Risk-Neutral Density
The full probability distribution the options market is pricing into expiration.
The probability distribution the options market is pricing
The second derivative of call price with respect to strike is the risk-neutral probability density: f(K) = erT · ∂²C/∂K². Read it and you can see exactly how much probability the market assigns to every landing zone for the underlying at expiration.
Differentiating raw quotes twice is hopelessly noisy, so this tool first fits a smooth volatility smile across strikes, converts that back into a clean call-price curve, and only then differentiates. Spot and the ±1σ implied range are marked, and you can read the implied probability of finishing inside any price band.
Risk-Neutral Density
Cumulative Distribution
probability of finishing at or below each priceProbability of finishing in a range
How to read this
- The peak is the market’s modal expectation; the spread is its uncertainty.
- A left tail fatter than the right is the equity-index signature — crash insurance is bid, so downside outcomes carry more risk-neutral probability.
- ±1σ markers use the ATM implied vol scaled to this expiry; roughly 68% of the density sits between them in a symmetric world, less on the downside when skew is steep.
Risk-neutral, not real-world: these probabilities embed the risk premium investors pay for protection, so they overstate true crash odds. Smile fit uses a kernel-smoothed IV curve; r assumed 4.3%.
Learn
The big-picture guide to what the implied distribution is and why it matters.
Read →FoundationsHow option prices encode a full probability distribution of future prices.
Read →FoundationsA practical walkthrough of the curve: peak, tails, skew and width.
Read →AdvancedThe math that turns a strip of option prices into a density curve.
Read →IntermediateA case study of the density during the COVID crash and its fat left tail.
Read →AdvancedHow a binary catalyst splits the density into two distinct humps.
Read →AdvancedTurning the density into actionable views on skew, tails and pricing.
Read →FoundationsQuick answers and key term definitions for the implied distribution.
Read →AdvancedSpotting and reacting to meaningful changes in the implied density.
Read →FoundationsA regular read of the current implied distribution for the market.
Read →