Extracting Risk-Neutral Price Densities from Option Prices

Summary

The discussion asks how to infer a stock price probability distribution from option prices and track changes in that distribution over time. One response identifies implied probability densities, also called state price densities, as the relevant framework. It notes that the density can be derived by differentiating call option prices with respect to strike, connecting the shape of the option price curve to the market-implied distribution.

A second suggested approach uses adjacent butterfly spreads across the option chain. Their prices or payoff characteristics can be used to estimate probability mass over the intervals between strikes, with the resulting values normalized so they sum to one. These are presented as ways to extract a distribution from option prices, not as a demonstrated daily forecasting method. The discussion gives no data, derivation, or validation of how changes in the estimated density should be interpreted. In practice, the estimates depend on the option chain and the assumptions used to turn prices into probabilities; the answers do not address those details.

Key ideas

• Option prices can be used to infer an implied state price density for the underlying asset.
• Differentiating call prices with respect to strike is described as a route to the implied density.
• Adjacent butterfly spreads can estimate probability mass between neighboring strikes.
• The estimated masses should be normalized to sum to one.
• Changes in the extracted density are proposed for monitoring market views, but no predictive evidence is supplied.

Tags

# How do I calculate probability distribution of stock prices given option prices?


# How do I calculate probability distribution of stock prices given option prices?












I'd like to calculate a probability distribution for prices given the option prices for that stock? Any ideas how to do this?

My desire is to do this daily and then see how the price PD changes over time and see if that can give any insight to the market's evolving view. I couldn't find any prior art on the later, any suggestions would be appreciated.

## Answer by Probilitator (score 7, accepted)

https://quant.stackexchange.com/a/10638

you should have a look at implied probability densities. They do exactly what you are asking - extracting the pricing density from option prices.

This is done by differentiating the option price with respect to the call.

Here are two links. The first one explains the procedure the second one deals with where such densities can be applied

- Implied state price density (Question 1 - derivation of the formula)

- Implied probability density (Question 2 - Applications and Interpretation)

## Answer by delta hedge (score 7)

https://quant.stackexchange.com/a/10637

One approach is to take the entire option chain, and calculate the prices for adjacent butterflies along the chain. The risk / reward of each of the butterflies represent the empirical probability that the market is pricing for the underlying to move between the strikes of the butterfly.

To make sure it is a proper probability distribution, you will want to normalize these empirical probabilities so that the sum of the entire probabilities from all the butterflies equals 1.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.