Option Delta Calculator
Estimate the delta of a call or put with Black–Scholes — how much the option premium moves per $1 move in the underlying, plus the hedge ratio.
How it works
Call delta = N(d1) · Put delta = N(d1) − 1 · d1 = [ln(S/K) + (r + σ²/2)T] ÷ (σ√T)
Delta is the first derivative of the option price with respect to the underlying — the slope of the payoff curve today. Call deltas run 0 to 1, put deltas −1 to 0, and both approach the extremes deep in the money and zero far out of the money. Traders also read delta as a rough probability the option expires in the money, and as the number of shares needed to hedge one contract.
Worked example
A $105 call on a $100 stock with 25% vol, a 4% rate and half a year to expiry gives d1 ≈ 0.02, so N(d1) ≈ 0.508. The call delta is about 0.51: the premium moves ~$0.51 per $1 of stock, and one contract behaves like 51 shares.
Sources: Greeks (finance) — Wikipedia
Frequently asked questions
Is delta the same as the probability of finishing in the money?
It is a close proxy, not exact. The true risk-neutral probability of finishing in the money is N(d2) for a call, which is slightly below the delta N(d1). Traders often use delta as a quick estimate anyway.
Why is put delta negative?
A put gains value when the stock falls, so its price moves opposite to the underlying. A −0.40 delta means the put rises about $0.40 for each $1 the stock drops.
How many shares hedge one option?
Multiply delta by the contract size (usually 100). A 0.51-delta call is hedged by shorting roughly 51 shares, though the hedge must be rebalanced as delta changes (gamma).
Related calculators
Educational estimate using a simplified Black–Scholes / options model — no dividends, no early-exercise, no fees or slippage. Options are high-risk. Not investment advice.