Delta

What delta measures

Delta is the first thing to know about any option position: how much it gains or loses when the underlying moves a little. A delta of 0.53 means that if BTC rises by $1,000, the option gains about 0.53 × $1,000 = $530.

• Calls have positive delta, between 0 and 1: they gain when the price rises.
• Puts have negative delta, between −1 and 0: they gain when the price falls.
• Deep in-the-money options approach ±1 and move almost like the underlying itself; far out-of-the-money options approach 0 and barely react (see moneyness).

A worked example

BTC at $100,000; a $100,000 call with 30 days left and 50% implied volatility is worth about $5,714, with a delta of 0.529. The $100,000 put has a delta of −0.471.

• If BTC rises to $101,000, delta predicts the call gains about $529, to $6,243. The model price is actually $6,256: the extra $13 comes from delta itself growing as the price rises, which is gamma.
• The call's delta minus the put's delta is 0.529 − (−0.471) = 1. That always holds for a call and a put at the same strike and expiry: together, long call and short put behave exactly like the underlying (a synthetic position).

Delta of a position

Deltas add up. A strategy's delta is the sum of its legs' deltas, each with its sign and size, and it tells you how much of the underlying the whole position is equivalent to right now. Buying one $100,000 call and selling 0.529 BTC of a perpetual future gives a position with zero delta: small moves in either direction barely change its value. That is called being delta-neutral, and it is how market makers isolate the other risks an option carries.

A covered call (holding the underlying, delta 1, and selling a call, delta −0.53) has a delta near 0.47: it still gains when the price rises, but about half as fast as the underlying alone.

Delta as a rough probability

Traders often read a call's delta as the chance that it finishes in the money: a 0.25-delta call is "about a 25% chance". It is a useful shortcut, not an exact figure. In the model the probability of finishing in the money is a slightly different number (for the example call, 47% versus a delta of 53%), and both depend on implied volatility being the right forecast, which it rarely is.

Delta changes

Delta is not fixed. It moves as the price moves (gamma), as time passes (an out-of-the-money option's delta drifts toward 0, an in-the-money one's toward ±1), and as implied volatility changes. A position that is delta-neutral now will not stay neutral without adjusting.

On a strategy page, BOSS shows the structure's live delta from the venue's own Greeks, and charts delta across a range of prices using the Black-Scholes model with each leg's current implied volatility.