nikolai-alexander.com

Research

Studies

Quantitative studies of financial markets and options, with the data and the numbers behind each result.

13 October 2026

Is delta the probability of expiring in the money?

The textbook model and four real option chains (SPY, Apple, Broadcom, Bloom Energy) show when delta works as odds, and why the skew can turn the textbook answer around.

Probability of finishing in the money for 30-delta puts on SPY, Apple, Broadcom and Bloom Energy, with each strike's own volatility and with the skew.