# 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:

1. 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.

1. 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?

• i havent looked at details but GME has ridiculously high OTM Put IVs. I wouldnt be surprised if the implied discributions goes to the left (even if most buying is for calls). Commented Jul 5, 2023 at 17:37
• @AKdemy, from what I can see the OTM puts do catch up with the calls, but it doesn't happen until an absolute delta of less than 0.05, which I would think would have a minimal effect on the overall distribution. Calls are dominant for all other OTM deltas. Would you agree at face that the two phenomena (risk-neutral distribution and the put-call relationships) are in conflict? Commented Jul 5, 2023 at 20:07