Option sensitivities and what each one tracks

Explain what the greeks are and what each of the main ones measures.

The greeks are partial derivatives of an option's value with respect to its inputs. Delta $\partial V/\partial S$ is the hedge ratio; gamma $\partial^2V/\partial S^2$ is how fast delta changes; theta $\partial V/\partial t$ is time decay; vega $\partial V/\partial\sigma$ is volatility sensitivity; rho $\partial V/\partial r$ is rate sensitivity. Traders use them to hedge and to see which risks a book carries; higher-order ones like vanna and volga matter for volatility trading.