How to create a prediction market to calculate the standard deviation of a normal model
Exactly what it sounds like
First, housekeeping: this is my first real post original to Substack. I put it here because it is too technical for the Horn, not developed enough for the Orator, and too niche for everywhere else. Also, this will be a bit to informal for any of those places. But prediction markets are very interesting, and are getting more interesting by the day, so I needed to write this somewhere.
For context, I am participating in the Salem Center’s Prediction Market Tournament, where participants are given a $1,000 in play money to bet on the outcome of various events. For instance, there is a market on whether Republicans will retake the Senate, which you can buy “yes” and “no” securities on, which will pay out, say, $100 if the outcome you bought occurs. The probability of the event happening is the price of the yes security.
For reference, here is the market
One of the organizers noted something strange, the market, at the time, gave Republicans a 45% chance of retaking the Senate in November, but a 29% chance that 538 would believe there was a 50% chance as of Election Day. One way of interpreting this is that the market thinks 538 is going to undershoot the odds, but I’m not sure this is correct. Suppose the market believed that 538 would, on Election Day, give an accurate estimate of the odds of Republicans retaking the Senate; that is, 538 and the market give the same probability for the same question. But suppose a few months out Salem created a prediction market about a prediction market: asking “what is the probability that, on Election Day, the Salem market asking whether Republicans will take the Senate will be trading over 50%? Surely this market wouldn’t be trading at the exact same price (except by coincidence) as the original market. After all, suppose some scandal pushes the chance of Republicans taking the Senate up to 80% tomorrow, in that case we would expect the secondary market to be trading at far higher than 80%, since it seems unlikely there is a 20% chance the market drops to 50%, as would have to be the case if the secondary market was indeed trading at 80%.
But this leads to a really interesting observation, which is what motivated me to write this post. Assuming we are dealing with a normal distribution here (an assumption which is probably not quite true, but I may write another post on why that probably doesn’t matter in this case) a combination of the primary and secondary market can get us the standard deviation of the market. For example, if the primary market says that there is a 45% chance of Republicans taking the Senate, and the secondary market says there is a 29% chance that the market is over 50% on Election Day, then, again assuming normality, we can calculate the standard deviation of the market as about 9 percentage points— that is, the market expects the market to move by an average of 9 points between now and Election Day. We could also use this to make more generalizations, such as saying that the market gives a 95% chance that, on Election Day, the market is between 27% and 63%. This confidence interval feels a little wide to me, which may be an indication that the market overestimates the SD and thus overvalues the chance that the market will be over 50% on Election Day.
[Update 5/26/23: Reviewing this piece now knowing much more about markets, I realize that there are already much simpler ways to do what I am proposing below, and that what I am proposing is somewhat just a more complicated version of the VIX but for oil futures. Nonetheless I leave it in because I think the idea itself is illustrative.]
Here, I must confess my ignorance of the finer points of future markets, because I have no idea whether the security I am about to propose already exists. But, at the risk of reinventing the wheel, I will suggest the obvious application of the above math. Suppose a firm engaged in trading oil futures wanted to know how likely the market was to vary of the next year, or to hedge against wide variation in the market. The firm could then create markets regarding, say, the probability that oil would be over $100 a barrel on X date. (If the possible future prices weren’t normally disrupted, or if they were not sure if this was the case, they would have to create several markets, such as one giving the probability that oil would fall below $40 a barrel, and do more complicated math to them. I can describe this math if another post if anyone is interested.) They could then calculate the standard deviation as we did above, and have a decent idea of how volatile oil markets would be.
A cursory search could not uncover any evidence such securities exist, but maybe they, or something similar, does. If anyone knows of them I would appreciate it if you let me know.
The obvious next step is to create a market to bet on what the standard deviation will be in the future, with the price of the security tied to the prices of the standard deviation formula and the two original markets, but that seems like too many levels of recursion. Unless, I guess, you really want to know whether oil prices will get more or less volatile in the near future, or hedge against this volatility, which actually seems pretty useful