The Greeks and delta hedging
By SuperdayReps · Michigan Ross Undergrad | Wharton MBA · ex-Citi | ex-Greenhill · Former Head Mentor at a Leading IB Prep Service
12 min read · updated November 3, 2026
A client calls the equity derivatives desk and buys one-month calls on 100,000 shares of a stock. The trader who sold them does not want a view on the stock; she wants to earn the spread and give the risk back. So within seconds she buys shares to offset the calls, and for the next month she adjusts that hedge as the stock moves. Whether the trade made money comes down to one question: did the stock move more or less than the volatility she sold it at?
The Greeks are how she sees that risk, and the hedge is how she manages it. Interviewers for any derivatives seat expect the signs of the Greeks cold and want to hear that a hedged option book is a bet on volatility, not a riskless one. This lesson builds both, with one option priced and hedged all the way through.
Delta tells you direction, and a market maker hedges it away with the stock. What is left is gamma against theta. An option seller who hedges collects theta every day and pays it back in rebalancing losses whenever the stock moves, and the two break even when the stock moves by exactly the amount implied volatility priced. So a delta-hedged option is a bet on realized volatility against implied volatility.
The Greeks, one at a time
An option's price depends on the stock price, time to expiry, volatility and interest rates. Each Greek measures how the price responds to one of those while the others stay still.
- Delta: the change in the option's price for a 1 dollar move in the stock. A call's delta runs from 0 to 1, a put's from 0 to minus 1. At the money it is about 0.5 for a call (minus 0.5 for a put); deep in the money it nears 1, far out of the money it nears 0. A delta of 0.52 means the option behaves like 0.52 shares. Delta is also a rough guide to the chance the option ends in the money.
- Gamma: the change in delta for a 1 dollar move in the stock. It is positive for anyone who owns an option, call or put, because owning optionality means your delta always moves your way: you get longer as the stock rises and shorter as it falls.
- Theta: the value the option loses as one day passes, all else equal. It is negative for an owner, because time value drains away toward expiry.
- Vega: the change in price for a 1 point move in implied volatility, the volatility the option's price assumes. It is positive for an owner of either kind, because more expected movement makes any option worth more.
- Rho: the change in price for a 1 point move in interest rates. Higher rates help calls and hurt puts a little, because a call lets you hold off paying for the stock and earn interest meanwhile. It matters mainly for long-dated options.
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Frequently asked
- What are the option Greeks?
- They are the sensitivities of an option's price to each input. Delta measures the change for a 1 dollar move in the stock, gamma how fast delta itself changes, theta the value lost each day, vega the change for a 1 point move in implied volatility and rho the change for a move in interest rates. Traders add them up across a book to see what risk it really holds.
- Is gamma positive for a long put?
- Yes. Anyone who owns an option, call or put, is long gamma, because their delta always moves in their favor as the stock moves. A long put has negative delta but positive gamma, positive vega and negative theta. Selling either option flips every sign.
- What is delta hedging?
- Trading the underlying stock so that a position's total delta is zero, which means a small move in the stock up or down leaves the value roughly unchanged. Because delta changes as the stock moves, the hedge has to be rebalanced, and the profit or loss from that rebalancing is set by how much the stock actually moves compared with the volatility the option was priced at.
- Why do gamma and theta move together?
- Because they are two sides of the same trade. Owning an option gives you gamma, which pays you when the stock moves, and the price of that is theta, the value the option loses each quiet day. For a delta-hedged option the two cancel exactly when the stock moves by the amount its implied volatility predicted.
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