Delta measures how much an option's price changes for a $1 move in the underlying stock. A call option with delta 0.60 increases by $0.60 for each $1 rise in the stock price. Delta ranges from 0 to 1.0 for calls and -1.0 to 0 for puts. Deep in-the-money options have deltas near ±1.0 (they behave almost like stock); far out-of-the-money options have deltas near 0 (they barely respond to small price moves). At-the-money options have deltas near ±0.50. Delta also approximates the probability that the option expires in-the-money — a 0.30 delta call has approximately a 30% probability of expiring in-the-money.

Gamma measures the rate of change of delta as the stock price moves — it is the second derivative of option price with respect to stock price. High gamma options are near-the-money with short time to expiration; their delta changes rapidly with stock moves, creating non-linear behavior. Long options positions have positive gamma (delta moves in your favor as price moves your way), creating a convex payoff. Short options positions have negative gamma (delta moves against you), creating a concave payoff. Managing gamma is the central challenge of options market-making and complex options strategies.

Theta measures the daily loss in option value due to the passage of time, all else equal. A theta of -0.05 means the option loses $5 (100 × $0.05) per day from time decay. Theta accelerates as expiration approaches, particularly in the last 30 days — the curve of time value decay is exponential near expiration. This is why options sellers prefer short-dated options (maximum theta decay rate) while options buyers prefer longer-dated options (time decay is slower relative to total time value).

The relationship between theta and gamma is fundamental: options positions that generate gamma (long options) pay theta as the cost of that convexity. Options positions that generate theta (short options) sacrifice gamma. A straddle buyer pays daily theta for the positive gamma — if the stock moves enough (more than theta compensation), the position profits. If the stock stays still, theta erodes the position. This theta-gamma trade-off is the central risk-reward dynamic in almost every options strategy.

Vega measures how much an option's price changes for a 1% change in implied volatility. A vega of 0.25 means the option gains $0.25 in value for each 1% rise in IV. Long options (calls or puts) have positive vega — they benefit from rising volatility because higher IV means more probability of large price moves, making the option more valuable. Short options have negative vega. Vega is highest for at-the-money, longer-dated options — these have the most time value to be affected by IV changes. Managing vega is critical for options portfolios because IV can change dramatically (VIX can double in a week during crises), creating large P&L swings independent of price moves.

Rho measures sensitivity to interest rate changes — less important for most short-dated options but relevant for long-dated options (LEAPS) where the present value of the strike becomes more sensitive to rate levels. Portfolio-level Greeks management involves aggregating individual position Greeks to understand the portfolio's total sensitivity. A delta-neutral portfolio (near zero aggregate delta) makes money from volatility (if long gamma) or from time decay (if short gamma) rather than from directional stock moves. Professional options market makers delta-hedge continuously, buying or selling underlying shares to maintain near-zero aggregate delta as stock prices move.