Friends,

In this issue:

• streetlight effects
• Investment Beginnings Class 5 was a delight
• thoughts on the current market cycle and options

I’ll be short up here with one rec since the investing sections are a little longer. Just have extra exhaust since last week’s graduation issue skipped finance stuff.

everything is a nail, or at least it ought to be | 5 min read

This is a short book review by Dan Davies of The Irrational Decision by statistician Ben Recht.

Recht argues that if a hammer is genuinely your best tool, it’s actually smart to look for nail-shaped problems or reshape problems them so they’re more hammer-friendly.

In Recht’s case, the hammer is mathematical optimization (especially linear optimization). The book traces how, over the past century, people haven’t just used optimization to solve problems, they’ve restructured problems to fit optimization’s strengths. He gives the example of the feedback loop between chip design and optimization algorithms where better chips enable better optimization, which enables better chip design, repeat.

If you recall, in my post slashing away parts of their humanity, Davies plugs C.Thi Nguyen’s book The Score (which I bought as I enjoyed Nguyen’s first book about games). Davies’ latches on to the manner in which metrics ultimately create “streetlight” effects by optimizing not necessarily for what really matters but for what’s easy to measure.

Well, as you might expect from any self-respecting James C. Scott fan, Davies praises Recht for resisting the profitable route of hyping this trajectory into AI singularity cheerleading. Instead, Recht asks: what do we lose when we reshape problems to make them algorithmically manageable? Once you’ve defined what to measure and set a success metric, an optimizer will always confess an answer. But all the real judgment happens upstream, in choosing what counts as success in the first place. That same process that advances chips is a convenient way to persuade overly left-brain, ahem, “data-driven” decision-makers.

Recht’s book ends with a thoughtful chapter on how we should make decisions, but Davies also flags an argument that knocks you a bit off balance. He makes the case that randomized controlled trials are better understood as a regulatory tool than a scientific one. I found that a bit haunting. It’s a dark thought that also casts a shadow on our accepted standards of rigorous epistemology.

Recht, Nguyen, and Davies are all quantitatively minded scholars converging on nuanced skepticism of data in what I’d describe as a next-gen version of How To Lie With Statistics. In the first gen, inferences were distortions of what the data told us (i.e. sleights of hand like substituting causation for correlation), but is the new deception to do inference honestly but use proxy measures that, to use a trading term, have a lot of basis risk to the thing we actually care about. And then the bigger question is how much of that is laziness vs trying to manipulate the terms of discourse.

Money Angle

This week I taught Class 5 of the Investment Beginnings series I’ve been doing with the 12+ year olds.

It’s the last class in the series before we do “labs” in July. During lab, we’ll convene when the market’s open and I will give each kid individual attention as they execute an investment. I want to make sure they know how to read the screen, navigate their broker site, see the confirmation of the execution etc.

This last class was special. I’ve been posting all the materials online and there are families following along remotely. One dad sent me an app that consolidated and vibecoded the slides and games. He and his son worked on the project together:

https://investment-class.vercel.app/

And this next part blew people’s minds in the class, not to mention my own. A mom brought her son from Miami because he’s been obsessed with the class and wanted to be here in person with the other kids. I’m speechless. Supermom and superkid.

We took them to dinner with my family and brought along my son’s good buddy so our visiting friend would know a few people before stepping into the class. I can’t gush enough about how nice this all was.

When Class 5 ended, a lot of parents came to talk to me and said all this kind stuff and gave me totally unnecessary, generous gifts (I would have done the same so I get it but also just feels like too much). The most important thing is how all these kids’ gears are turning. It feels like a no-brainer to really clean this up (I learned a lot from doing it and know how I’d mod it in the future) and turn it into something. Maybe a well-produced YT thing, but I’m stretched pretty thin. We’ll see, I guess. Famous last words.

Anyway, here’s the outline of class 5 and link to all the materials from the classes.

Class 5 — Making Trades & Reading Markets

• Different kinds of auctions and how markets are continuous auctions
• The order book: bids, asks, spread, and what “depth” actually looks like
• Price discovery as consensus — the price aggregates what everyone knows
• Market hours, plus what pre-market and after-hours really are (and why beginners should avoid them)
• Public vs. private markets, with real estate as the bridge example
• How an IPO turns a private company into a publicly traded one
• Why baskets exist: the easy button for diversification (callback to Class 4)
• Three kinds of baskets — index, themed/sector, manager-picked
• ETF vs. mutual fund: same idea, different checkout (auction all day vs. one daily NAV)
• Index construction math: cap-weighted vs. equal-weighted, with four real stocks
• Why SPY and RSP — the same 500 names — can produce materially different returns
• 🔨 Homework: talk to parents about a brokerage account, ahead of the July lab where students place their first real trades

We spent much of the class doing a mock trading game:

How the game worked

There are 16 kids.

• Each gets 2 cards — that’s private info
• There are 3 “stocks”: Hearts, Spades, and Red
• At the end of the game, each stock settles to the sum of the cards held collectively in that category across all 32 dealt cards
• Card values run 1 to 13 (ace to king)

Kids bid, offer, and trade with each other based on what they think final settlement will be. They log transactions on index cards they carry.

Every few minutes, news hits — I reveal some of the remaining 20 cards. These are cards that will NOT contribute to the value of the 3 stocks.

The Teaching Moments

Basic valuation

1. What’s the maximum value of each of the 3 stocks? (Also a fun way to teach someone to quickly compute the sum 1 to N.)
2. What is the fair value of the stocks at the start of the game, when no common information has been revealed?

Information and private signals

1. What is the fair value of Hearts if you’re holding the 9 of Hearts?
2. Ask the kids: what’s a good hand to be dealt, and why? (A very simple exploration of what “information” actually is.)
3. After news is revealed, how do you update fair value? Walk through the exact math.

Reading flow

1. Your fair value is always subject to adjustment based on flow. What is Alice’s bid generally saying about Spades? What is Mike’s offer suggest about Red?
2. At the end, computing P/L is a big exercise — marking trades to settlement.

We didn’t go into crazy depth on any of these. Just getting a basic understanding easily takes a group of 16 kids an hour, and even then some are lost. Totally expected. Honestly, many adults are too.

It’s super interesting to see who gets it very quickly though.

The origin of the game

This was the first trading game I remember doing as a trainee at SIG. All the new hires in NYC played while the trainees who had been around for 3-9 months traded options on these “stocks.” Their hedge orders would get sent into our trainee market!

Money Angle for Masochists

Not to deter any stubborn bears, but just understand your history. In 1999, the Nasdaq returned 86%.

If we ignore the small 3.2% down year in 1994, that run looks even crazier and capped with an insane blow-off top.

The 1999 blow-off top is pretty interesting from a how-do-I-reconcile-option-pricing-with-real-world lens.

I’ve written a bunch on how volatility measures are sensitive to sampling periods.

See:

• Volatility Depends On The Resolution
• The Coastline Paradox in Financial Markets

Volatility scaling can feel unintuitive.

If an asset’s annual standard deviation (ie volatility) is 16%, then its daily standard dev is ~1%

Well, from that, it’s clear that you can have say a 10 sigma move in a day, but not in a year. That alone seems to point to a weakness in how we scale volatility through time.

But this is mostly resolved by measuring distances in lognormal space correctly. When you do that, you find that +86% is MUCH closer than -86%.

Just screenshot the formula in the post and treat Gemini like a calculator:

This explains why calls that are 50% OTM calls are worth more than puts that are 50% OTM (even if the spot and forward were the same).

To consolidate knowledge, it’s why collars can look so attractive in high vol stocks:

“Stock pickers market”

I’m old enough to remember when investment managers complained that the Fed drove the market, everything was correlated, and there was little reward for discerning between companies.

Well, we’re in the opposite world.

This is showing put skew falling and call skew rising in QQQ:

This is front-and-center to the options market:

There’s a record disconnect unfolding in the trading pits right now

A chart from the article shows the spread between the weighted avg stock vol in the SPX vs the index vol. It’s another proxy for cross-correlation as the index vol is dampened relative to the stock vols because the stocks are doing a great job diversifying each other.

A stretched relationship can get more stretched. But as it stretches, there is a mathematical reason why the spread would revert. Think of the limit. If 1 company achieved singularity and ate all the other companies, its weight would increase relatively as each dollar it made was a dollar less for the others until it was the index. This is NOT a tradable idea. That is a make-believe world. I’m only being pedantic to help you move the pieces around in your brain to help you see how they fit together.

But the price action of anything related to AI ripping vs everything else is a giant singularity trade, and from that context, the low correlation makes superficial narrative sense for now…a handful of companies are expected to eat the rest.

But I’ll pose this one…if instead this handful of companies are becoming the COGS of all other companies, wouldn’t that look more like the stories we’ve heard that I had Claude reconstruct by asking it to describe the circular revenue phenomena in bubbles:

Company A buys ads on Company B’s site. Company B uses that revenue to buy servers/software from Company C. Company C buys ads on Company A’s site. Everyone books revenue, everyone’s growth numbers look great, valuations rip higher — but no net new money is entering the system from outside customers. It’s just the same dollars chasing themselves around a closed loop, with each pass inflating reported revenue.

The poster children were the late-90s telecom and dot-com names. Global Crossing and Qwest got nailed for swapping fiber capacity with each other and booking both sides as revenue (”capacity swaps”). AOL was accused of round-tripping ad deals. A lot of dot-coms were essentially selling ads to each other, with VC money funding the ad budgets — so the “revenue” was really just recycled venture capital.

The concern isn’t fake deals today, but the circularity possibility means index vol will have its revenge. Good luck with timing though.

SpaceX

It’s interesting to hear Mitchell step through the numbers of how much day 1 shares need to be absorbed and how unprecedented this is. Recall in PTJ’s interview on Invest Like The Best:

2000 was the easiest bear market I’ve ever seen in my whole life. It’s got so many similarities to right now, in the sense that the bear market of 2001 and 2002 were a consequence of all the IPOs in ’99 and 2000. And then as they unlocked, you just had this never-ending cascade of selling, that’s a great way of putting it. And we’re getting ready – I want to say that the contemplated IPOs for next year are going to be five or six percent of market cap. So why are we where we are right now? Because we’ve been retiring two or three percent of market cap, probably a little less than 2% of market cap, every year without fail for the past 10 years [through buybacks.]

And so now all of a sudden, you’re gonna completely reverse that math. And so, I don’t think it necessarily happens instantaneously with the IPOs, but then there’ll be the unlocks. So you can see a situation where, okay, maybe we go through some kind of rolling top. And then 18 months from now – six months, we’ll have to look at the unlock schedule. But you’re going to want to watch those because that’ll just be adding equity supply. And you’ve already gonna be diminishing the buybacks because of all the commitment to capex from the hyper-scalers. They’re already gonna be eating into their cash flow.

The current set-up feels very strange. It seems too easy to think it’s going to be a local top right? But it’s a hard idea to resist. It feels like it’s a negative for gross returns and then under the hood, possibly chaotic for sector flows (like to absorb the IPO do people rebalance out of what went up the most recently? That feels like a pretty natural idea).

You have pockets of extreme bullishness manifesting in options with cheap put skew and risk reversals and left-for-dead implied correlations but the bearish factors are also common knowledge:

• IPO issuance
• Midterms (assuming Dems are the bear choice)

  

• elevated bond yields

   

In other words, current prices are NET of everyone knowing about these factors. By the way, this is always the problem with markets. If you see something on the horizon and think it’s bearish, whose to say the current market wouldn’t just be higher if that thing you’re latching on to is observable by others.

What’s the more contrarian position right now, to be bullish or bearish?

The option point spreads have shifted in a way that suggest bullishness is consensus.

Moontower agent

The moontower agent has been in the wild for a month now, and it’s been an awesome companion to help you reason through trading questions. It’s connected to the data we buy and process, and it’s tuned and under ongoing reinforcement learning for trading contexts.

=> 🤖Ask the agent a question

It’s powered by AI harnesses, of course, so the output is non-deterministic, so it’s helpful to report back on what it does well and poorly so we can keep course-correcting.

Thursday webinar

Been a lot of recent chatter about volatility funds as a strategy and as a business. There’s a lot of misconceptions about them. Which makes sense, it’s an opaque world.

I’ll host a live webinar on Thursday for paid subs to share some thoughts on the topic.

Details to follow this week.

This Week In The Options Trench

Stay groovy

☮️

Moontower Weekly Recap

• nunchuck
• what new grads should focus on

In this issue:

• smart vs wise
• adverse selection when landing a senior role
• cost of carry as a conviction hurdle
• KenKen

Friends,

Happy Mother’s Day!

It’s a nice day for an enjoyable read. I loved this post from Adam Mastroianni:

Infinite Midwit | 12 min read

The essay is insightful and typical of Adam. It’s just so fun to read. Which is itself an exhibit of his thesis.

I stitched together these excerpts, but I encourage a full reading.

The opening

The better AI has gotten, the less anxious I’ve become.

A few years ago, when the computers first started talking, it was reasonable to believe that we would soon be in the presence of omnipotent machines. For someone like me, whose job is to produce words on the internet, it seemed like only a matter of time before I would have to fill my pockets with stones and wade into the sea.

Two intelligences

Some problems have clear boundaries and verifiable solutions, like “What’s the cube root of 38,126?”. These problems require objective intelligence. Other problems are vague and squishy and it’s not clear whether you’ve solved them, or whether they exist at all, like “How do I live a good life?”. These problems require subjective intelligence. Objective intelligence can be trained, reinforced, and validated. Subjective intelligence cannot.

[Kris: smart vs wise]

It’s unfortunate that people use one word to refer to both of these capabilities, when in fact they have nothing to do with each other. It is also, ironically, a case of objective intelligence overshadowing subjective intelligence: these skills are obviously and intuitively different, but a century of psychological research has “proven” that only one of them exists. Over and over again, psychologists have found that all intelligence tests correlate with one another, even when you ostensibly try to test for “multiple intelligences”. Numbers don’t lie, and they all say that there’s only one intelligence, the so-called g-factor.

The problem is that any test of intelligence is only ever a test of objective intelligence. “How do I live a good life?” is not a multiple-choice question. “Discovering” the g-factor again and again is like being surprised that you find the same patch of sidewalk every time you look under the same streetlight.

The Catan fantasy

The promise of artificial superintelligence is based on the idea that objective intelligence is the only intelligence. Or, even if there are multiple forms of intelligence out there, that they are fungible. To be an AI maximalist is to believe we are playing under Settlers of Catan rules, where if you have enough of any one resource, you can trade it for any other resource. If you have infinite objective intelligence, then you have infinite everything.

It’s not just that objective intelligence can’t be transmuted into “emotional” intelligence or social savvy or whatever we want to call it. It appears to be very difficult, if not impossible, to transmute objective intelligence into any other cognitive ability.

When you meet a human who can do quadratic equations in their head but can’t hold onto a job or a relationship, you know they’re missing something upstairs.

Nerds

When I was growing up, this paradox was an endless source of sitcom plot lines—if you’re so smart, nerds, why don’t you figure out how to make yourselves popular? The entrepreneur/essayist Paul Graham took up this question 20 years ago and came to the conclusion that the nerds must not want to be popular. They’re too busy with their Neal Stephenson novels and their D&D campaigns to spend a single brain cycle figuring out how to keep their heads out of the toilet.

I disagree. The nerds I knew in high school—myself included—were always hatching harebrained schemes to increase our social status. They just didn’t work. (”All the girls will want to go to the Homecoming dance with me once they see how many state capitals I’ve memorized!”) We couldn’t use our smarts to make ourselves popular because we had the wrong kind of smarts.

For example, I went to college with a guy who was super smart, but he also couldn’t do anything on time. He would be late to exams. His grades would tank because he would finish his essays but forget to turn them in. He would set meetings with his professors to sort everything out, and then never show up.

I always used to wonder: why doesn’t this guy just use his big brain to make himself more conscientious? Isn’t life one big role-playing game, and isn’t intelligence just experience points that you can assign to any of your Big 5 skills?

Why AI writing vaguely sucks

It’s cool that AI can fold proteins, create websites, fact-check journal articles, etc. but it can’t write anything that I am interested in reading. The problem isn’t that it hallucinates or makes mistakes. It’s that everything it writes vaguely sucks. I drag my eyes across the words and I feel nothing. That’s not quite right, actually—I feel like, “I would like this to be over as soon as possible.” …

Words themselves don’t contain feelings—they are a recipe for creating that feeling inside your own head, to assemble the right set of emotions out of the experiences you have at hand. If I do a good job, the subjective experience that results inside you might resemble the one that originated inside me, but it will never be identical, because we’re working with different ingredients.

The computer doesn’t know any of this. It can’t know any of this. It can only read the cookbook; it can’t taste the meal. Objective knowledge can make your sentences true, but it can’t make them alive. Without access to subjective knowledge, you quickly hit a wall. And unlike all previous walls that AI has surmounted, you can’t overcome this one by scaling—either in the literal or metaphorical sense—because it’s a wall with a width you cannot describe and a height you cannot see.

Money Angle

I unlocked this older post, which found a lot of agreement from senior professionals who have transitioned from a pure trading job to building a business or strategy for asset managers. The target audience is seasoned practitioners. It’s written specifically for option traders but the feedback is that its warnings extend well beyond that market.

We know that there is adverse selection when hiring but this is about avoiding a lemon when getting hired.

Enjoy: adverse selection in the option job market

Money Angle for Masochists

Brent and Alf have a regular macro podcast. This episode from late 2025 has a number of evergreen trading concepts.

On carry costs

Alf: In our fund, we’ve recently added an alert before you put a trade on. It looks at how much it’s going to cost you to own this position in a relatively vol-adjusted way if nothing happens for the next 30 days. It’s essentially like some sort of carry-to-vol screener.

[Kris: notice how naturally this forces a “thinking in bets” discipline on your opinions]

“Am I really sure this is going to happen or not?”

A Hungarian Forint Example

We had this example where we had all this theory on Hungary. We looked at Hungarian forint and said, “Okay, Orbán is going to go nuts, win the elections, spend money, and the market will punish the Hungarian forint.” Sure, you can be right. You can be wrong.

So how do you trade this? Well, you buy the euro and sell the Hungarian forint. You do the trade, then you look at the carry differential between the two currencies. It’s about 4% yearly carry differential in the forwards. That comes down linearly to about 30 basis points a month.

You think: “30 bips? That’s nothing. I can pay 30 bips in a month. No problem.”

But then you look at the monthly volatility of this thing. It basically doesn’t move. So at some point you realize: the static ex-ante Sharpe ratio of this thing for the next month is like -0.6 Sharpe against me. Holy crap. I need to be not only right—I need to be right with a high Sharpe ratio just to start offsetting the negative carry adjusted for volatility in this trade.

Analogizing to options where the costs are framed more explicitly

A lot of trades are very expensive to own. With options, I think sometimes you fool yourself because you pay for the vol once, and then you say, “Okay, I paid for the vol. That’s what it is. Now I can go long, and this thing can drop to zero, but I did pay for the vol.”

But if you buy 50 options in a year, you pay for the vol 50 times. In options, people do it very lightly but in linear trades, people don’t sometimes think about this, but I think it’s important.

[Kris: I’d broaden this discussion to be a reminder that the cost of carry in addition to opportunity cost is hurdle that your conviction must clear. If it’s not explicit in your in your process consider making it so]

Extending to Hard-to-Borrow Stocks

Brent: It’s a different but similar situation where normally you’re dealing with a very high-vol instrument, but you’re paying a crazy amount of money to borrow the stock.

There’s quite a bit of research showing that hard-to-borrow stocks do tend to go down—they’re hard to borrow for a reason. Someone knows a secondary is coming, or it’s a really bad company. There’s usually a reason why it’s hard to borrow and why everyone’s short.

But functionally, the cost to borrow negates the negative return of the stock almost perfectly in aggregate. If you short all the hard-to-borrow stocks, you’ll make money on the shorts and lose about exactly the same amount on the borrow.

A CoreWeave Example

Take CoreWeave—there’s no borrow right now, but there has been off and on, and the borrow was like 150%. Even if you catch a pretty decent move, if you’re not catching it instantaneously, you’re probably not making money.

One simple thing I’ve been doing: when there is a borrow, just selling it at 9:30 AM and buying it back at 4:00 PM. You actually made more money being short that way than holding it—not even counting the borrow—because CoreWeave has a little bit of positive drift in the overnight session.

Extending to currency roll costs

Even buying EUR/GBP today, I bought some at 0.8573 and rolled it to September 17th. It was 17 pips to roll it. For six weeks on a currency that’s moving 20-30 pips a day, you’re paying 17 pips of roll. That’s not very fun.

Spurious Correlations: When Copper Takes Down Silver

Here’s something huge: spurious correlations. Copper tariffs were actually removed. All of a sudden there are no tariffs on copper, and copper goes down like 30% in two days.

Then, do you know what happens to any other high-beta metal? Doesn’t matter what it is—palladium, silver, you name it—goes down 3-4-5 percent. It’s just cascading risk and basically some exogenous event hitting you, not even directly, but laterally, and you lose money.

The Sympathy Trade Strategy

Brent: Generally, I feel like there’s a lot of good trades like that. Being short silver when copper collapses—but these sympathy trades tend to be very short-lived.

For me, it’s always like: if you’re looking at AppLovin and The Trade Desk, they’re similar companies. When AppLovin craters 50 bucks on earnings, you can sometimes sell The Trade Desk and make four bucks in 20 hours.

But like you said, there’s really no economic reason for the correlations a lot of the time. Sometimes with earnings there is, but with copper and silver, it’s just that a lot of the same people who have that sort of inflation and debasement trade are long copper, long silver, long gold, long Bitcoin. When one part of the portfolio gets a nuclear bomb, they’re forced to liquidate the other stuff.

I do those trades as sympathy trades, but I go in knowing: most of the time, there’s no economic sense to this. It’s more of an endogenous unwind story. So you got to be quick.

This Week In The Options Trench

Erik has a soft spot for GME so in light of Ryan Cohen’s bid for eBay, we talk about GME options. A fun thing about this casual weekly podcast format is, while we have a loose calendar of discussion topics, we might just talk about whatever grabs our attention. This week was one of those.

📺GameStop and eBay Acquisition What the Options Markets Say

From My Actual Life

My 4th grader loves the KenKen puzzles his teacher gives. KenKen is a Sudoku-style puzzle invented in the early 2000s.

In this 4×4, grid each row and column must contain each of the digits 1,2,3,4.

The digits in the bold cages must satisfy the arithmetic operation in the cage’s upper left. For example, in the cage with “5+”, the only valid combinations are (4,1) and (3,2).

In this example, I know the bottom right corner must be a 2 because the 7+ cage must include (3,4). 2 is the missing number in the column.

We got the kid a KenKen book for his birthday and my mother saw it while visiting and got addicted herself so we sent her a copy as well.

The puzzles can range from easy to hard and they can have larger grids (ie 6×6 or 7×7). My son likes to make them as well. At first, that sounded difficult but brain fart. I asked him how he did it and he said he just fills out the grid so the row/column rules are satisfied and then you simply cherry-pick your cages and display the operation and target that makes the cage work.

Again Happy Mother’s Day, may the simple pleasures pamper.

Stay groovy

☮️

Moontower Weekly Recap

• there was a black bum named “Bad News Brown”
• face ripper

In this issue:

• AI leaves the hardest problems to humans
• The triggering implication of arbitrage theory
• options rapid fire
• Nine Inch Noize

Friends,

A thought sparked by the digital id known as X…

AI anxiety takes many forms. Doomerism, FOMO, a sense that the value you may have associated with your efforts is now solved by power cycles, or simply being overwhelmed by its usefulness, making it difficult to triage.

AI forces us to price even more tradeoffs, mostly in the form of explore/exploit problems. In other words, lots of tasks became easier, leaving diabolically hard ones to focus on, namely, “what should I be doing?”

My answer is mostly to be doing more of what I’m doing faster. I admit this is incremental and I’m a little ashamed of that, but I’ve already told you I feel like I’m the American in 1920 who saw a car and thought, “I could use that to go buy food for my horse”.

If you do your work faster, you can do more work. I heard this is called Jevon’s Paradox, which is some economic idea that caught fire like “Baumol’s disease” or “network effects” or “EBITDA”.

Gemini defines Jevon’s paradox:

The Jevons paradox is an economic observation that increased efficiency in using a resource tends to increase, rather than decrease, its total consumption. Proposed by William Stanley Jevons in 1865 regarding coal use, it occurs because higher efficiency lowers the cost of using the resource, driving higher demand that outweighs the initial savings.

Well, ain’t that the truth. Don’t mind me, I’m just over here paying off “intention debts” when I should be…[trails off]. If you feel the same, I see you.

Money Angle

As I expected, the post how to get arbed with perfect info would trip people up. I didn’t expect confusion because I thought Professor Doug Costa, whose explanation is featured in that post, was itself confusing. But because the concept of replication is hard and feels like it violates the good. It’s triggering. It means you can know the truth and still get arbed. Again, this is why I call it the pons asinorum of finance.

A reader brought it up in our Discord so I’m going to share the discussion here as he felt like our back and forth helped.

Before getting to the conversation, let’s refresh the problem Doug set up:

A company issues contracts based on a provably fair coin. The contract pays $150 on heads, $75 on tails. It trades at $100. Interest rate is 0%.

You calculate the true expected value of the 110 call using the true probability of 50%.

It’s worth $20 because it has a 50% chance of being $40 in-the-money.

You pay $20 for it (but even if you paid a bit less for an unambiguously positive EV trade, this analysis will hold. I just want to stay with Professor’s example)

The dealer sells it to you, hedges with 8/15 of the underlying contract, and locks in $6.67 profit in both states. Pure arb.

You had perfect information about the true probability and you still got arbed. The dealer made money in all scenarios, trading the call at fair value with you.

Doug is showing how the real-world probability doesn’t matter to the derivatives trader if they can also trade the underlying. In this case, the underlying is mispriced, but the dealer doesn’t know that. All the dealer cares about is whether the relationship between the derivative price and the underlying price is mispriced. In this contrived example, the mispricing was more profitable than knowing the true probabilities.

And to add something Doug doesn’t mention…if the investor knew the stock was underpriced and bought that instead, they’d have a positive EV trade (the fair price of the stock is $112.50) but they are still worse off than the dealer who knows the relative value of the 2 securities is wrong and gets to make a profit in all scenarios.

This is a good place to insert the chat.

Reader: I see, so the main point is we can converge a spread by trading two things instead of betting on one.

Kris: In a world with no derivatives, you’re left with having to be good at guessing real-world probabilities, but derivatives are their own source of possible edge that doesn’t inherit from knowledge of the future but from relative mispricings between the derivative and the underlying.

It’s obvious that being able to handicap probabilities would be a source of edge, but it’s quite subtle that once you introduce derivatives and the idea of replication, there becomes a source of profit that doesn’t rely on such an ability.

Reader: Right, so it’s instructive in giving one more spread to look at. If you used a put in your example, then the dealer would lose because they’d be too short. Then a dealer that actually has no information and sells both sides ends up $0. This example is picking the (long) side where it wins.

Kris: Yeah, the underlying in this example is too cheap RELATIVE to the call option.

If the call option was $13.33, then from the vantage point of real-world probability, both the underlying and call are too cheap, but they are priced correctly with respect to each other.

Which makes the point — if a derivative and underlying are correctly priced to each other, then the real-world probability is not important to the dealer. The dealer only cares about the relative values.

You can just compute the Sharpe of buying one vs the other I suppose to see which is better (that’s one lens). The call is more underpriced in % terms, 3.33 when it’s worth 20. But it’s more volatile as it will lose all of its value when it loses.

I’d just stick them both in a Kelly calculator in Claude or something and whichever one it says bet more on is the better one lol.

There are some important implications here. And brain damage — investor brain and derivatives brain collision.

The goal is that derivativesbrainskill.md becomes something one calls as needed, like Neo downloading kung fu. But you don’t wanna get carried away with it and shoot it at everything in life. It’s this weird artificial thing that works in a replication context, but it’s also not artificial in that its violation presents hard cash arbitrage!

That’s the end of the chat, but let me add one more thing to make you feel better if it’s still foggy.

I’ve seen this subtlety trip up seasoned options traders where they take B-S pricing to mean that the forward for a stock is stock grown at the risk-free rate (RFR), but this is ONLY true in a world where you can trade the underlying AND the options. Outside the context of replication, you cannot make that assumption.

Struggling with this idea is entirely forgivable. I mean, the realization that you could use RFR as the discount rate was a revolutionary breakthrough. Bachelier figured out option pricing in the early 1900s, but he and his contemporaries were stumped by what rate to discount the payoffs.

Later academics wondered if you should use something like the required return from CAPM or something, but it was the whole idea that if you trade a derivative vs the underlying against one another, then you can have equivalent payoffs and therefore it’s riskless to go long one and short the other. If it’s riskless, then RFR is the appropriate discount rate.

Warren Buffett sees the necessity of agnostic dealers using the RFR to price options in arbitrage-free ways as an opportunity. He asserts that put options are overpriced because they use too low of a discount rate, but the dealers don’t care so long as they can trade the underlying, they can arb any other rate assumption. Again, so long as “they can trade the underlying.”

This single idea allows derivatives traders who know nothing about the fundamentals of securities to make money in a sea of people who do. It’s quite profound and not a small part behind why I think vol trading is easier than directional trading.

Money Angle for Masochists

Rapid-fire format today:

1) A reminder that trading vol is not trading a line on a chart

Selling vol when it’s high because it’s mean-reverting is not like selling a stock that is going to fall. Option performance depends not just on what happens to implied vol but on how much the stock moves. If it were as simple as selling an IV number before it went down, just sell equity options just before earnings come out. I’m 99.9% sure the implied vol is going to fall after the announcement.

Just remember:

If you are trading near-dated option just think “I’m not trading implied vol I’m trading straddles”. The further you go out in time, the more it’s like trading implied vol since those options are dominated by vega, not gamma and theta.

2) Avis stock

Ticker CAR has been squeezing higher as activist owners with locked up float collide with heavy short interest.

As you might guess, I’m getting messages about the positive delta puts post.

Refreshing a perma-disclaimer: I’m not an advisor. You are responsible for your own actions.

3) ICYMI

Thursday’s post was especially well-received perhaps because it’s highly relevant for option sellers. Just to bring it to your attention again:

A devilish question for option sellers: Which VRP is higher?

It is a fuller response to the quote-tweet from a month ago:

To be blunt, if it were as easy as “sell vol when IV is high” you’d need to believe that option traders, a cohort known for being stupid, never noticed the obvious.

And finally, a reminder of the Moontower Review of option-based ETFs ISSB and ISBG:

• IncomeSTKd: Anatomy of a Next-Generation Options-Income ETF

This Week In The Options Trench

From My Actual Life

My wife tells me the internet is calling the Nine Inch Noize set last weekend the best in Coachella history. Nine Inch Noize is a mash-up of Nine Inch Nails and German DJ Boyz Noize

These are subjective things, of course, but I’m not surprised by the sentiment. I’ve seen NIN 3x in the past 3 years and the last 2 were with Boyz Noize on the Peel It Back Tour. I’m tired of raving about them. I don’t listen NIN regularly (well, more so now), but the recordings don’t do the in-person experience justice. You can watch YT clips to get a sense of the lasers and effects which are impossibly creative. The Coachella show was different than the tour shows so that’s something to appreciate of its own.

The tour shows have sections where it’s just Trent, sections with Trent and the band without the drummer (who by the way the alien Josh Freese), the full band, and then the tracks with Boyz Noize which thump real hard.

On Friday they dropped the first Nine Inch Noize album. The Pitchfork review is worth a read if any of this is interesting to you. It gives a lot of context not just to the album but the Peel It Back Tour as well.

🔗Pitchfork Album Review of Nine Inch Noize

The song I’m currently obsessed is an old one from the With Teeth LP. He opens his shows with it solo on the piano, which has a different feel than the recorded version. The recorded version has that velcro static fuzz sound that I f’n love, but it’s not for everyone.

Recorded version:

Live:

And my favorite cover of it has a beautiful orchestration and recording:

Stay groovy

☮️

Moontower Weekly Recap

Posts:

• IncomeSTKd: Anatomy of a Next-Generation Options-Income ETF
• A devilish question for option sellers: Which VRP is higher?

In this issue:

• Investment Beginnings Class #3 and the game we played
• What if gasoline futures roll up to the current spot price all year?
• Checking in on the TIPs replication trade
• Music for the journey

Friends,

I taught lesson 3 of the Investment Beginnings course this week for 14 middle and HSers. That link includes the materials and video.

I opened the class with a timely story. The day before the class, I found a statement for an account I forgot about. 25 years ago, I spent $800 buying FedEx and Motorola shares. The account value according to the current statement…$14k!

I told them about studies claiming dead investors outperform the living, but I actually think it’s apocryphal. Sometimes it’s spun as “investors who lost their password” made the most money. In any case, it basically happened to me. It gave me an excuse to foist math on them by making them calculate my CAGR. You want those calcs to be second nature. I saw an interview where an investment manager said their target is to return 2 to 4x the fund in 7 years. Immediately, in your head, that’s 10-20% CAGR.

We sprinkle little bits of investing math as we traipse around the day’s lesson, but the heart of this particular class was a game. We broke into teams to construct investment portfolios. I used historical data but changed the names of 15 real companies to local towns and showed various metrics like margins, FCF%, earnings, revenue growth, etc.

We played 4 rounds starting from 2014. The teams lock in their portfolio weights to the different companies. There’s also the choice to invest in T-bills. There’s also a benchmark portfolio that is 20% T-bills and the remaining 80% is equally allocated across the 15 stocks.

Every 3 years, we reveal returns and allow the teams to re-jigger their portfolios. The 4 rounds take us through the start of 2026. Then I revealed the actual companies and we discussed what drove the returns. The most interesting aspect is how the metrics are loaded with pitfalls, so we talk a bit about the nature of the information that is available. The most overpriced stock with terrible margins was the best performer. Wear a helmet kiddos.

Introducing Erdo

Emi, my moontower.ai cofounder and a small team who I have the privilege to chat and work with as well, launched Erdo. I’ve been using it for months connected to the moontower.ai infrastructure. By now, I know enough to just go along with anything Emi’s up to.

Check out his brief post if interested:

Introducing Erdo: The AI Workforce for Business

Money Angle

Trailing 1-year inflation per the CPI index has been ~2.5%

Prompt CME gasoline futures (RBOB) are up 80% this year but the curve is strongle backwardated (deferred futures are trading much lower).

Gasoline is about 3% of CPI. If the futures roll up all year to prompt levels, this alone will add about 2.5% inflation for the next year.

The bond market has added 25 bps to the 10-year yield since the start of 2026. It sits at a 9 month high(via CNBC):

The put skew is also starting to kick in with the risk reversals on IEF making 1-year highs. This is the 1-month maturity for reference:

Bonds are in a weird spot. If the economy sputters, you usually want bonds as a hedge but not if it sputters because of supply-side inflation. Kinda makes me want to sell bond vol as they are might whip around but not really go anywhere but even though IEF vol is relatively pumped, how much fun is it to sell 9 vol?

Anyway, all the commotion did get me to pull up TIPs. The 10-year yield is 2%. The purple checkmark is the last time I bought them (and I wrote a big post on that decision and how to understand TIPs generally).

10-year breakevens look a tad elevated but not especially compelling so if you don’t like bonds, TIPs don’t look like an extra cheap alternative.

Been a while since I pulled this up. My TIPs replication “symphony” on Composer comprised of oil + bonds, inverse vol weighted:

The replicator has been underperforming TIP (the TIPs ETF) for years but just “caught up” on cumulative gain thanks to the recent oil surge.

Explained here:

💡Inflation Replicator | 8 min read

Finally, for the yield hogs with a stomach for swings, M1 WTI trades about >3% premium to M2, so if you think the futures keep rolling up, that’s a 36% annualized roll return if a prompt barrel maintains a market premium (formally called a “convenience yield” in futures parlance).

Money Angle for Masochists

Put skews normalizing and then some

We already saw IEF put skew coming to life.

Silver put skew is coming back. The 25d risk reversal on 30d options has been grinding back toward zero and is now turning positive.

Gold put skew is ripping higher.

Apparently, no asset is safe. Maybe those private ones that don’t have prices. Oops, scratch that…

Energy and the dollar vs everything else (Street Fighter voice) Ready, FIGHT!

And for the metals enjoyyyers…gold vols are way off the curve after prices crashed 15% in a month (that’s ~52% annualized realized vol but who’s counting?)

Erik (Outlier Trading) and I record a podcast each week. We usually discuss an evergreen idea but we also sprinkle in topical episodes based if something current is on our minds. This is one of those:

I step through my thinking and the price of oil call spreads on the pod but here’s a summary:

If CPI trends toward 5%…

My market on real yields: somewhere between 0 and 1%.

→ That puts the 10-year at ~5.5%, or about 100 bps higher than today’s 4.4%

→ IEF (duration ~7) drops about 7% — which is in line with current higher implied vols in IEF.

→ Jan IEF 90/89 put spread: ~6-to-1 payout

Now the equity side.

Current SPX forward earnings yield: ~5%. If investors accept just 50 bps of risk premium over a 5.5% 10-year, then the earnings yield needs to be ~6%, which implies a P/E of ~16.7. That’s roughly 17% lower from here, assuming forward earnings don’t contract.

→ Jan SPX put spreads at those levels: also ~6-to-1

Both trades land in the same neighborhood.

(You could go to lower strikes for fatter payouts if you think the market is genuinely asleep at the wheel on inflation risk.)

Oil call spreads suggest that the chance of the oil prices rising to current levels through the end of the year are about 25% so you can take or lay 3-1 odds. Steeping through the chain, if there’s a 25% chance of dropping 20% and the current price is fair then the upside to SPX is:

.75*SPX_up – .25*20% = 0

.75*SPX_up = .25*20%

SPX_up = 6.66% which would take the market back to unchanged for the year.

It’s quite reductionist to think this is binary and to reduce the valuation of equity to inflation —> higher interest rates —> multiple falling, but the art of market-making is essentially sense-making between prices and probabilities quickly.

If there are aspects you disagree with, I’ve shared what some of the prices are in the market so let me know what the trade is.

In the vein of Thursday’s post, you could think about stuff like “Buy IEF put spreads and buy SPY shares on a ratio of X” if you think SPY has more upside because its current pricing is coming from a >20% downside (do you see how that math works?). Trading is like a sudoku puzzle with prices as the given numbers. It’s like you have to find the hedge ratios that solve the grid.

I thought this thread was interesting, but not to scare you, it thinks my downside scenario is quite conservative if gasoline prices stay stubbornly high:

If SpaceX and OpenAI want to go public, I wonder if Elon and Sam call Trump…”bruh you’re ruining our picture”. And then they could all sit down and work something out. Art of the Deal.

You know how in Monopoly when you are a bystander to 2 other people trade you are sad? It’s because regardless of who got the better of it, you know YOU are worse off.

From My Actual Life

I discovered the band King Buffalo while watching the Lost In Vegas guys…

…and now I’m obsessed. Spacy, bangin’, psychedelic grooves. If you’ve heard of Elder or All Them Witches, you’re familiar with the style. It’s kind of like Tool’s last album, but more chill. Built for journeys.

I’m not the only one feelin it…

There’s no band called Otter and it sounds like the name of a band that would make music like this. I gotta get the kid on this, I’m stretched too thin.

(I’m playing with the music school band class today actually. Setlist is Your Love by Outfield, Complicated by Avril, and Yellow by Coldplay).

Stay groovy

☮️

Moontower Weekly Recap

Posts:

• a market-making project you can do today
• how to get arbed with perfect information

In this issue:

• AI scheduled task example
• A rare, honest trading post-mortem
• Sorting through the bluster of the SpaceX IPO controversy

Friends,

⏰Claude can now run scheduled tasks | 3 min read

Khe explains how laypeople can easily schedule cron jobs. Put your “daemons” to work.

I used the scheduler to create a morning brief from my emails. For the past several years, I’ve been using autofilters in gmail that star (⭐) and apply a “Newsletter Subscriptions” label to senders I sub to.

At 6:30 am, I have a new gmail draft (Claude is not authorized to send emails so it’s stored as a draft):

 

By the way, anyone else notice that AI means we work even more? I think there’s something to that and it’s underdiscussed relative to the clickbait extremes of “post scarcity” utopia and a Skynet uprising.

The “something to that” is a mix of 3 things:

1. fomo
2. being busy “paying off our intention debt”
3. something we can’t see yet

I suspect both #1 and #2 are temporary and characteristic of a transition period on a compressed timeline.

#3 is a force that is probably positive and appears once our intention debts are paid. Which means the place to look for answers regarding #3 is young people embracing AI (although not in a Cluely, hollow your humanity, spirit). Young people have less intention debt, less to “renovate”, less to backfill, and a smaller corpus to consolidate. More of their use should be moving forward.

I like watching people like Nat use AI because they are plugged into the right layer of abstraction. The layer of recursion where every action is coupled with instructions to learn from the action so next time the action is improved. Infinite loop until you find a limit of perfection that would satisfy Zeno himself.

[My mathematical metaphor would be doing a Newton search, a common technique for estimating implied volatility, except the ε tolerance term isn’t predefined but shrinks as the calculation itself gets more efficient.]

In contrast, when I use AI, I feel like I’m living during the debut of the automobile and thinking, “Wow, I could drive that to the granary to buy food for my horse.”

 

I’m doing old things faster which is now par, but if you stop there it’s a failure of imagination.

Anyway, as I keep an eye on Nat’s quest, I’m curious how he breaks through this plateau.

Money Angle

📺”Ben Lost Everything”| 15 min

A few issues ago, I pointed out Ben’s YT channel, which is equal parts educational and hilarious. My wife is even working through the catalog. She and I were both very impressed with his most recent video, which continues to teach despite a brutal context. Ben blew his account up.

The post-mortem, reflection and honesty on display is rare. Given what I’ve seen from him, I expect him to bounce back stronger and wish him the best.

The follow-up to building an option chain in your head

🔗A Deeper Understanding of Vertical Spreads | 12 min read

 

If you are interested in prediction markets, that post’s discussion of binary probability is fertile soil for cross-pollination. Enjoy.

Money Angle for Masochists

From at least make the conspiracy make sense, we acknowledge an enduring difficulty between information and infohazards. In that issue, I called out Zerohedge for a disenguous conclusion based on NVDA earnings. But chastizing Zerohedge might as well be a pro wrestling ref admonishing the Macho Man for grabbing a folding chair.

Even as you move up the chain of credibility, it can be hard to distinguish degrees of expertise. Hell, even the idea of expertise itself has been in retreat. Freddie deBoer’s recent post Overlearning ($) collects several examples where a justifiable backlash to putting too much faith in expertise has led to its own form of blindness:

Each of these examples of overlearning began with a real grievance and a defensible insight, and each got driven by the normal human hunger for clean conclusions one or two steps further than the evidence actually supported. The result was backlashes to backlashes. The trouble with overlearning is that it inoculates people against correction. Because the original observation was right, any challenge feels like an assault on hard-won clarity, like a regressive attack. The overlearner has usually endured some version of being fooled (by the audiophile YouTuber, by the diet industry, by institutions that failed them) and so they’re constitutionally committed to not being fooled again. That commitment becomes its own kind of blindness, arguably more intractable than ordinary ignorance because it comes armored with a legitimate grievance.

Today I have an example of famous professional investors touting viral views that seem to be more smoke than fire.

I’m not an expert on the topic at hand, but an important skill in a complex world is being able to identify who is.

Let’s start with the scene.

In February, Nasdaq opened a public consultation on changes to its Nasdaq 100 methodology. The subtext being SpaceX’s plan to IPO at a $1.75 trillion valuation. There were two proposed rule changes.

1. a “fast entry” provision letting large new listings join the index after just 15 trading days
2. a multiplier that inflates how low-float stocks are weighted

This struck observers as suspiciously tailored to the occasion.

Fund manager George Noble called it “the most SHAMELESS structural manipulation of a major index I’ve ever seen.” Michael Burry called Noble’s piece a “Must Read” and sent it to his >1 million followers. The posts went viral, sparking the latest outrage: Musk bends yet another institution to serve his interests, and your 401(k) is the exit liquidity.

I don’t know who Noble is, but apparently, he’s a famous investor with all the credentials and job history.

My perception of Burry is that he has that autistic cocktail of persistence, intelligence, and disinterest in norms that enabled him to make a fortune betting the Don’t Pass (and got lucky enough on the timing which famously bedeviled a wider pool of investors who saw the same thing but were early).

Since the GFC, it’s been a bear market for bear outlooks, and from what I can tell with some googling Burry has done just fine since securing his bag over 15 years ago, so I’m not throwing tomatoes here. In fact, that he is such a smart guy devoted to the craft of investing, but has not performed notably one way or the other since his big score, illuminates how difficult alpha is.

But the bar to say stuff is simply lower. Their outrage is the kind of message built to travel, while expertise and nuance are boring and, in this case, going to throw cold water on the outrage festival.

I happen to know that when it comes to anything related to the details of index constitution, from arbitrage to legal frameworks, you find out what the account I call the “sensei” has to say.

Sensei gave a masterclass on this topic on my X timeline, then decided to resurrect his sleeping substack to publish a full rebuttal to Noble and Burry.

This Is Not The NASDAQ 100 Consultation Fight You Are Looking For | 20 min read

This is sensei. His pushbacks in descending levels of importance from my point of view.

1. Fast entry isn’t new. MSCI, FTSE, TOPIX, and most major global indices already have fast entry rules for large IPOs, some even faster than 15 days. It’s not a Nasdaq invention.
2. The “up to a year” claim is wrong. Nasdaq 100 already allows entry with as little as 16 weeks of trading history under existing rules.
3. The 5x multiplier actually lowers the index weight. Under current rules, a 10%-float stock is weighted at its full market cap. Under the proposed rule, it’s weighted at 50% of market cap. The math runs the opposite direction from what Noble and Keubiko claimed.
4. Keubiko’s lockup-expiry conspiracy doesn’t calendar out. He argued Musk engineered the IPO so a 180-day lockup would expire at the December reconstitution. For that to work, the IPO would need to be in late May, not the mid-June date Keubiko himself cited.
5. Noble misread S&P 500 methodology. He wrote that S&P weights a 5%-float stock at 5% of market cap. A stock with less than 10% float is simply ineligible for the S&P 500 entirely.
6. They ignored the Russell consultation. Russell US Indices was running an identical consultation on fast entry and float rules at the exact same time. None of the three mentioned it.
7. They complained after the window closed. Both Noble and Keubiko published their pieces weeks after the consultation period had already ended on February 27th, too late to actually influence the outcome.

Bau acknowledges the legitimate underlying grievance that Nasdaq uses total market cap weighting at all rather than free-float weighting like the S&P 500. But that objection predates this whole consultation.

You could take a consequentialist position that the ricochet effect of these folks’ rage-laundering brings a warranted, overdue wider awareness of Nasdaq’s weighting bugs, but I’m old-fashioned and think intentions do matter even if they sometimes backfire in effect. Let’s be clear, the intention here was to rile you up with a villain story, not emphasize Nasdaq or apparently many index providers’ methodologies.

I’m not one to strain to make an apology for Elon. But if you call fouls where there aren’t any, you lose credibility.

Baus exist. There’s credibility out there. But it often sounds like a whisper because it gets drowned out by what’s built to travel. Nuance is heavy and takes up lots of space. It’s the first thing to get thrown overboard when you want to get somewhere fast.

This Week In The Options Trench

Erik and I talk options earnings:

Stay groovy

☮️

Moontower Weekly Recap

Posts:

• Fair…I can see how you might think I lied
• does revenue seasonality translate to vol seasonality?
• a cleaner way to compute seasonal vol

In this issue:

• math in the car with kids
• trader quick math
• from straddle to gamma

Friends,

My older kid is getting braces in a few weeks. Based on the expected time he has to wear them, it’ll cost about $350/month. That’s a car lease. I’m not complaining (God: “he’s complaining”), I just suffer from chronic numeraire substitution. I’ll come back to the braces thing in a bit, but let’s chat some other stuff for a bit.

My sons are in 4th and 7th grade. A nuisance I will one day miss is shuttling them and their friends all around. We talk about lots of stuff, but stuff is often made of numbers, so I end up teaching them how to reason numerically about real-life stuff in an organic way in the context of things they find interesting. Yay. Except they groan because they know it’s coming. But I believe in osmosis and their future selves will be thankful. Or at least have endearing stories at my funeral about their old man being a crank who also happened to love them. And since they might have kids of their own one day, appreciate, just as I do now when I think about my parents, that we’re all just making shit up as we go.

Where were we before my inner monologue took over, ah yes, car convo. I got the boys in the car with another friend headed to practice. The 7th graders said they were learning scientific notation. Shouldn’t have told me that. Immediate quiz. Represent 1/50th in scientific notation.

I was impressed. I listened to his friend reason aloud for about 20 seconds before getting 2 x 10⁻²

Zak got the answer faster than I did. The Math Academy lessons are paying off.

Why is scientific notation useful?

To torture us.

Besides that.

“I don’t know, [proceed to fumble around for explanations before landing on something that tracks]. Because we need to measure stuff in micrograms? Is there even such a thing as micrograms?”

Very good. That makes sense. From the stars to bacteria and atoms scientists deal with things that are really big or really small. It’s right there in the name: SCIENTIFIC notation. We talked about how insane the idea of a light year is for a bit before arriving at the gym but not before I told them next time they watch YouTube, instead of watching Jesser we’re gonna learn about the Fermi Paradox which they theorized naturally but didn’t realize it was a famous contradiction.

On the way home from practice, the kids started talking about IQ. I forget what the comment was, but it indicated that they did not understand that an IQ of 100 is normalized to be the average. Sweet. We get to learn about bell curves right now.

I explain that 15 points is 1 standard deviation which encompasses 68% of the population. So to be greater than 1 standard deviation means being in the top 16%, since the 32 remaining percent have to be split between the lower and upper parts of the population, leaving 16% ABOVE 1 standard deviation.

2 standard deviation outperformance means top 2%.

I note that my scientific notation quiz asked for 1/50. Your father is psychic.

[Between that and the fact that I predicted that Axl Rose, who’s friends with AC/DC and lives in LA is probably at the Rose Bowl concert we were at last May, only to have him walk out from backstage about 60 seconds after I said that, they might think I’m a witch.]

Then we do 3 standard deviations. That encompasses 99.73%. For just the upper, it’s about 1.3 per 1000; let’s call it 1 in 750.

Given the size of your middle school, there are probably 2 kids that smart.

Except for that your school isn’t a random draw from the population.

We’re a long way from where I grew up. That night I explained to them that the test they took in 3rd grade, where they got 2 standard deviations above the mean, wasn’t even close to getting accepted to the local GAT program. Sorry boys, you’re not Asian enough and that’s on me.

Wanting to change the topic from IQ, I brought up height. After all, we just left hoops. I invented some numbers. The average adult American male is 5’9 with a standard deviation of 3”.

We stepped through the progression.

A 6’0 man is taller than 5 out of 6 men. (1 st dev)

6’3 and you’re 1 in 50. (2 st dev)

6’6 and you’re 1 in 750. In the running for the tallest boy in H.S. (3 st dev)

(Although selection effects need another nod here).

7-footers are 5 sigma. Using just the right-tail probability that’s 1 in 3.5 million.

This was a chance to apply their newfound knowledge of scientific notation.

How many 7-footers do you expect in the world if there are 3.5 billion adult men?

A million is 6 zeros. 10⁶. A billion is 9 zeros.

9 zeros divided by 6 zeros leaves us with 3 zeros.

We expect 1,000 7-footers.

Google says it’s estimated that there is “2,800” 7-footers in the world which the CDC statistically extrapolated using a standard deviation of 2.9 to 3. Small differences add up when you start adding sigmas such that our final estimate is off by a factor of 3. But hey, the right order of magnitude.

While we were countin’ sigmas the 9-year-old wants to know how Wemby exists. Wemby is officially listed as 7’4. There’s online debate as to whether his height is underreported and if it’s really 7’5. We’ll use that since it’s 6 standard deviations.

Siri, what’s the probability of an event beyond 6 standard deviations? 1 in 500mm. One-tailed, 1 in a billion. Wemby.

Statistically speaking you wouldn’t expect to have enough 7’4 mutants to assemble a starting 5 lineup but in reality you there’s enough of them to at least field a football team. Waves hand in the shape of epsilon.

Anyway, in service of handy takeaways, it’s useful to remember that a 3 standard deviation extreme on 1 side of a bell curve occurs about 1 in 750. For quick math, call it 1 in 1,000 or 10³. So if you’re talking about the American population of 3 x 10⁸, the number of 3 sigma people on a particular trait would fit in an MLB stadium.

Or about the same number of people who subscribe to moontower. See, you’re all 3 sigma! ❤️

Speaking of…

Moontower is 7 years old.

The first issue was March 17, 2019. This is Moontower #307, Munchies is up to #146, there’s been 96 paywalled posts, plus possibly the single largest archive of options blog posts on the internet. (ChatGPT mentions Larry McMillan and Kirk DuPlessis as being similarly if not more prolific.) Fyi, nearly everything I’ve ever published is indexed here and religiously updated so when robots erase me it is in totality. Thanks for following. I never expected to be writing this long. I didn’t expect anything.

Addendum on braces:

I wore braces from freshman to senior year of high school. My son will get his off about a month into freshman year. How’s that for generational progress.

The braces thing conjures something of a subway platform riddle for demographics where I can’t tell if the world is moving or me. My little guy got “spacers” in 3rd grade and will wear a retainer for 2 years. I’m like, is getting braces twice a new thing, or something I just never would have seen in my strata growing up?

I’ve noticed 2 other versions of this demographic subway platform riddle.

The older kid is now past the halfway point of middle school and I still never hear of fistfights. Growing up, at least every other week, the beacon went up, “FIGHT!!!”. Social class or changing times?

Finally, skiing. This one isn’t a riddle but it’s so jarring. I was 20 years old the first time I stepped on a mountain. Here’s school in the winter feels optional. Everyone has a cabin in Tahoe, all the dads are metereologists, and an expert on MTN stock.

Cold, heights and ski lifts, driving on dangerous roads?

I think I’ll just binge Nelly & Ashanti: We Belong Together thank you very much.

[We did knock this out in 2 nights. Plenty of time to cancel the 1-week Peacock subscription it required. I friggin’ love Nelly and so much more after watching the show. He comes off as an amazing father, raising both of his own as well as his sister’s kids when she passed at a young age. The only thing that bugs me is how good he looks at age 50. Save something for the rest of us bruh.]

Money Angle

Dean Curnutt graciously invited me to be on his outstanding podcast. His prompts led the conversation towards useful stuff. The description:

We begin with developments in commodity markets, particularly crude oil, and silver, where geopolitical tension and speculative flows have led to sharp changes in volatility surfaces. Kris explains how option skew in underlyings like oil can reprice rapidly during shock events, leading to inverted termstructure and a well bid call skew. These dynamics create unusual behavior in vertical spreads and probabilities implied by option prices.

Kris describes how the relationship between spot moves and volatility changes across market environments, emphasizing that traders must continually recalibrate their models. What appears to be a stable relationship—such as the familiar beta between the S&P 500 and the VIX—can shift quickly depending on positioning and market structure.

A major focus of our conversation is on the mental math traders use to interpret option prices without relying on models. Kris walks through several shortcuts that allow traders to move quickly between volatility, straddle prices, and probability estimates. These approximations help traders identify when prices look unusual and whether options markets imply probabilities that diverge from other markets.

Finally, we discuss the work Kris is doing on financial education. Inspired by teaching his own children about investing and compounding, he has begun running small classes for students and sharing the materials publicly. The goal is simple: introduce younger investors to concepts like time value of money and long-term compounding earlier in life.

If you are interested in a step-by-step breakdown of how I found an estimate of an out-of-the-money put like I did in that interview this post is for you:

🔗building an option chain in your head

Money Angle for Masochists

A topic I could have rattled on for much longer in that interview with Dean is trader mental math devices. By now y’all know option traders have the ATM straddle approximation burned into their retina:

straddle ≈ .8 Sσ √T

A useful approximation I did not explain in the interview is the similar-looking ATM gamma formula for a Black-Scholes straddle:

Γ ≈ .8 / (Sσ√T)

The three things that shrink gamma are in the denominator:

Higher S (price): The same $1 move is a smaller percentage move on a more expensive underlying.

Higher σ (vol): The option is already “priced for action.” The curvature of the price function gets spread over a wider range of expected outcomes. More vol → flatter curvature near the money → less gamma.

Higher T (time): Same logic as vol. More time spreads the curvature out. The more time to expiry the less a given move influences the delta of the option. The delta of 10-year option is not going to change much based on how the underlying changes day-to-day.

A couple of educational points:

• Take note of the scaling. Double the vol, gamma roughly halves. You need to quadruple DTE to get the same effect.
• As always, a good habit when trying to understand greek levers, is to take examples to extremes. If you raise DTE or vol to infinity, all options go to their maximum value. For calls, that’s the spot price itself. For puts, it’s their strike price. That means calls go to 100% delta since they move dollar-for-dollar with the spot. Puts go to 0 delta. It doesn’t matter where the spot price goes, the option is already at its max value. It doesn’t change. If a call is 100% delta and a put is 0% delta, the option has no gamma. Its delta doesn’t change with respect to the spot.

Going back to those formulas for a moment:

straddle ≈ .8 Sσ √T

Γ ≈ .8 / (Sσ√T)

The denominator of gamma = straddle/.8

Substituting:

Γ ≈ .8 /(straddle/.8)

Γ ≈ .8 /(straddle/.8)

Γ ≈ .64 /straddle

So when you want to do mental math you take “2/3 of the inverse of the straddle.”

This might sound obtuse, but taking inverse or “1 over” some number should be one of the fastest mental math operations anyone dealing with investing does. After all, when you see any ratio like P/E or P/FCF you are immediately flipping that to a yield where it can be compared with things like interest rates or cap rates.

If a straddle is $5, the gamma is 2/3 of $.20 or ~.13

And we know that doubling the straddle halves the gamma so you can just memorize that a $10 straddle has ~6.6 cents of gamma and linearly estimate gamma for any straddle price relative to that (ie $20 straddle is about 3.3 cents of gamma and $15 straddle is in the middle of 3.3 and 6.6).

And of course there’s time scaling. To find an option that has double the gamma you need to cut the DTE by 1/4.

Keep flipping this stuff over in your head, it’s satisfying, and it thickens the myelin around whatever brain cells you sacrifice to options damage.

If the 9-year-old can do it, so can you.

(I’m kidding. I just found this moment of deep thought cute. Between both kids’ basketball lives, the gym has become my office. Max does his Math Academy after his practice while waiting for the bro to finish. He recently discovered my Kindle is a scratchpad which has made my “no math without scrap paper” rule less of a nuisance. I adopted that rule from Math Academy’s recommendation, my affinity for mental math notwithstanding.)