Risk-Neutral Density Validation

Article Quant Q&A · Author: v.y.

Summary

The document describes a proposed validator for a risk-neutral distribution recovered from option prices. It uses the distribution to compute expected discounted put payoffs across strikes, then checks whether each theoretical price lies between the corresponding market bid and ask. The author reports that many generated prices fall outside those spreads and asks whether this is a suitable validation method or evidence of a faulty density generator.

The approach offers a direct price-consistency check, but the document supplies no results that establish whether the method or the generator is at fault. Interpretation depends on details such as the density's support and normalization, the mapping from returns to prices, discounting, and the quality and timing of the quoted spreads. The text presents a question and implementation rather than a resolved evaluation, so out-of-spread prices alone do not identify the source of error.

Key ideas

A recovered risk-neutral density can be used to calculate expected discounted option payoffs.
The proposed check compares generated put values with market bid–ask intervals at matching strikes.
Prices outside quoted spreads may indicate model or implementation issues, but the document does not diagnose them.
Validating the method requires checking density construction, pricing conventions, and quote quality.

Tags

Options Derivatives pricing Statistics

Full text

# Validator for Risk-Neutral Distributions Derived from Option Prices


# Validator for Risk-Neutral Distributions Derived from Option Prices












I've developed a validator for risk-neutral distributions. I did this for the purpose of testing the risk-neutral distributions generated by a Spectral Analysis risk-neutral density recovery method, implemented in MATLAB here.

My validator calculates the expected payoff for each put option across the provided strike prices, thereby generating a set of theoretical option prices. These are compared against the market's bid and ask prices to assess if the generated prices fall within the acceptable range.

Here is the MATLAB code for the validator:

```
currentPrice = F * exp(-r * tau);
    
FPrices = currentPrice + currentPrice * result2.returns;

densityR = result2.densityR; 

% Calculate the payoff for each option
optionPrices = [];

% Loop through each strike price in X
for i = 1:length(X)
    % Calculate payoff for the current strike price against all FPrices
    % This calculates the put option payoff: max(0, K - S)
    % FPrices and densityR are extremely dense - array length = 1387
    optionPrices(i) =sum(max(0, X(i) - FPrices) .* densityR)* exp(-r * tau);
end

invalidPricesResult = validatePrices(X, bid, ask, optionPrices);

% Display invalid prices, if any
disp('Invalid Prices for Result:');
disp(invalidPricesResult);

function validationResults = validatePrices(strikePrices, bids, asks, pricesToCheck)
    validationResults = [];
    for i = 1:length(strikePrices)
        K = strikePrices(i);
        bid = bids(i);
        ask = asks(i);
        priceToCheck = pricesToCheck(i);
        isValid = bid <= priceToCheck && priceToCheck <= ask;
        if ~isValid
            validationResults = [validationResults; K, priceToCheck, bid, ask, isValid];
        end
    end
end
```

The validator code is to be pasted at the end of the spectralExample.m file from the MATLAB project linked above.

However, using this validator, an enormous amount of the risk-neutral-distribution-generated put option prices lie outside the bid-ask spread. Here's a screenshot showing this: the 1st column is the strike, the 2nd column is the generated-price, the 3rd column is the bid, the 4th column is the ask, the 5th column is the outcome of the validation (0 = invalid):

Therefore, I have to ask - is this a good way to validate a risk-neutral distribution?

If so, does it mean that there is a problem with this risk-neutral-distribution generator?

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.

Testing Risk-Neutral Densities Against Option Bid–Ask Prices

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