First-order Greek · ∂V/∂S

Delta: how much an option moves with the underlying

Delta is the change in an option's price for a $1 change in the underlying. A call with a delta of 0.40 gains about 40 cents when the stock rises a dollar.

The formula

d₁ = [ ln(S/K) + (r + σ²/2)·T ] / (σ√T)
d₂ = d₁ − σ√T

Δ_call = N(d₁) Δ_put = N(d₁) − 1

Black-Scholes, European options, no dividends. S spot, K strike, T years to expiry, r the risk-free rate, σ implied volatility, N(·) the standard normal CDF and φ(·) its density.

Calls have delta between 0 and 1, puts between −1 and 0. At the money a call sits near 0.50. Deep in the money it approaches 1 and the option moves like the stock; far out of the money it approaches 0.

Three ways to read it

Sensitivity. Dollars of option value per dollar of underlying.
Hedge ratio. Short one call with delta 0.40 on a 100-share contract and you buy 40 shares to be flat. That is delta hedging, and it is what market makers do all day.
A rough probability. Delta is often quoted as the odds of finishing in the money. It is close but not exact: the risk-neutral probability is N(d₂), not N(d₁).

Delta moves, and that's the whole game

Delta isn't fixed. It changes with price (that's gamma), with time (charm) and with implied volatility (vanna). A hedger who is flat now won't be flat after any of those move, so they trade again. Summed across every option dealers hold, those re-hedges are the flows Zanshin Flow tracks.

Dealer delta exposure (DEX)

Add up delta × open interest × contract multiplier across a chain, signed for the side dealers are assumed to hold, and you get dealer delta exposure: the share-equivalent position market makers carry before hedging. Zanshin Flow reports it in the Exposures window alongside gamma, vanna, charm and theta.

Questions

What does a delta of 0.50 mean?

The option gains or loses about 50 cents for each $1 move in the underlying, and a hedger short one 100-share contract would hold about 50 shares to be neutral.

Is delta the probability an option expires in the money?

Approximately. Under Black-Scholes the risk-neutral probability of finishing in the money is N(d₂) for a call, slightly below the call delta N(d₁).

Why is put delta negative?

A put gains value when the underlying falls, so its value moves opposite to price. Put delta equals call delta minus one for the same strike and expiry.

Not financial advice. This article explains how options and dealer hedging work. It is not a recommendation to buy or sell anything.