paid off” ultimately depends on whether the realised volatility was larger or smaller than the volatility priced in.

Vega: the bet on volatility#

Vega measures how strongly the option price reacts to a change in implied volatility – per percentage point. Our call has a vega of $0.20. If implied volatility rises from 25% to 30%, it gains about $1 without the stock moving at all.

Calls and puts both have positive vega: more expected volatility makes every option you own more valuable. Long-dated options react more strongly than short-dated ones.

All the Greeks at a glance#

The table shows a basic pattern: buyers have positive gamma and vega and pay theta for it. Sellers have exactly the opposite. There are more Greeks – rho for interest-rate sensitivity, or vanna and charm for interactions – but the big four are enough to understand what is going on.

Summary#

Delta tells you how much price risk your option carries, gamma how fast that changes. Theta is the daily price the buyer pays for the right, vega the bet on future volatility. Every option position is therefore a combination of bets on direction, size, time and volatility. In the next lesson we look at where the prices themselves come from – from a simple binomial tree to the Black-Scholes formula.