Delta (Options)

Delta, in options trading, is the rate at which the price of an option changes when the price of its underlying asset changes. A delta of 0.60 means that if the underlying rises by $1, the option is expected to gain about $0.60, other things being equal. Delta is the first and most widely used of the option Greeks, the set of sensitivity measures used to manage option risk.

The delta of a call option lies between 0 and 1, because a call gains when the underlying rises. The delta of a put option lies between −1 and 0, because a put gains when the underlying falls. The underlying asset itself has a delta of exactly 1. Traders use delta to express an option position as an equivalent holding of the underlying and to decide how much of the underlying to trade as a hedge.

Formula

Delta is the first derivative of the option's value with respect to the underlying price:

\Delta = \frac{\partial V}{\partial S}

Under the Black-Scholes model, for an option on an asset that pays no dividends:

\Delta_{\text{call}} = N(d_1)

\Delta_{\text{put}} = N(d_1) - 1

d_1 = \frac{\ln(S / K) + \left(r + \frac{\sigma^2}{2}\right) T}{\sigma \sqrt{T}}

$V$ is the option's value and $S$ is the price of the underlying.
$K$ is the strike price.
$r$ is the risk-free interest rate and $T$ is the time to expiry in years.
$\sigma$ is the volatility of the underlying.
$N(\cdot)$ is the cumulative standard normal distribution.

Worked example

A share trades at $100. Consider a call and a put with a strike of $100 and three months to expiry, with an interest rate of 5 percent and volatility of 20 percent. Then $\ln(S/K) = 0$ , and the numerator of $d_1$ is (0.05 + 0.02) × 0.25 = 0.0175. The denominator is 0.20 × √0.25 = 0.10. So $d_1 = 0.175$ and $N(0.175) = 0.57$ .

The call has a delta of 0.57 and the put a delta of 0.57 − 1 = −0.43. If the share rises to $101, the call should gain about $0.57 and the put should lose about $0.43. A trader holding 10 call contracts, each covering 100 shares, has a position delta of 10 × 100 × 0.57 = 570. For small moves the position behaves like 570 shares.

How delta behaves

Delta depends on where the underlying price sits relative to the strike.

At the money. An at-the-money option has a delta near 0.5 for a call and −0.5 for a put.
In the money. A deep in-the-money option moves almost one for one with the underlying, with delta approaching 1 or −1.
Out of the money. A far out-of-the-money option barely responds, with delta near zero.

Delta is not fixed. It changes as the underlying moves, and the rate of that change is gamma. It also changes with time and volatility. As expiry approaches, the deltas of in-the-money options move toward 1 and those of out-of-the-money options toward 0, so the transition around the strike becomes abrupt. Higher volatility pulls all deltas toward the middle. The sensitivity of delta to volatility is called vanna.

Uses

The main use is hedging. A dealer who has sold calls with a total delta of 570 can buy 570 shares so that small price moves no longer affect the combined position. This is delta hedging, and a position with a total delta of zero is called delta neutral. Because delta changes, the hedge has to be adjusted as the market moves.

Delta is also used to add up risk. The deltas of all options and underlying holdings on the same asset can be summed into one number that shows net directional exposure. Many traders use delta as a rough guide to the probability that an option will finish in the money: a call with a delta of 0.25 is loosely described as having a one-in-four chance. The figure is an approximation, since the exact risk-neutral probability in the model is a slightly different quantity. In the FX option market, strikes are commonly quoted by delta, so that a "25-delta put" identifies a strike by its sensitivity instead of by a price level.

Limitations

Delta describes the effect of a small move only. For a large move the actual change in option value differs from the delta estimate, and the gap is the effect of gamma. A position that is hedged for small moves can still gain or lose substantially on a jump.

Delta is also a model output. It depends on the volatility put into the formula, and different assumptions about how implied volatility shifts when the underlying moves give different deltas for the same option. Where there is a pronounced volatility skew, the Black-Scholes delta can misstate the true sensitivity, and dealers often apply an adjusted figure. Delta says nothing about exposure to time decay or to changes in volatility, which are measured by the other Greeks.

Last updated 15 Nov 2026