By the FinLingo Team | Capital markets practitioner, front office experience at a major European investment bank. FinLingo covers 342 lessons from bonds to exotic derivatives. About · Last updated:
Delta (Δ) is the first and most important Greek. It measures how much an option's price changes for a $1 move in the underlying asset. A call with delta 0.60 gains approximately $0.60 when the stock rises by $1. A put with delta −0.40 gains $0.40 when the stock falls by $1.
The primary interpretation of delta is as a hedge ratio. If you are short 1,000 call options with delta 0.52, you need to buy 520 shares of the underlying to be delta-neutral. This combined position has no directional exposure to small stock moves. The hedge must be rebalanced as delta changes — which is where gamma comes in.
A common shortcut says "delta approximates the probability of expiring in-the-money." This is misleading. N(d1) is the delta of a European call, but the risk-neutral probability of exercise is N(d2). They differ by σ√T. For a 6-month ATM call with 30% vol, that difference is about 10 percentage points. On a trading desk, nobody thinks of delta as a probability. They think of it as the number of shares to trade.
Deep in-the-money calls have delta close to 1.0 — they move almost dollar-for-dollar with the stock. Deep out-of-the-money calls have delta near 0 — they barely respond. ATM calls sit around 0.50. As expiry approaches, this curve steepens: ITM deltas snap to 1, OTM deltas snap to 0. Near expiry, ATM delta becomes extremely sensitive to small stock moves.
Stock at $49, strike $50, r = 5%, σ = 20%, T = 20 weeks. Compute d1 = 0.054, giving N(d1) = 0.522. A trader short 100,000 of these calls must buy 52,200 shares at $49 to hedge — a $2.56 million position. If the stock rises to $50, delta increases to roughly 0.57, and the trader must buy another 4,800 shares. That rebalancing cost is the price of maintaining neutrality.
Delta tells you how much the option price moves for a $1 change in the underlying. A delta of 0.60 means the option gains $0.60 per $1 stock increase. It also tells the trader how many shares to hold to hedge the option position. It is the most fundamental Greek for risk management on any options desk.
No. Delta equals N(d1) for a European call, but the risk-neutral probability of exercise is N(d2). The two differ by sigma times the square root of time. For practical hedging, delta is the hedge ratio, not a probability. The probability interpretation is a common shortcut that breaks down for longer-dated or higher-volatility options.
Because the option moneyness changes. As the stock rises, a call becomes more in the money and delta increases toward 1. As the stock falls, the call becomes more out of the money and delta decreases toward 0. The rate at which delta changes is measured by gamma. High gamma means delta is unstable and the hedge needs frequent rebalancing.
FinLingo covers delta and all five Greeks in Level 3 — 10 units with interactive charts. Explore delta curves in The Lab. Level 1 is free.
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