Option Greeks

Option Greeks are measures of how the price of an option responds to changes in the factors that determine it. Each Greek isolates one factor, such as the price of the underlying, the passage of time or the level of volatility, and states how much the option's value changes for a small change in that factor with everything else held fixed. They take their collective name from the Greek letters used to denote most of them.

The Greeks are the working vocabulary of options risk. A trader holding hundreds of different contracts cannot reason about each one, but can add up the Greeks across the book to see its exposure to each source of risk and decide what to hedge. They are calculated from a pricing model, most often the Black-Scholes-Merton model, and so depend on that model's assumptions.

Delta is the change in the option's price for a one-unit change in the underlying. It runs from 0 to 1 for a call and from -1 to 0 for a put, and it is also the number of units of the underlying needed to hedge one option. Gamma is the rate at which delta itself changes, and it is highest for options near the money and close to expiry. Theta is the amount of value lost each day as expiry approaches. Vega is the change in price for a one-percentage-point change in implied volatility. Despite its name it is not a Greek letter. Rho is the change for a one-percentage-point change in the interest rate and is usually the least important for short-dated options.

Mathematically each Greek is a partial derivative of the option's value. Together they give an estimate of the change in price for small simultaneous moves:

• $\Delta V$ is the change in the option's price.
• $\Delta S$ is the change in the price of the underlying.
• $\Delta\sigma$ is the change in implied volatility, in percentage points.
• $\Delta t$ is the time elapsed, in days.
• $\delta$, $\Gamma$, $\nu$ and $\Theta$ are the option's delta, gamma, vega and theta.

Suppose a call has a delta of 0.50, a gamma of 0.04, a vega of 0.12 and a theta of -0.05 per day. Over one day the share rises by 2 and implied volatility rises by one point. The estimated change in the option's price is $0.50 \times 2 + 0.5 \times 0.04 \times 2^{2} + 0.12 \times 1 - 0.05 \times 1 = 1.15$. Delta contributes 1.00, gamma 0.08, vega 0.12 and theta -0.05. The estimate is accurate for small moves and becomes less so for large ones, when the Greeks themselves change.

Because the main Greeks change as the market moves, traders also track the sensitivities of the sensitivities. Vanna measures how delta changes with volatility, or equally how vega changes with the underlying. Vomma, also called volga, measures how vega changes with volatility. Charm measures how delta changes with time. These matter most for large books, for options far from the money and for products whose value depends heavily on the shape of the volatility surface.

• Hedging direction. A market maker who sells options offsets the delta by trading the underlying, a practice called delta hedging. A book whose deltas sum to zero is delta neutral.
• Trading volatility. With delta hedged away, what remains is exposure to gamma, vega and theta. A position that is long gamma profits from large moves and pays for it through negative theta. Gamma scalping is the technique of harvesting that gamma by re-hedging.
• Setting limits. Firms cap the net delta, gamma and vega a desk may carry, alongside scenario tests. This is the core of options risk management.
• Choosing contracts. Buyers and sellers use the Greeks to compare options, for example by choosing a strike by its delta or weighing the theta earned against the gamma risk taken.

The Greeks are linked. An option with high gamma also has high theta, because the features that make it responsive to moves in the underlying are the same ones that make it lose value quickly as time passes. A buyer of options pays theta every day in exchange for gamma, and gains only if the underlying moves more than the option's price implied. A seller collects theta and loses if the moves are larger. The option premium can be seen as the market's price for this exchange.

The Greeks are local measures, valid for small changes around current conditions. In a sharp move they shift, sometimes quickly, so a book that was hedged can become badly exposed. They are also model outputs. Different assumptions about volatility, dividends and interest rates give different Greeks for the same option, and the basic model's assumption of constant volatility is contradicted by the market itself. Each Greek also treats its factor in isolation, whereas in reality the underlying and its implied volatility tend to move together. For these reasons the Greeks are supplemented with full revaluation of the book under large hypothetical moves.