Options Greeks

The Options Greeks are a set of metrics that quantify the sensitivity of an option's price to changes in various underlying factors. Named after Greek letters (with the exception of vega, which is not actually a Greek letter), they are the essential risk management tools for anyone who trades options.

An option's price depends on multiple inputs: the underlying stock price, time to expiration, implied volatility, interest rates, and dividends. When any of these inputs change, the option's value changes. The Greeks tell you how much it will change, allowing traders to quantify and hedge their risk exposures.

The five primary Greeks are:

• Delta (Δ) — sensitivity to the underlying price
• Gamma (Γ) — rate of change of delta
• Theta (Θ) — sensitivity to time (time decay)
• Vega (v) — sensitivity to implied volatility
• Rho (ρ) — sensitivity to interest rates

These are all derived mathematically from the Black-Scholes model (or other option pricing models) by taking partial derivatives of the option price with respect to each input variable.

Delta (Δ) is the most important Greek. It measures how much the option price changes for a $1 change in the underlying stock price. A call with delta = 0.60 gains approximately $0.60 when the stock rises $1 and loses $0.60 when it falls $1.

• Call deltas range from 0 to 1; put deltas range from -1 to 0.
• At-the-money options have delta near ±0.50.
• Deep in-the-money options have delta near ±1.0 (behave like the stock).
• Deep out-of-the-money options have delta near 0 (barely sensitive to the stock).
• Delta also approximates the probability that the option expires in-the-money.

Gamma (Γ) is the rate of change of delta — it measures how quickly delta shifts as the stock moves. Gamma is highest for at-the-money options near expiration. High gamma means your delta can swing rapidly, creating large P&L swings. This is why options market makers closely monitor gamma exposure — being short gamma near expiration is one of the most dangerous positions in options trading.

Theta (Θ) measures time decay — how much value an option loses each day, all else being equal. Theta is always negative for long option positions (options lose value as time passes). At-the-money options have the highest theta, and theta accelerates as expiration approaches. If you are long options, theta works against you; if you are short options (selling premium), theta works in your favor.

Vega (v) measures how much the option price changes for a 1-percentage-point change in implied volatility. An option with vega = 0.20 will gain $0.20 in value if implied volatility rises by 1 percentage point (e.g., from 25% to 26%).

• Vega is positive for both calls and puts — higher volatility makes all options more valuable (more uncertainty = more upside potential for option buyers).
• At-the-money options have the highest vega.
• Longer-dated options have higher vega than shorter-dated ones.

Vega is arguably the most important Greek for volatility traders. When a trader says they are "long vol," they mean they have positive vega exposure — they profit when implied volatility rises. When they "sell vol," they have negative vega and profit when implied volatility falls.

Rho (ρ) measures sensitivity to the risk-free interest rate. A call with rho = 0.05 gains $0.05 for a 1-percentage-point increase in interest rates. Rho is typically the least important Greek for short-dated options but becomes more relevant for long-dated options (LEAPS) and in environments where interest rates are changing rapidly.

Suppose you are a market maker at Optiver and you've sold 100 call options on stock XYZ (strike $100, expiring in 30 days) at $3.50 each. The stock is currently at $100. Your Greeks are:

• Delta: -100 × 0.52 = -52 (equivalent to being short 5,200 shares)
• Gamma: -100 × 0.04 = -4.0
• Theta: -100 × (-0.08) = +8.0 (you collect $8/day in time decay)
• Vega: -100 × 0.15 = -15.0 (you lose $15 per 1% increase in IV)

Delta hedging: To neutralize your delta exposure, you buy 5,200 shares of XYZ. Now your portfolio is delta-neutral — a small move up or down doesn't affect your P&L much.

What you're left with: Short gamma, long theta, short vega. You profit from time passing (theta), but you're exposed to large moves (gamma) and rising volatility (vega). If the stock stays near $100 and volatility declines, you win. If the stock makes a big move or volatility spikes, you lose.

This illustrates the fundamental tradeoff in options: theta and gamma are inversely related. If you earn time decay (positive theta), you must accept gamma risk (negative gamma), and vice versa. Managing this tradeoff is the daily reality of options market makers.