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Reading Risk-Neutral Density Alongside Option Skew and Open Interest

Article Quant Q&A · Author: SuperCodeBrah
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Summary

The document compares a Breeden-Litzenberger risk-neutral density estimated from GME options with the option chain’s out-of-the-money prices, implied volatility, and open interest. The author observes that OTM calls appear more expensive than puts at comparable absolute deltas, and that call open interest is substantially higher, mostly at OTM strikes. Yet the estimated density has a mean below the last stock price, prompting a question about whether the distribution should instead suggest a rise.

The material frames a useful interpretive issue rather than resolving it: risk-neutral probabilities inferred from option prices are not straightforward forecasts, and open interest alone does not establish directional demand. The density and skew also depend on option prices across strikes and on the estimation choices. The observations are a snapshot from July 3, with the author noting that conditions had changed by July 5. No full chain, calculation details, or answer is provided, so the apparent contradiction remains open.

Key ideas

  • The author derives a risk-neutral density from option prices using the Breeden-Litzenberger approach.
  • GME OTM calls appear more expensive than puts at comparable absolute deltas in the cited snapshot.
  • Call open interest is higher than put open interest, with much of it in OTM options.
  • The estimated density has a mean below the last stock price, which motivates the central question.
  • The document does not resolve whether these observations imply a directional forecast.

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# Risk-neutral density versus put-call skew and open interest


# Risk-neutral density versus put-call skew and open interest












I've been experimenting with the Breeden-Litzenberger formula in Python based on some code obtained here:

https://github.com/robertmartin8/pValuation/blob/master/ProbabilisticValuation/OptionImpliedPDF.ipynb

I first looked at SPY and got something that looks similar to the usual right-shifted distribution that most BL examples show, indicating that the market expectation is a marginal price increase between the SPY price and the price at expiration:

However, after looking at several stocks, I noticed something interesting for GME in that there was an apparent contradiction between the pricing of calls and puts and the distribution produced by BL (note that this data is as of 7/3 but has changed over the trading day on 7/5).

I'm looking at these two aspects of the GME option chain:

- Pricing of out of the money options: In the first chart, the price of OTM puts drops off much faster as the strikes move away from the last traded price compared to calls. For clarity, the vertical line is the last price with OTM puts to the left and OTM calls to the right. The second chart effectively shows the same information but shown as IV plotted for different absolute deltas. For absolute deltas < 0.5, calls are priced above puts for all strikes. There might be minor differences in the pricing of OTM calls and puts with equal IVs based on the underlying math, but for SPY, AAPL, etc., calls and puts are much more balanced by comparison.

- There is significantly more open interest for calls than for puts and most of the OI is for out of the money options.

Here is the risk-neutral distribution for GME derived from Breeden-Litzenberger plotted against a lognormal distribution based on the last price and ATM volatility. If I understand correctly, it implies that the expected price at expiration (orange vertical line) is below the last price (blue vertical line) based on the mean of the distribution.

This seems to be in contradiction with the fact that people are paying more for OTM calls (usually seen to be a better signal of aggression than ITM calls) and there is significantly more open interest for calls, most of which is OTM.

So is there something about the risk neutral distribution or concepts of skewness, put-call relationships, etc. that I'm missing? Shouldn't we expect to see the distribution shifted to the right of current price based on pricing of calls and puts?

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.