Delta: sensitivity to the share price
Delta is the change in the premium for a one-dollar change in the share price. A call with a delta of 0.45 gains roughly 45 cents per share if the stock rises a dollar, which is 45 dollars on the contract. Calls run from 0 to 1, puts from 0 to −1, and the sign simply records the direction of the relationship.
Delta is also read as a rough probability of finishing in the money, and as a share-equivalent exposure: a 0.45-delta call behaves, for small moves, like holding 45 shares. Both readings are useful and both are approximations that decay as the move gets larger, which is exactly what the next greek measures.
Gamma: how quickly delta itself moves
Gamma is the change in delta for a one-dollar change in the share price. It matters because delta is not constant: a contract that behaved like 45 shares can behave like 70 after a two-dollar move. Gamma is largest for contracts near the money and close to expiry, which is the arithmetic behind the warning attached to short-dated positions.
For a holder, gamma works in your favor: gains accelerate and losses decelerate. For a writer it does the opposite, and that asymmetry is the honest reason an option sold near expiry is a different proposition from the same option sold with two months to run.
Why the second derivative earns a nameDelta alone would be enough if the relationship were linear. It is not, and gamma is the size of that curvature. A position hedged on delta and ignoring gamma is hedged for a move that does not happen and unhedged for the one that does.
Theta: the cost of time passing
Theta is the change in the premium for one day passing, everything else unchanged. It is quoted per share per day and it is negative for anyone holding an option, because the extrinsic part of the premium is compensation for remaining time and there is one day less of it. A theta of −0.04 means the contract sheds about four dollars a day if nothing else moves.
Theta is not constant either. It grows as expiry approaches for contracts near the money, which is why the last days of a position lose extrinsic value fastest. Writers receive what holders pay, and the same acceleration is what makes the final week attractive to sell and dangerous to hold.
Vega: sensitivity to implied volatility
Vega is the change in the premium for a one-percentage-point change in implied volatility. Implied volatility is not a measurement of the past; it is the figure at which the market is currently pricing the possibility of movement. When it rises, options become more expensive without the share having moved at all.
This is where a position can be right about direction and still lose. A call bought before an earnings date, when implied volatility was elevated, can fall in value when that expectation deflates afterwards, even if the share moved the way the buyer expected. Vega is the quantity that makes that outcome predictable rather than mysterious.
Rho, and what the greeks cannot do
Rho is the sensitivity to interest rates, and it is last on the list because for short-dated equity options it is usually the smallest of the five. It becomes worth attention on long-dated contracts, where the cost of carrying the underlying is a real component of the price.
The general limit is worth stating plainly. Every greek is a partial derivative: it describes what happens if one input moves and the others hold still. Real markets move several at once: the share falls, implied volatility jumps, a day passes. The greeks then have to be added up, each one already slightly wrong. They are an instrument panel, not a forecast, and reading them as a forecast is the one mistake they invite.