Greeks
The Greeks are a set of risk sensitivity measures for options and other derivatives that quantify how the price of the derivative changes with respect to changes in underlying market variables, including the price of the underlying asset, time to expiration, implied volatility, and interest rates. Each Greek is named after a letter of the Greek alphabet and represents a partial derivative of the option pricing formula.
Key takeaways
- Delta measures the sensitivity of option price to a $1 change in the underlying asset price; it also approximates the probability that the option expires in the money.
- Gamma measures the rate of change of delta with respect to the underlying price, reflecting the curvature of the option's value function.
- Theta measures the daily time decay of option value, representing the cost of holding a long options position.
- Vega (not a Greek letter) measures option price sensitivity to a 1% change in implied volatility.
- Rho measures option price sensitivity to a 1% change in the risk-free interest rate, generally more significant for long-dated options.
Explanation
The Greeks emerge naturally from the Black-Scholes-Merton (BSM) partial differential equation framework, which prices options as a function of five variables: the current price of the underlying (S), the option's strike price (K), time to expiration (T), the risk-free interest rate (r), and the volatility of the underlying (σ). By taking partial derivatives of the BSM pricing formula with respect to each of these variables, one obtains the Greeks, each providing a distinct dimension of risk measurement.
Delta (Δ) is the most commonly used Greek and the starting point for options risk management. A call option with delta of 0.60 will, for a small change in the underlying price, change in value by approximately $0.60 per $1.00 move in the underlying. Delta ranges from 0 to 1 for calls and -1 to 0 for puts. Deep in-the-money options have deltas near ±1, at-the-money options near ±0.5, and far out-of-the-money options near 0. Delta is central to delta hedging, where a market maker holds an offsetting position in the underlying to neutralize directional price risk.
Gamma (Γ), the second derivative of option price with respect to the underlying, is the rate at which delta changes. Options near expiration and near the money have the highest gamma because small price movements can dramatically alter the probability of expiration in the money. Long gamma positions (long options) benefit from large moves in either direction, while short gamma positions (short options) are hurt by large moves. Gamma and theta have an inherent trade-off: long gamma positions bleed theta daily, while short gamma positions collect theta but are vulnerable to gap moves.
Vega quantifies exposure to changes in implied volatility — the market's consensus estimate of future realized volatility embedded in option prices. A vega of 0.20 means the option price changes by $0.20 for a 1% increase in implied volatility. Long options are long vega (benefit from rising volatility) and short options are short vega. Volatility trading strategies such as straddles, strangles, and calendar spreads are fundamentally vega trades, with delta and gamma hedged away.
Rho measures interest rate sensitivity and is most significant for long-dated options (LEAPS), currency options, and futures options where the cost-of-carry relationship between futures and spot prices makes rho economically material. Advanced practitioners also work with second-order and cross-partial Greeks: vanna (change in delta with respect to volatility), volga/vomma (change in vega with respect to volatility), and charm (change in delta with respect to time), which are particularly important for exotic options and structured products.
Formula
Δ = ∂C/∂S; Γ = ∂²C/∂S²; Θ = ∂C/∂t; Vega = ∂C/∂σ; Rho = ∂C/∂r
Example
An options market maker holds a short position in 1,000 call options on a stock trading at $100, with the following aggregate Greeks: Delta = -60,000, Gamma = -800, Theta = +$2,500/day, Vega = -$15,000 per vol point. To delta-hedge, the market maker buys 60,000 shares of stock. If the stock rises $1 to $101, the hedge offsets approximately $60,000 of option losses but the short gamma position means the new delta is approximately -60,800, requiring the purchase of an additional 800 shares. If implied volatility rises 2%, the short vega position loses $30,000 regardless of the stock price. The $2,500/day theta income partially offsets these risks over time.
Related terms
American Option At The Money Call Option Charm Delta Exchange For Physicals Exotic Options Gamma Hedging Implied Volatility In The Money Interest Rate