Friends,
First of all, here’s the slides I shared in this vid:
I hope that video was helpful. I watched it again because I think the topic is complicated, not technically, but because it’s counterintuitive. Trades like that were bread-and-butter through my pro career (and very much why I “spotted” it despite it not sticking out in the tools other than the Trade Ideas filter which is highly validating of our algorithm! I’ll allow myself to be happy about my own work for 1 second. Ok moment’s passed, let’s continue.)
Before we head on to Money Angle stuff I’ll share 2 things I enjoyed.
Fermat’s Last Theorem Documentary (link)
This 45-minute video follows the story of Andrew Wiles’ quest to prove Fermat’s Last Theorem. I didn’t know the history of the theorem or even what it stated. It claims that for any whole number exponent other than 2 there are no values of A,B,C that would satisfy Aˣ + Bˣ = Cˣ. You can’t prove this computationally because there’s an infinite number of possibilities. The proof requires mathematical logic.
Fermat allegedly had a proof but hadn’t written it down. Turns out Wiles’ proof would have been impossible for Fermat to have known (this will make sense when you watch it). Anyway, I loved it. Yinh fell asleep. YMMV.
Holiday coziness — games!
I taught both my boys backgammon this past week. This is the younger guy playing with mom.
I used to play a lot with my colleagues during the year I was a broker on the NYSE floor. Every time I came back to the “booth” (that’s the area with the phones on the sides of the exchange floor where clerks take orders from customers), I’d make a move on the Jellyfish software. When I was at Parallax, I worked with 2 of the best backgammon players in the world (one of them was the best in the world iirc) and Parallax was actually founded by Roger Low, who was a world-class player himself. Roger had retired by the time I joined so I never got to know him.
Backgammon is especially neat if you like options because the doubling cube begs you to price volatility.
💡Fun fact
When I had my phone interview with SIG (this is the round that happens in between campus interview and the final on-site) I was asked a basic question regarding the cube. Since accepting the cube doubles the stakes of the game and rejecting it forfeits the current value of the game, what is the minimum probability of winning you need to accept it?
When looking at the board you need to evaluate how risky the position is. Like how likely is the tide to turn? When you offer the cube you are selling your opponent an option and its value depends on their chance to come back. But if you offer it to them when they have no chance they will reject it and be bailed out because you have now sacrificed your own possibility of getting a gammon (double points) or backgammon (triple points) if you beat them by a large margin.
It’s a great game for kids because they get to practice dice math. I was playing last night and knew I’d win as long as my next roll wasn’t a [1,1], [1,2], [2,1], [3,3]. So the kids would first need to figure out that’s what they are rooting for and then I ask them the probability. 4 out of 36 or 1/9. Plus the little guy figured out that the 7 is the most common roll and how to count the number of ways to get each number (he was delighted by the pyramid pattern when he realized it).
We always play more games during the cozy holiday season and I’m irrationally pleased that this season it’s backgammon. I recommend it.
[Also, if you are just learning, play a bot on your phone. You learn the game very quickly getting trounced by bots. Ultimately, I’m just a casual player, I never studied it, but would just play for $5 a game with my NYSE squad. Of course, with the doubling cube and possibility of gammons that can multiply pretty quickly. The real hitters play for four and five figures per game.]
Money Angle
On Nov 26th, Imran asked his followers would what more likely to double in the next year — gold or BTC?
I looked at the result of the poll just a few hours after it was posted and it was BTC 52% to gold 48%.
By the time the poll closed with 750 votes, BTC had garnered 2/3 of the votes.
I don’t know if me a jerk had anything to do with this but when I saw that the vote was almost a coin flip I chimed in.
Focus on the last part.
The poll should be nowhere near 50/50 because you would be able to lock in a great trade by selling gold in this proposition, buying gold call spreads financed by even more expensive BTC call spreads.
This is a classic difference between markets and democracy. It’s a perfect example of the Dinosaur Markets post in real life. The markets in the options reflect the volatility and the cost of replicating these bets. Money-weighted votes are interested in the truth where opinions are cheap as sand.
It’s very difficult to have opinions that are above replacement value about liquid assets. If you’re truly good at this, then being Scrooge McDuck rich based on consistently betting on these fantastic opinions is the only proof of such a skill. Few people are rich because of a crystal ball.
A good way to make a living in finance is to find the people that voted for gold in this poll and offer to trade with them. You need to do this in the dark because if you tried to do it on a public exchange, you’d be undercut by traders competing to sell the gold proposition to these opinionated people and it would drive the price down to a non-arbitrageable price.
Public markets protect overconfident people with arbitrageable opinions from their own ignorance and stupidity. Private or non-transparent markets are nice ways to shove a vacuum hose into their bank account.
There are many places where there’s alpha in projecting your opinion. This is the stuff you spend your time on in life. Where you have self-knowledge, private info, competitive advantage, skills, taste and so on.
But when it comes to markets, remember what we learned here just a few weeks ago:
the arbitrage reflex is more profitable than the opinion reflex
Money Angle For Masochists
Let’s continue on the same theme because these threads are going to become far more frequent for anyone who cares about markets. “Thinking in Bets” has a long runway, the way this country is headed, before it jumps the shark, so you might as well get used to it. From “Everything computer?” to “everything casino?”
Prediction Market Arbitrage: Using Option Chains to Find Mispriced Bets
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 likely 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 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
Stay groovy
☮️
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