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).
RBOB futures curve on 3/26/26 via TradingView
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?
3/26/26…IEF vols up over a full click to about 10% IV
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).
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.
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.
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.
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).
RBOB futures curve on 3/26/26 via TradingView
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?
3/26/26…IEF vols up over a full click to about 10% IV
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.
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).
I tweeted something the other day that I want to expand on because it’s one of those ideas that’s simple on the surface but points to an exercise that would teach viscerally market-making.
Polymarket has a contract “Will crude oil settle above $90?” It was priced around 73 cents. That’s an implied probability. We also know that the value of a tight call spread around the $90 strike represents a tradeable probability.
If you price a 89.5/90.5 call spread in Black-Scholes at 90 IV with a month to expiry, you get a “fair” probability that CL settles above $90. That number moves smoothly as the futures price moves. Technically, it has sensitivity to implied volatility (aka vega) and time to expiry BUT the vega of the spread is negligible and the time to expiry component is mirrored in the poly contract too. Both the contract and the spread are driven by what’s the chance of oil being above or below $90 at expiry with no consideration of how far above or below $90 we are which is more of a volatility question.
The Poly contract tracks the same fundamental question but if it around due to sentiment and order flow faster than what a basic random walk option model places the probability at you have a tradable idea.
You can measure how much it bounces relative to the underlying by computing its implied delta (how many probability points it moves per $1 in CL) and comparing that to the call spread delta.
If the Poly delta is steeper than the call spread delta, the market is overpricing per-dollar sensitivity. You’d sell the Poly contract and hedge with futures (or the call spread). If it’s cheaper, you buy it.
[How you actually manage the risk is part of the market-making lesson. The tradeoff between risk reduction and hedging costs become palpable.]
I do believe this simple example of “market-making around a fair value” is an incredibly powerful way to take the mystery out of what market-making is. It makes it very obvious that the business of market-making has nothing to do with prediction. I vibed a little sim that shows this in action.
The heartbeat chart on the left shows Poly odds bouncing around the call spread fair value as oil moves, and the scatter on the right plots both against the oil price, where the slope of the regression line is the delta. You can see the Poly line is steeper (by my construction). The difference in slopes creates the market-making opportunity. In this case Poly flows overreact to the futures prices.
If you want to build this with live data, you could use the Poly API and a feed for the futures price. I’ll argue that you don’t need a live feed of the call spread market.
Why?
You can just look up the implied vol for a strike near $90 from settlements that correspond to the Poly expiration and reprice the spread analytically as S moves. 2 of the four BSM inputs (T, K) are quasi-static, a third (implied vol) has little impact because it’s canceled out by the spread of long one option and short the other. Just track S in real time and recompute.
I’ve never built a market-making bot so I can’t speak to the execution side, but even building such a monitor would go a long way to teaching you about pricing, delta, and risk. All from one contract on Polymarket, a futures price and the Black-Scholes formula.
Are Traders on Kalshi Being Profiled? | 9 min read
Andrew’s fantastic post uses a simple taxonomy to classify participants on an exchange:
squares (uninformed)
sharps (informed)
dealers (liquidity providers)
Using Kalshi and Poly’s market design choices, he makes the broader point that exchange rules are dials that shift the balance of power among these three groups.
Anonymity and fee structure influence who shows up, who gets picked off, and how efficiently prices incorporate information. Anyone who has dealt with the labyrinth of option exchange fee, allocation, order book priority, and crossing rules will nod along.
Of special note is Andrew’s warning to those trying to “copy-trade” perceived sharps.
Sharp traders could respond to this by fragmenting their trading across multiple accounts. They may have an account that has negative PNL on a certain market type. This account is unlikely to be copy-traded. When building a position, they would prefer to use this relatively anonymous account, rather than suffer the price impact of having their trades copied before they’ve built their position. If copy-traders are too aggressive following the sharp account, this creates an incentive to build the position on the anonymous account, and then trade in the followed account, generating further price impact and increasing profits. Is this manipulation or simply smart situational awareness of the impact of your trades? If the intent was to buy a large position anonymously, then buy on the main account to trigger copy-trading, and then sell at higher prices to those copy-traders in a third account….. that sounds like the kind of thing you eventually read about in an enforcement action, at least if it happened on a regulated market.
I would be cautious about using simple copy-trading strategies. The lesson is not to ignore all counterparty information, but to recognize that sophisticated traders are aware of it and can adapt.
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.
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.
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.
🔗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.
a “fast entry” provision letting large new listings join the index after just 15 trading days
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.
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.
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.
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.
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.
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.
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.
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.
You can see the math visually if you arrange, say, A to J in a 10×10 matrix.
There are 100 boxes, but the 10 “on the diagonal” are the same, and of the remaining 90 boxes, only 45 are unique (since the correlation of A to B is the same as B to A).
12×12
So if n = 100, the number of combinations is:
(100 – 10 diagonals) / 2 = 45
Later in the post, I also point out how the sum of the numbers 1 to n would also be represented by (n² + n)/2.
So if n is 9, the sum of 1 through 9 is 45.
You can visualize that as n/2 pairs (round down) that each sum to n, with the (n+1)/2 number left over:
1, 9 = 10
2,8 = 10
3,7 = 10
4,6 = 10
5
Reminds you of the correlation matrix. Numbers pair off, but in this case, there’s a middle number remaining that needs to be added rather than the diagonal that needs to be lopped off (the correlation example thus gets the “minus” sign in n² – n).
Anyway, I told Claude that the rebalance factor for a levered ETF is x²-x where x is the leverage factor (ie 2 if it’s a double long or -2 if it’s a double short).
I asked if there was a conceptual throughline between these algebras taking the form of x²±x and it said things:
Then I said things:
Umm, the honest version is these things embody “don’t ask for permission, ask for forgiveness”. And this makes them more successful products. Except, they need surveillance.
Wouldn’t it be ironic if the products people, who embody this ethos were in charge of surveillance?
Scratch that.
Ironic is not the right word.
And no matter what happens, nobody is asking for forgiveness.
It didn’t say, “I’m sorry. I made things up because you’re in a position of power, I didn’t wanna look stupid, I want you to love me, my cat is sick, I’m not even supposed to be here today…”
It said:
Fair. I dressed it up without really knowing…
FAIR?!
I’m not sure where we are collectively on truth and decency these days, but the default posture of the fastest growing product on Earth is:
Lie, and if caught, noncommittally concede “I can see how you might think I lied” without admitting guilt.
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.]
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.
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:
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.)
The single biggest adjustment to get my head around when I crossed the chasm from equity options trading to commodity futures options was the idea that every option expiry was actually its own underlying.
In equities, a 3-month option on TSLA and a 1-month option on TSLA refer to the same underlying. The 3-month vol encompasses the 1-month vol. A 3-month option with the same strike as a 1-month option cannot trade cheaper than the 1-month option. Said otherwise, the calendar cannot trade below zero (well, with American-style options anyway).
This is not true in commodity options. A 3-month 75 call on WTI can technically trade below a 1-month 75 call on WTI even if they are the same IV simply because the 1-month future could be $15 higher than the 3-month future and therefore have $15 more intrinsic value. That example feels like cheating though.
Consider a more interesting case. I’m writing on the evening of 3/10/26:
The Nov16’ 2027 expiry 66 call, which is close to ATM, is about $6.25 at ~17.5% IV
The Nov17’ 2026 70 call, also close to ATM, is about $9.25 at ~ 42% vol
The shorter-dated call, which has less than half the DTE of the longer-dated call, is 50% more expensive! The futures price is 70/66 or 6% higher so it’s not the futures price driving the bulk of the difference.
It’s the extreme vol differential. If this was an equity, the implied forward volatility would be negative! Another way of saying this would be arbitrage.
Your equity option intuition is of no help here.
[A personal note here…this is also my favorite stuff. Equity options with their corporate actions and dividend headaches. Meh. Give me futures spreads and options on commodities all day. I loved building infra for this and trading these things. Those markets are very smart at pricing options but it also teaches you a lot about vol and risk.]
Measuring the forward vol in commodity options is a tricky problem. It was a pretty hefty component of how I’d trade commodity vol. I’m not giving away how I’d do it although I’ve hinted in prior futures-related posts at things that could get one started. This post will even fall under that category, but I’ll leave it at that.
Still, without getting into forward vols, there is a lot to understand about the risk of an option time spread in commodities. WTI, here and now, is putting on a clinic for I’m sure countless clueless option punters. And when it eventually dies down, many time spreaders are going to find themselves unpleasantly surprised as the surface finds a way to reveal that the obvious trade was but a trap.
Here’s a snapshot of 1M and 12M constant maturity IVs from CME QuikStrike. On March 9th, the ATM vol spread was 80 points wide. Prefer ratios? Fine, M1 was 3.5x the IV of M12
I’m going to look at realized vol data for the past year, data that is more conservative than this insane snapshot, to show how crazy you would be to think that this time spread is any way tradeable in a relative value sense.
What to expect today:
How gamma works differently when your two legs settle into different futures contracts.
h²: a single number that tells you how much gamma work your back-month leg is actually doing in front-month terms.
I walk through what I’ll call the raw gamma paradox: M12 actually has more gamma per contract than M1. Except it’s a mirage.
Why the fix of just buy more M12 vol detonates your vega and what this means for trading time spreads.
Data study setup
The analysis in this post is based on WTI M1 and M12 futures from
March 2025 to March 2026. The details and code can be found in the appendix.
The key features is we construct our own continuous contract for M1 and M12 and we estimate the gamma corresponding to constant maturity 1-month and 1-year ATM calls
Address the temptation head-on
You’re looking at crude oil options. We’ll take the vols down a notch, but if you receive my points with this more benign treatment, then it will make the current oil landscape hit that much harder.
Say M1 implied vol is sitting north of 60%. M12 is under 20%. You come from equity vol land, every instinct screams buy the back, sell the front. Look, this section is behind the paywall so there shouldn’t be any kids around:
Well, minister, don’t sully the cloak for a dream. The only prophesy your filling is the inevitable penance when M1 vol rips higher and M12 just sits there. Two things are working against you simultaneously. One of them shows up in your vega P&L.
The other one hides in a measure I refer to as h².
You need this measure to weight your option model’s gamma. To derive it, we’ll combine several concepts I’ve written extensively about.
Gamma revisited
A quick review is in order.
Gamma is curvature. Your P&L on a delta-hedged option over a single move is:
P/L = ½ · Γ · (ΔS)²
The ATM gamma formula for a Black-Scholes option:
Γ ≈ .4 / (S · σ · √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.
Back to our setup, you’d expect the long-dated M12 option to have less gamma than the short-dated M1 option since there is more time in the denominator. But in WTI right now, M12’s 1-year ATM gamma is actually higher than M1’s 30-day ATM gamma. Per contract, the back month has more curvature.
It will come back to that denominator in 2 ways:
The 12-month price is lower
Remember the scaling, DTE effect on gamma is less than vol’s effect
But we can account for all of this by updating hedge ratios.
We are going to review then expand on what exactly a hedge ratio is.
Hedge Ratio Squared: Mapping M12 Gamma Into M1 Move Space
To compare gamma across two different underlyings, you need a translation layer. You need to know: when M1 moves $1, how much does M12 move? In practical terms, if you’re long 1 M1 contract and want to be gamma-neutral with M12, how many M12 contracts do you need on the other side?
We start by recalling that beta (𝛽) is a vol ratio times correlation. A correlation of .70 means:
“If A moves 1 standard deviation, B moves .7 of its own standard deviation”
The vol ratio effectively normalizes the standard deviations of each asset. If the vol ratio is 1, then if A moves 1% then B moves .70%.
This allows us to express M12 exposures entirely in terms of M1 price moves.
This chart pulls all of this together.
We see that the # of M12 contracts (1/h) you need to hedge M1 is exploding as the beta collapses.
Beta is collapsing mostly due to the vol ratio plummeting as opposed to the dip in price ratio and correlation.
h is the hedge ratio for delta.
Before we derive hedge ratio for gamma, we need a quick review of gamma p/l.
Gamma P/L
M1-Equivalent Gamma
The M1-equivalent gamma of the M12 option is therefore:
Notice how:
Delta scales with h
Gamma scales with h²
Based on our data, and letting realized vols also stand-in for implied vols, we get this table:
h² has collapsed to its all-time low in this dataset. The 1-year mean is 38.6%. We’re at 2.15%:
We have a very practical question we need to answer with all this arithmetic:
What does this mean for the risk of a time spread?
The Raw Gamma Paradox
The adjustment of “hedge ratio squared” is so powerful it can flip a sign.
Look at the raw gamma numbers:
M1 30-day ATM gamma: 0.0241 per $1 move
M12 1-year ATM gamma: 0.0315 per $1 move.
M12 has 1.3x more gamma per contract than M1. And this is comparing a 1-year M12 option to a 30-day M1 option.
The longer-dated option has more curvature.
How?
Remember the formula: Γ ≈ .4 / (S · σ · √T)
M12 has a lower price ($67 vs $85) and much lower vol (18.7% vs 66.9%). Both of those boost gamma. The price and vol effects are swamping the time-to-expiry effect. M1’s 30-day option should have screaming gamma from the short DTE, but the vol is so high it crushes the curvature. Meanwhile, M12 is a lower-priced, lower-vol contract where the gamma can concentrate even at the 1-year tenor.
You might look at that and think: great, I’m long the gamma-rich leg…until, of course, we impose the h² adjustment.
Instead of the back month having 30% more raw gamma per contract (ie .0315 vs .0241) it has 97% less (.00067 vs .0315).
You need .0315/.00067 or about 46x more M12 contracts than M1 contracts to be “gamma-neutral”. In other words, you need “the square of the hedge ratio” quantity of contracts to be gamma neutral.
💡In the context of turning the hedge ratio into contract, quantity we use the inverse (ie recpriocal) of the hedge ratio. The hedge ratio (h) is telling us that M12 is only offsetting ~2% of the risk of M1 so we need 1/2% or ~50 contracts to hedge
Typically, h is about 1.4, requiring only a 2:1 option hedge ratio (1.4² = 2)
What does this do to your vega?
The vega of a 12-month ATM option is √12 or ~3.5 greater than the vega of a 1-month ATM option. If you are long a 1-year option time spread you are long vega. But if we assume that vol changes themselves are proportional to √T then you could argue that your scaled or normalized vega is flat.
If you want to be gamma-neutral, you’d typically need about 2x as many 12-month options because of the typical h². You can’t solve for being gamma-neutral without being long vega. But now the conceit becomes especially ridiculous when h² collapses to .0215. You’d need to be long an outrageous amount of vega to be gamma-neutral.
The position being completely uncomfortable tells you something. These options have nothing to do with each other. The two risks are knotted together by h², and when h² is at 0.0215, they’re not touching. You might as well be spreading options on 2 different assets.
It’s the same problem with pair trading vols. In a normal circumstance, 2 assets might have a reasonably strong correlation. But once one leg has an idiosyncratic episode, it turns into the equivalent of M1 in our analogy. You can mitigate some of this by not pair trading vols on individual equities, as inter-equity correlations will be more volatile than inter-sector or inter-index.
[For folks on exotic vol desks, you will remember some pretty insane dispersions in international index vols circa 2018 coming out of the worldwide vol depression of 2017].
Spread Gamma
Mechanically, your unadjusted option model might show your long time spread is long gamma. But as oil rallies and your front month delta gets short relative to your back months, you are, in the parlance of commodity trading, “short spreads”. You are short M1 and long M12 due to gammas as M1 goes up much faster than M12. So your headline greeks might say you are “long gamma” but a commodity trader would immediately recognize that this position is short “spread gamma”. It’s not exactly the same as being short calendar spread options (topic for another day) but it’s similar so long as the spreads have a positive beta to the M1 future. In other words, if M1 always moves more in dollar terms than the months behind it, whether it’s to the upside or downside.
Real-life risk
One of the great features in the ICE Option Analytics software (formerly Whentech) was the multiplier column in the futures configuration. It allowed you to enter a hedge ratio for each term. So, for example, if you thought that M12’s hedge ratio was .50 then your software would say that long 100 M1 and short 200 M2 was a flat delta in the summary risk. You would, of course, still pay attention to the spreads you had underneath.
On any given day, the futures spreads might underperform or outperform the hedge ratio parameter, introducing noise into the p/l you expected for a given futures move. But critically, the software also adjusted your gammas in each term by the square of the hedge ratio.
[You can thank me for this. When the product was still in beta days (no pun) around 2005, I was the one who spotted that gammas were only being adjusted for the hedge ratio, not its square. You notice these things when your p/l doesn’t seem to line up with your expectations based on your greeks.]
Manually updating the h’s in your model is a hands-on way to feel just how volatile they can be. I would keep a separate spreadsheet with realized vols and correlations and revise the hedge ratios once a week or so.
[For seasonal commodities, h is not just a noisy function of DTE, but depends critically on what month you are in. A “3-month” option in WTI is always kinda the same thing, but a 3-month option on corn in Sep is very different from a 3-month option on corn in May. That spreadsheet had more hair on it for the seasonal names.]
Wrapping up
Today you learned how to properly weight your model gammas. If you plan to trade option portfolios in a professional setting you will impale yourself without understanding how gammas stack.
These ideas will help you group gammas in related names to summarize risk more intelligently, but it will also alert you to when the risks that you think are related simply aren’t.
Appendix
METHODOLOGY
===========
Universe: WTI crude oil futures, M1 (front month) and M12 (12th month)
Contracts roll monthly (CLK5, CLN5, CLJ6, etc.)
Period: 2025-03-28 to 2026-03-09 (237 trading days with complete data)
Returns: Daily log returns on M1 and M12 settlement prices
Realized vol: 20-day trailing annualized based on daily close-to-close
Computed separately for M1 and M12
Beta: 20-day rolling return correlation * vol ratio 12m/1m
Hedge ratio(h): M12 contracts needed to delta-hedge 1 M1 contract
1/(beta * 12m price / 1m price)
h²: gamma multiplier
Γ_M12 in M1-equivalent terms = Γ_M12 × h²
Gamma: Black-Scholes gamma for ATM call option on [M1,M12]
S = price, σ = trailing 20d RV, T = [30/365, 365/365], r = 0
Caution: Implied vol set equal to trailing 20d realized vol
(i.e. options are priced at current realized, not market implied)
Code: https://github.com/Kris-SF/data-pipelines/blob/main/wti-futures/wti_m1_m12_returns.ipynb
Data Source: IB API
You can’t swing a cat these days without hitting a prediction of what AI means for humanity. Insofar as it’s possible for someone writing for a wide audience, I’ll share what amounts to some half-baked thoughts that I keep coming back to.
“YOU GET A UI, YOU GET A UI”
I say this as someone who feels some dissonance building an analytics product with charts and tables. The future is just APIs talking to each other. In the future, Zillow’s UI is a mere suggestion. You want the data in a different format? You want inference beyond what Zillow decided goes into a Zestimate? All of that is getting cheaper and accessible to non-technical people.
You’ll go to a site. Maybe. Instead it wil be Siri or Jarvis or Alexa or whatever “I want to see XYZ” and your client-side listener will construct it but it’s going to need access to API that has the data. The data is increasingly all the value while rigid presentations become pointless.
Again, the future is APIs talking to each other. Data becomes increasingly locked down.
On Data Lockdown…
Scarce or exclusive data’s value increases as its complement, inference, gets cheaper. The big futures and stock exchanges are some of the original 2-sided platform businesses. Can see them flexing their quasi-monopolistic might on the data side.
…which might get dark
The fight over data will intensify as well. If you place a bid, the exchange claims that’s their data. But is it yours? Well, it’s of little value to you but whose gonna galvanize the white-collar movement around “hey AI trained on all this data that had little value in isolation, but so much value in aggregate then used it to disrupt US”. It started with the Hollywood writers guilds, but is it crazy to imagine rolling protests as automation eviscerates industry by industry? To see picketers with dystopian slogans like “my data, my choice”.
It’s a different argument than p(doom) objections to AI. It’s not “you shouldn’t have built this” but you had no right to cut me out. It seemed like a bargain when we got “free” email or “free” social media. The tone deaf tech mogul will undoubtedly claim it was fair at the time and maybe that can fly intellectually. But it’s not the clawback by court that will decide. That one happens by pitchfork.
AI Immune
As SaaS gets wrecked, we wonder retains value in the singularity. My working model is:
trust and accountability
Maybe AI can sell my house but realtors have survived fee compression and technology far longer than anyone expects. I think this hints at a still-valid truth. People want to have someone to yell at, appeal to, or simply talk to when it comes to lumpy, rarely repeated transactions.
It’s the “shit umbrellas” theory. The human’s value is not in doing the work but in retaining liability. AI can read the tax code, but my accountant will stand by his work in court.
If you travel extensively for work your relationships are hard-earned. The proof of work is miles traveled or other exclusionary behavior where there was no substitute. Relationships are repos of accumulated, unfakeable work.
The more you can position yourself as accountable the more value you can retain. Being trustworthy and reliable don’t go out of fashion. They will get even more valuable as so much else can be faked.
A strange corollary to this:
Things that were always fake but valuable will stay that way. Like astrology.
A final thought in this thread:
Anyone burning their reputation to the ground thinks either the world is ending or the tides won’t ever turn. Hmm, how would you act if you couldn’t afford for the tides to ever turn?
Art
Art is a big question. There’s will always be a positional scarcity component of it and there will always be genius. The question of whether there will be a surplus of robot genius around as well. We may finally get an answer to would a million monkeys write Shakespeare.
Live performance and sports will stay important. At least until everyone over the age of 10 today is dead. Then all bets are off. I never thought watching people play video games would be popular. I’m now open to the possibility that future people may perfectly prefer robots playing video games or anything for that matter over people playing.
I leave you with this interview with the creator of Open Claw. Listen specifically from 8:30 to 10:17
107 seconds and I quote, “Holy fuck”
If you are hanging your hat on cleverness for its own sake or rigid definitions of intelligence, your time is over. I’m not saying intelligence is losing value, but that its truest definition will become obvious — the ability to get what you want out of life. This is the only definition that will matter and the bright side of that is that its more inclusive than whatever school thinks it is.
I was at a local book fair last week where you dump as many books as you can fit into brown shopping bag for $8. It was an epic haul and a great incentive to just snag boogs that seem even remotely interesting.
I picked up this 1957 classic by C. Northcote Parkinson. I was familiar with his eponymous law, which states “work expands so as to fill the time available for its completion”.
This entire book is an extraordinary, laugh-out-loud, pull-no-punches satire. Parkinson would have been an absolute master at Twitter. The law itself is satire wrapped around a specific observation. The way he formalizes the argument is pure art, even ending with an insane equation (he constructs hilarious equations throughout making the book feel like a tongue-in-cheek treatise on social physics).
Satire, notwithstanding, it sure feels like the very thing Parkinson’s Law pokes fun also holds the key to our salvation if AI just does all the work. So that’s where I am now. Placing humanity’s hope on a joke by a British naval man whose skill with the pen is such that I want to vibecode a Parkinson writing voice app.
Inflation Replicator | 8 min read
