Recovering Risk-Neutral Density from Vanilla Option Prices

Summary

The note gives a direct method for estimating an asset’s risk-neutral probability density from quoted vanilla call prices. The density is obtained by taking the second derivative of the undiscounted call price with respect to strike. Put prices can also be used for the corresponding portion of the implied volatility smile.

The method requires a complete, smooth price curve across strikes, so the observed option quotes must first be interpolated and extrapolated. The document does not compare fitting methods or discuss their advantages and drawbacks, despite the question asking for them. It also provides no empirical example, and the resulting density depends on the quality and assumptions of the curve construction.

Key ideas

• The risk-neutral density is obtained from the second strike derivative of the undiscounted call price.
• Put prices can be used to represent the put portion of the option smile.
• A smooth, sufficiently complete option price curve must be constructed through interpolation and extrapolation.
• The note does not specify how curve-building choices affect the inferred density.

Tags

# Suggestions of papers for computing market implied probability distribution function


# Suggestions of papers for computing market implied probability distribution function












I need suggestions of papers that propose simple and fast methods (not heavily dependent on simulations, nut can depend on simulation) to derive the market implicit probability distribution function of an asset (under risk neutral measure) from plain vanilla option (on this same asset) quoted prices.

I would also appreciate to read the main advantages and drawbacks of each suggested model and a summary of it.

Thank you!

## Answer by user34971 (score 1)

https://quant.stackexchange.com/a/42350

The market implied probability density function is $\partial^2 C(K,T) / \partial K^2$, where $C$ is the un discounted option call price. You can also use puts instead of calls for the put part of the smile.

Hence given vanilla options prices (and hence the smile) you need to take the second derivative. Note that you will need to have constructed the full and smooth smile first using interpolation and extrapolation of your preference.

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.