Delta (options)

What is delta

Delta is how much an option's price is expected to change when the price of the underlying asset moves by a small amount, with everything else held constant.

An option is a contract that gives the right, but not the obligation, to buy (a call) or sell (a put) an asset at a set price (the strike) before or on an expiry date. Delta is one of the option Greeks, a set of measures of how sensitive an option's price is to different factors. For the basics of a call, see Call option.

Delta has a sign and a size:

• Calls have positive delta. The option tends to gain value when the underlying rises.
• Puts have negative delta. The option tends to gain value when the underlying falls.
• Deep in-the-money options move almost one-for-one with the underlying. Far out-of-the-money options barely react.

Worked example (hypothetical)

This example uses letters, not real prices.

• A call option on a share has delta D.
• The share price rises by a small amount m.
• The option price rises by about D × m.
• A put on the same share has a negative delta, so the same rise in the share price lowers the put's value.

For a whole position, multiply the delta by the contract size and by the number of contracts. The result is the position's delta exposure: the amount of underlying the position behaves like. A trader who holds the opposite amount of underlying is "delta-neutral" for small moves. This is why delta is also called the hedge ratio. See Exposure.

The estimate only holds for small moves. For larger moves, delta itself changes, so the result drifts away from D × m.

Formal definition and edge cases

Formally, delta is the partial derivative of the option value V with respect to the underlying price S: Δ = ∂V/∂S. In the Black-Scholes model, the delta of a European call on a non-dividend asset is N(d1). The put delta follows from put-call parity.

Points experienced traders watch:

• Gamma. Delta changes as the price moves. Gamma measures that rate of change. It is highest for at-the-money options close to expiry, where delta can swing sharply.
• Time and volatility. As expiry approaches, in-the-money deltas move toward the full underlying exposure and out-of-the-money deltas shrink. Higher implied volatility pulls deltas toward the middle. See Annualized volatility.
• Model dependence. Delta comes from a pricing model and a volatility input. Different inputs give different deltas. Volatility skew adds a further question: does implied volatility stay fixed by strike, or does it shift as the price moves?
• Probability shortcut. Delta is often quoted as a rough chance that the option expires in the money. Strictly, that probability comes from a different term of the Black-Scholes formula, and it is a risk-neutral figure, not a forecast.
• FX conventions. Currency options quote spot delta, forward delta or premium-adjusted delta. Check which one a quote uses. See Forward contract.
• Dividends and early exercise. These change delta for American-style options and for options on dividend-paying shares.

Where you see delta

• Option chains on exchange and broker platforms, shown next to gamma, theta and vega.
• Risk reports, which show the net delta of a portfolio.
• Market news about "delta hedging" or dealer hedging flows, where large option positions push dealers to buy or sell the underlying.

Options are derivatives, and leveraged derivative positions can lose money fast. For position sizing and controls, see Risk management.

Related terms

Derivatives · Call option · Expiry date · Counterparty · Trading glossary

Frequent questions

What does delta mean in options, in simple words?

Delta is roughly how far the option price moves when the underlying moves a little. A positive delta means the option tends to rise with the underlying. A negative delta means it tends to fall.

Is delta fixed for an option?

No. Delta changes with the underlying price, time to expiry and volatility. Gamma measures how fast delta changes when the price moves.

Why do traders hedge delta?

Holding an offsetting amount of the underlying cancels out small price moves. This leaves exposure mainly to volatility and time decay. The hedge has to be adjusted as delta changes.

Is delta the probability of profit?

No. It is only a rough proxy for the chance of expiring in the money, and expiring in the money is not the same as making a profit after the premium paid.