Option Delta

Option delta is the estimated change in an option's price, per share, for a $1 change in the price of the underlying security, with every other input to the price held constant. FINRA's investor page on options defines it as "the amount an option price is expected to change based on a $1 change in the underlying stock," and the Options Industry Council, the educational arm of the Options Clearing Corporation, calls it "a theoretical estimate of how much an option's premium may change given a $1 move in the underlying." Delta is one of the options Greeks, the set of sensitivity measures produced by option pricing models.

Signs and ranges. FINRA states the ranges for the option itself: for call options delta is "a positive number between 0 and 1," and for put options "a negative number between 0 and -1." That describes someone who bought the contract. The writer of the same contract holds the mirror image. The council explains that a seller "essentially has a negative quantity of contracts," so short puts have a positive delta and short calls a negative one. A stock position has a delta of exactly 1.00 per share held long and −1.00 per share held short, which is what makes delta comparable with shares.

Delta as exposure. Because a standard equity option covers 100 shares, multiplying delta by 100 turns it into a share-equivalent figure, and adding those figures across contracts gives the delta of a whole position. A call with a delta of 0.49 behaves, for small moves, roughly like 49 shares of the stock; four such calls behave roughly like 196 shares. This is the most practical use of the number for a holder, because it answers how much stock exposure a position carries, which the number of contracts alone does not.

What moves delta. Delta changes whenever the stock price, the time to expiration or implied volatility changes. As a call goes deeper in the money its delta approaches 1, and as it moves further out of the money its delta approaches 0; puts behave the same way toward −1 and 0. Time pulls in opposite directions on the two sides of the strike. The council notes that as expiration nears, "in-the-money call Deltas increase toward 1.00, at-the-money call Deltas remain around .50 and out-of-the-money call Deltas fall toward 0 provided other inputs remain constant." Higher implied volatility does the reverse of time: "With an increase in implied volatility, Delta gravitates toward .50," because a wider range of outcomes makes more strikes plausible winners. How fast delta itself changes as the stock moves is measured by gamma.

The probability reading. The council notes that some traders "view Delta as a percentage probability an option will wind up in-the-money at expiration," so a call with a delta of 0.30 would be read as roughly a 30% chance of finishing in the money. That is a trader's shorthand, not the definition, and it is best treated as a rough guide rather than a forecast.

Limits. FINRA notes that delta is technically "an instantaneous measure of the option's price change," so it describes the next small move rather than a large one. Each broker's figure also depends on the pricing model and the inputs behind it, so two platforms can show slightly different deltas for the same contract. Delta should not be confused with beta, which measures how much a stock tends to move with the overall market; delta measures how much an option moves with its own underlying.

Delta is quoted per share. To turn it into dollars for a position, multiply it by the size of the price move and by the number of shares the contracts cover. For small moves, the option's new price is about its old price plus delta times the move.

Consider an example, with illustrative figures. A stock trades at $84. A call with an $85 strike and 45 days to expiration is quoted at $3.00 with a delta of 0.49, and the put at the same strike and expiration is quoted at $3.60 with a delta of −0.51.

• If the stock rises $1 to $85, the call should be worth about $3.00 + 0.49 = $3.49, a gain of about $49 on one 100-share contract. The put should be worth about $3.60 − 0.51 = $3.09, a loss of about $51 a contract.
• If the stock falls $1 to $83, the signs reverse: the call should be worth about $2.51 and the put about $4.11.
• Someone holding four of the calls has a position delta of 4 × 0.49 × 100 = 196, so a $1 rise in the stock should add about $196 to the position.
• Someone who sold one of the calls has a delta of −0.49 on the contract, so the same $1 rise costs that writer about $49.

The estimates are only approximate. Delta itself changes as the stock moves, so after a rise the call's delta is higher than 0.49 and the actual gain on a larger move tends to exceed delta times the move; time passing and changes in implied volatility move the price at the same time.

Pros

• Turns an option position into a share-equivalent exposure that can be compared directly with owning the stock.
• Shows at a glance which way a position makes money and roughly how fast.
• Adds across contracts, so a position mixing calls, puts and stock can be summarized in one number.

Cons

• It is a model estimate for small moves and drifts as soon as the stock moves, so it understates or overstates the effect of a large move.
• It holds time and implied volatility fixed, and both change constantly.
• The common reading of delta as the chance of finishing in the money is a trader's shorthand, not what the number is defined to measure.
• Quoted per share, it understates dollar exposure by a factor of 100 for a standard contract if the multiplier is forgotten.