Interpreting Risk-Neutral Option Density Moments and Implied Volatility

Summary

The document asks how to interpret the mean and standard deviation of a risk-neutral density inferred from one-week AMD options using the Breeden-Litzenberger result. It treats the discounted density’s expected underlying price as a forward-price estimate and compares the standard deviation of strike-based returns with quoted implied volatility. The reported mean is close to spot, while the calculated return standard deviation is much smaller than the options’ implied volatility.

No answer or supporting derivation is included, so the apparent discrepancy remains unresolved. The question does not specify density normalization, the precise return transformation, or how implied volatility is annualized and mapped to a one-week horizon. Those details matter when comparing distribution moments with quoted volatility. The options are American, and the document assumes they can be treated as European, which may also limit interpretation.

Key ideas

• A risk-neutral density inferred from option prices can be used to calculate an expected underlying price.
• The document compares that expectation with spot and interprets it as a forward-price estimate.
• It asks why the standard deviation of returns derived from the density is below quoted implied volatility.
• No response is provided to resolve the discrepancy or validate the calculation.
• Treating American AMD options as European is an assumption that may affect the inferred density.

Tags

# Interpretation of first and second moments of Risk Neutral Densities for AMD


# Interpretation of first and second moments of Risk Neutral Densities for AMD












Suppose I have correctly computed the option-implied risk neutral density for AMD options expiring in exactly 1 week, and discounted this expectation with the correct risk free rate, using the Breden Litzenberger result. Let us pretend that these AMD options, while American are basically equivalent for European Options.

Now suppose I calculate the following moments:

(1) Expected-value over the entire density. Sum(f(Xi)*Xi)/N for all i=1...N

(2) Transform the X axis (strikes) into a return by computing (K-s0)/S0. Then compute the standard deviation over the entire return density (volatility).

I would like to check if my intuition is correct regarding these statistics.

(1) This gives the forward price of AMD under the risk neutral measure. When I do this calculation, the expected value is very close to current trading price.

(2) This gives me the volatility of the return distribution under the risk neutral measure. It turns out to be around 3%. However, most amd options of this maturity have IV of 50% or higher. Why is there such a large discrepancy?

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