Horse tweeted:
The moment you see a bet on a platform like Kalshi, Polymarket, or the soon-to-be Robinhood+SIG exchange, your mind should jump to the options chain.
The tweet says the Kalshi market is pricing a 9% chance of BTC hitting $250k
The options market can offer a quick sanity check. BTC is about a 55% vol. We are just being very approximate so not worrying about the term structure. I just want to show you my automatic mental response to the tweet.
Without hesitating I pulled up the calculator on my phone and entered:
ln(250k/89k) / (.55 * sqrt(13/12))
Why?
We want to compute how many standard deviations away the $250k strike is to get a z-score which we can then convert to probability. Standard deviation depends on volatility and time. The more time or volatility you have the “closer” some percent return is. A strike that’s 100% away is extremely “far” if the asset needs to get there by tomorrow. If you have 10 years to get there, it’s not super far at all, you only need to go up 7% per year. Likewise, if an asset only varies by 5% a year, 100% is “far”, but if it moves 50% per year, 100% feels much “closer” or possible.
The formula above is simply dividing the percent return to get to the strike by the annualized volatility scaled by root(time) to find the distance.*
*Standard deviation or volatility as a quantity is proportional to the square root of time. Or you can say variance, the square of standard deviation, is proportional to time. The easiest way to remember this is to recall that when you compute the standard deviation of anything, you have an intermediate step of summing the squared deviations to get the variance, then divide by N. But to get back to the standard deviation, you take the square root of the ratio. The ratio in the intermediate step was variance/N. The final answer, the standard deviation, was the ratio of sqrt(variance) / sqrt( N). In our computations, N is replaced by time.
At the time of the tweet, BTC was 89k and there was 13 months until 2027. I assume 55% volatility.
Solving:
ln(250k/89k) / (.55 * sqrt(13/12)) = 1.80 standard deviations
We then use a standard normal table or normdist in Excel to see that 1.80 standard deviations encompasses about 96.4% of the cumulative distribution. Therefore the probability of BTC going HIGHER than 1.8o standard deviations must be 3.6%
🔗This is fully explained in Using Log Returns And Volatility To Normalize Strike Distances
The computation of this distance, besides being dependent on an estimate of volatility which we can borrow from the options market, assumes the asset is lognormally distributed. If you believe, as the options market certainly will if you look not at the at-the-money vol, but the far out-of-the-money call vols, that there is more positive skew than a lognormal distribution then our 3.6% estimate is too low.
But that logic is moving us in the right direction. We want to take the intel embedded in the options market when considering the price in the prediction market. We expect the liquid options market with much more volume and money behind it, to be the best guess as to the “fair price” of a proposition. If there’s an edge, it will be in the mispriced prediction market.
A prediction market bet can take a binary flavor. For example, “Probability that BTC settles above X by some date”
It can take a “one touch” flavor. “BTC to touch but not necessarily settle above X by some date”
Of course, “touch” is more likey than “settle” because “touch” encompasses all the times BTC settles above X, but also includes all the cases where it breaches X and falls back below X by expiry.
We can get information about the price of both binary and one-touch scenarios from the option market.
1. The Binary Bet: Price the Terminal Outcome with Vertical Spreads
Pricing: To find the true market-implied probability of the event, use the price of the spread:
Vertical spread price/Distance between the strikes ~ probability of asset expiring above he midpoint of the spread
Potential arbitrage if…the probability implied by the options chain is lower than the price offered on the prediction platform, you can buy the vertical spread and take the under in the prediction market or vice versa.
Further Reading: A Deeper Understanding of Vertical Spreads
2. The Path Bet: Account for Skew and Volatility with the One-Touch Rule
Pricing: You can estimate the path probability using the trader’s rule of thumb: take the delta of the vanilla option at that strike and multiply it by 2. This naturally takes into the account the option implied skew because the delta is derived from the implied volatility at the strike.
The mechanics of an arbitrage here are complicated as it requires dynamic hedging. If that sounds interesting, perhaps you are born to be an exotic options trader. I have never tried replicating a one-touch option so while I could certainly “financially hack” a model, the main point I want to convey is that the pricing of the one-touch can be inferred from the vanilla options market.
Further Reading: one-touch