Recovering Risk-Neutral Probabilities from Option Prices
Summary
The article explains how an option chain can reveal the market-implied distribution of an underlying’s price at expiry. It uses the Breeden-Litzenberger result: the second strike derivative of call prices, adjusted for discounting, gives a risk-neutral density. Since raw option quotes are noisy, the described MetaTrader indicator first converts prices to implied volatility, smooths the volatility smile, and reconstructs call prices before estimating the density. A hybrid Newton and bisection procedure is used to invert Black-Scholes prices robustly.
The resulting distribution can be used to estimate probabilities above or below price levels, expected moves, skew, kurtosis, and confidence bands. The tool also compares the risk-neutral distribution with historical realized returns to illustrate the premium embedded in option prices. Its estimates depend on the quality and coverage of the option chain, and the risk-neutral distribution includes compensation for risk rather than representing objective real-world probabilities. The article describes native and CSV data paths and a flat-volatility control chain for validation, but the displayed estimates are only as reliable as their input data and modeling assumptions.
Key ideas
- A tight option butterfly approximates the probability mass near its central strike.
- The second derivative of call prices with respect to strike recovers a risk-neutral density.
- Smoothing implied volatility before reconstructing prices helps limit noise amplification during differentiation.
- The derived density supports probability, expected-move, skew, kurtosis, and quantile estimates.
- Comparing risk-neutral and realized distributions can expose market pricing of risk, but the former is not a forecast of objective probabilities.
Tags
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.