Comparing Option-Implied and Realized Return Distributions
Article Quant Q&A · Author: KaiSqDist
Summary
The document asks how to calibrate a stochastic discount factor so that a physical or risk-adjusted density derived from option-implied risk-neutral densities resembles a realized return distribution. It raises a practical alignment issue: realized densities are estimated over returns, while option-implied densities may be expressed over terminal prices or strikes. Treating zero returns as the at-the-money point is posed as a question, not established as a valid procedure.
Key ideas
- Option prices can be used to infer risk-neutral distributions, which can be transformed toward physical distributions using a pricing kernel.
- The question proposes calibrating pricing-kernel parameters by comparing adjusted and realized densities.
- Density comparisons require compatible variables and units, such as returns versus terminal price levels.
- A suggested exploratory exercise simulates paths under a chosen drift change and compares historical and risk-neutral distributions.
- Historical estimates are time-averaged, while option-implied distributions can vary from day to day, so the comparison may require averaging across dates.
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# Questions on calibrating risk-neutral/risk-adjusted densities to match realized densities # Questions on calibrating risk-neutral/risk-adjusted densities to match realized densities In Market Timing with Option-Implied Distributions: A Forward-Looking Approach by Kostakis et al. (2011), the authors imply a risk-neutral density (from a cross section of option prices) $\rightarrow$ physical/risk-adjusted density (via the pricing kernel aka SDF). They then compare this to the realized density (from the returns I assume, the authors did not really say what it is) to calibrate the parameters of the SDF through minimizing the differences between the two densities (physical/risk-adjusted vs realized). How does one go about comparing the two densities? If I used a non-parametric method such as the kernel density estimation (KDE) approach to estimate a realized density (plotted against returns) and compare it against a physical/risk-adjusted density (plotted against strikes/moneyness), would it make sense to equate zero returns as at-the-money? Are there any other pieces of literature that go specifically into comparing the risk-neutral or physical/risk-adjusted vs realized density in detail? I looked into other queries on this forum, but mostly talk about the measures in theory, but nothing much on the empirics: How to infer real world measure from risk neutral measure ## Answer by Andrea (score 1, accepted) https://quant.stackexchange.com/a/82004 More than an answer, this is what I would do - Select a model (Heston) of which you know the parameters (risk neutral) - Select a drift change (for S, or vol, or both, simple or less simple) - Generate 1 path (of some length) with the drift change - Apply "the" method (the vague one) to get historic distribution - Price options (risk neutral) to get risk neutral distribution - Plot them and see if you can make them overlap with some transformation (4) can only give you time-averaged distribution. (5) will give different distributions for every day (because of the different vol). So I guess you will have to average (5) over time to compare it to (4). Hopefully at the end, you will have learnt something that can be transferred to a real case.
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