Using the Inverse CDF for Risk-Neutral Density Probability Transforms

Summary

The document concerns evaluating density forecasts derived from GBP/USD options. The author has estimated a time series of risk-neutral densities nonparametrically and wants to apply the unconditional test described by Christoffersen and Mazzotta. The practical difficulty is obtaining probability integral transforms when the estimated densities are truncated.

The supplied answer recommends forming the cumulative risk-neutral distributions and using them to map observed outcomes into probabilities between zero and one. Reversing that mapping gives a way to sample outcomes from the estimated distribution using uniform draws, equivalent to inverse-CDF sampling. This explains the basic transformation but does not provide implementation steps, resolve how to handle truncation or probability mass outside the estimated range, or detail the unconditional test itself. The note offers a conceptual method, not a complete statistical procedure or evidence that the resulting forecasts pass the test.

Key ideas

• A risk-neutral density can be integrated to obtain its cumulative distribution function.
• The cumulative distribution maps outcomes to probability integral transform values.
• Inverting the cumulative mapping allows uniform draws to be converted into samples from the estimated density.
• The answer does not explain how to correct for truncated density support or implement the forecast test.

Tags

# Probability integral transforms of risk-neutral densities


# Probability integral transforms of risk-neutral densities












I've backed out a time series of risk-neutral densities of GBP/USD options using a non-parametric approach in Matlab and would like to assess their forecast ability by applying the unconditional test in section 2.3 (page 11) of Christoffersen and Mazzotta "The Accuracy of Density Forecasts from Foreign Exchange Options", JFinEmetr (2005). However, I am not sure how to go about obtaining a series of probability transforms from truncated RNDs. I am also very new to Matlab so any guidance regarding the implementation of Christoffersen and Mazzotta (2005) would be greatly appreciated.

## Answer by James Spencer-Lavan (score 1)

https://quant.stackexchange.com/a/36536

Calculate the cumulative RNDs - these form a bijection from the observed returns through to (0,1). Reverse the map so that a random uniform samples a return from the distribution. Non-parametric sampling that recovers the estimated terminal density goven by vanilla option prices.

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