The convention of reading an option's delta as roughly the chance it finishes in the money; useful, approximate, and not what delta actually measures.
Under Black-Scholes the true risk-neutral probability of finishing in the money is close to a call's delta, and for most listed options the two numbers sit within a few points of each other. That is why a 0.30 delta call is described as a 30% chance.
The approximation breaks in the places you care about. It ignores real-world drift, it is a risk-neutral number rather than a forecast, and it is distorted by volatility-skew — which is precisely why deep out-of-the-money put deltas overstate the market's genuine expectation of disaster.
Example: XYZ at $50, the 45-day $45 put shows a 0.22 delta. Reading that as a 22% chance of finishing below $45 is fine for sizing. Treating it as a forecast, and selling the put because you think the real odds are 10%, is a bet against the skew rather than against the market.
Original diagrams for the ideas on this page. Illustrative, not real market data.
Delta across the range of prices. Delta says how much a call's price moves for a one-point move in the stock. Far below the strike it is near 0 and the option barely reacts; at the strike it is about 0.50; far above it approaches 1 and tracks the stock.The volatility smile. Options on the same stock and the same expiry are not priced off one volatility. Strikes near the money carry the lowest implied volatility, and it rises towards both ends — usually faster on the downside, which tilts the smile into a skew.
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