Prince Charming
dp/dt
Disclaimer: I am not a financial advisor. I’m a blueberry. I possess no formal financial education, no advanced degrees, and possibly no brain cells. I am merely a moron with a Substack. Absolutely nothing in this article is truthful, factual, or even remotely accurate—it is pure, unfiltered parody. While I will reference various complex concepts, I will simplify them to a level that makes sense to me, which means they will become entirely nonsensical and wrong. Please do not take anything in this article seriously or as advice. I am making all of this up. None of it is true. Not a single word.
I think a rather common observation that I’ve had in prediction markets is the slight inconsistent application of ‘charm’ (even when filtering out outlier points).
Greek Refresher
I think that it’s worth noting that the real math behind these greeks generally isn’t needed, but more so just the high level ‘intuition’ behind them. In Black76 world:
To see why we expect charm, it’s probably best to think of delta as P(ITM) for a second. Say we have a 20 delta 0DTE call and nothing happens except for the passing of time (price and vol stay constant) is the probability of expiring in the money the same during the day? No. Clearly it has to decrease as time passes since now a larger move would have to happen in a shorter time for it to expire in the money. Similarly the delta of a 70 delta 0DTE call would increase as time passes (all else held constant). Now obviously delta isn’t actually the probability of expiring in the money (please see this previous post for an explanation1). And clearly no real firm actually uses Black76 greeks without incorporating vol dynamics2 but I’m just trying to use more standard trading greek terms for trying to explain how we would expect the behavior of prediction market contracts to behave.
It’s also worth noting that (theoretically) not even prediction market prices represent the true probability of the event happening even with rationale participants do to standard log utility functions of participants. See Lihong’s3 or Quantian’s4 post on this if you want a more detailed explanation.
P(Jimmy Carter Dies)
Now one could debate the ethics of prediction markets on someone’s death, but that’s not really what this post is about, and frankly is probably more suited for a LessWrong blog post. What I will be talking about is how the change in delta (as stated before I’m knowingly incorrectly using delta interchangeably with the price of the contract/probability of the event happening) is just a bit off here. Now I don’t claim to be a great doomsayer, mortician, or frankly a lot of things, but I AM a trader that (hopefully) knows probability.
Let’s treat each day as an independent event where Jimmy Carter could have or could not have died. Let’s also assume that the rough price where the market opened was the true yearly probability (80%). Well some rough mental math should tell you that the approximate probability of death on any given day should be about 0.4% but let’s actually calculate this out.
Ok so or original approximation of 0.4% was actually somewhat decent, but what we really want to look at now is how this should look over time (conditioned on the event not actually happening).
How To ‘Trader Math’ 0.4%
The probability of an event occurring when N trials are performed of something with 1/N independent odds is approximately 63% (detailed post on this5). So if there was a 1/365 chance on a given day the probability would be about 63% for the year. The fair value was 80% so we need to increase the daily odds. Something something rough mental approximation/intuition and 1/250 or 0.4% seems ~reasonable~.
Why The Graph Looks (Somewhat) Off
We should see a concave graph over time that looks something like this:
And in the graph for that prediction market, after the peak around May (maybe there was some additional news or something), even the outlier removed decay seemed to be somewhat linear and didn’t have the ‘curve’ you would expect from exponential decay; obviously the massive spikes from the trend were also…weird (with the exception of the final one of course).
Let’s assume that the probability of him dying is not perfectly constant over time and instead was uniformly increasing from 0.4% per day to 0.5% per day by the end of the year. We would still get something very similar:
Then again, I’m probably just being overly harsh with this contract, but I think that maybe some more exponential smoothness could be expected in the actual contract. Also, I think it’s worth noting that I can’t really point out the best examples of where the decay is incorrectly applied (think difference between weekday and weekend stuff, etc.) because that would be giving up still usable alpha.
Fun enough post. I was reading more on sports betting and data vendors... inspired by your previous post and thinking about event driven bets for NFL, MLB, ... but was not sure what latency I might find for free on the internet. Those high volume Kalshi markets seem interesting and I was surprised to learn that some Redditors allege television is sometimes 30-120s off live. I like the links out to other web logs, and will include even more textbooks via Giuseppe Paleologo in my Library.txt file.