Recall the levered silver flows post where we see the quick math of levered ETFs. For a fund to maintain its mandated exposure, the amount of $$ worth of reference asset they need to trade at the close of the business day is:
x(x - 1) * percent change in the reference asset * prior day AUM
where x = leverage factor
examples of x:
x=2 double long
x=-1 inverse ETF
x= 3 triple long
x= -2 double inverse
This isn’t just a levered ETF thing. The -1 leverage factor is exactly the same as just a vanilla short position. It’s a sneaky reason why the shorting is mathematically challenged.
The easiest way to think of this as an individual investor is to imagine you have an account value of $100. The account is holding $100 in cash, but it’s the proceeds from shorting a $100 stock (assume you don’t need any excess margin to maintain the short). If the stock falls to $50, your account value is now $150 (your cash + $50 mark-to-market profit on the short). You earned a 50% return on a 50% drop in the stock.
Now what?
If the stock falls another 50%, you make $25.
$25/$150 = 16.7%
If you want to maintain the same exposure so that you make 50% on your account on that second 50% drop, you would have needed to short more shares at $50.
How many more dollars’ worth of stock?
-1 (-1 -1) x -50% x $100 = -$100
You needed to sell an additional $100 worth of stock or 2 more shares at $50. Then on that last leg down, you would have made $25 on 3 shares total or $75.
a * b = Mean² − MAD² (where MAD = mean absolute deviation)
As soon as numbers deviate from the mean, their product is dragged down — even if the mean is unchanged. More deviation, more drag. And what is deviation? Volatility.
Bridging middle school math to investing math
In investing, we compound, or multiply returns. So even if the mean of two returns is identical, the dispersion between them matters. Not just matters. It matters quadratically.
No dispersion: The arithmetic mean of (8, 8) is 8. The geometric mean of (8, 8) is √(8×8) = 8.
With dispersion: The arithmetic mean of (5, 11) is still 8. But the geometric mean of (5, 11) is √(5×11) = ~7.4.
If you earn 10% on an investment and then lose 10%, your mean return is 0, but your actual compounded (geometric) return is 1 − √(1.1 × 0.9) = −0.50%.
Now increase the volatility: earn 40%, lose 40%. Mean return is still 0. Compounded return? 1 − √(1.4 × 0.6) = −8.3%.
The drag on your returns is a function of squared deviation. Put simply:
As I mentioned a few weeks ago, I’ve been re-publishing educational posts on X Articles which serves as spaced repetition practice for long-time readers or just bringing them to the attention of new readers who would be better served by a steady IV drip (no pun) of archival posts than attempting to raw dog the compendium.
I bought silver a year ago because of Alexander Campbell’s substack. He does a great job showing his thinking behind ideas with data and charts. This alone is helpful because it reveals “these are the datasets a smart guy pays attention to”.
AI tools are shortening the distance between “Hey, that’s neat, I should add that to my dashboard” and like actually adding it to your dashboard. Even if you stink at the world’s most popular coding tool —- Excel (see Will Claude Eat Excel?)
I used one of Alexander’s recent posts to whip up a silver dashboard. I’ll explain what I did, what I added, and share it with you so you can duplicate it as your own starting template. But the broader lesson is that agents are going to make all content “interactive”, we’re just not used to those patterns. Yet.
It is just another staple in my belief that as the cost of inference approaches zero the value of unique data increases. At one time oil was used for light and warmth. But when the automobile was born it claimed the largest cut of the barrel. If data is oil, more people everday are unlocking the ability to “refine” it by transforming it, building new logic and visualizations.
Let’s get to creating the dashboard.
One giant disclaimer:
Expectations are everything. AI is not going to one-shot this project. I’d estimate it reduced a 6 hour task to 90 minutes. Indulge my parental tone for a sec. It would be a mistake to permit this to let you work less in the spirit of that stupid Genspark AI Super Bowl ad. Instead, you should see this as “I can do 4x as many projects as I could before.” This may sound like hustle-porn (you know it when you see it, right?) but if that’s your attitude I offer 2 observations:
You probably don’t like your work. If you do, then giant increases in productivity allow you to get even closer to the the best parts of your work.
Regardless, this goldilocks period will end, everyone will know how to use the 21st century calculator, and 4x as productive will become the new baseline. Red queen. A very smart guy who used to work with me (he was the one who did a lot of the math and technical stuff that we’d need) works in real estate now. I suspect he’s in the top 1% of nerd in that industry. He recently applied for a job and failed a test that was intended to deomonstrate how resourceful he was in the context of AI tools. Knowing him as well I do, I found this shocking because he’s the kind of person that always does well on formal exams. Granted, he admits he’s not not using AI as much more than a google replacement. That this exam exists and a person like him failed, suggests the goldilocks period may already be drawing to a close. It’s not like real estate companies are living on the bleeding edge either.
On a positive note, I think you learn just as much in the compressed time as if you spent 6 hours. Instead of fumbling around with semicolons and syntax you learn how the internet is stitched together and how technologies talk to each other. Embrace manager mode.
We include a control tab for sourcing the relevant data. All of Alexander’s sources were public but whether you can automatically connect to them is another matter.
That’s why I like ot have a control tab which triages which sources are MANUAL, API, or SCRAPED.
Google App Scripts
This is the equivalent of VBA behind Google Sheets but it’s in Javascript which Claude will happily write for you whether you want to wire the sheet up to APIs or scrape.
Step 3
Troubleshoot. Claude’s sheet gets you 75% of the way in moments and then you spend 90 minutes on this step.
Most of the scraping failed. Sometime because Claude referenced a stale website. But even when you update the correct URL you quickly find that financial data websites tend to lockdown the ability to scrape.
I worked through each data source, iterating with Claude to find automatic (and free) solutions or writing AppScripts usually falling back to “manual” when necessary.
Finally, as I made changes to the spreadsheet there’s the expected debugging and tracing of formulas that happen whenever you delete stuff from a sheet someone else (in this case a bot) made. Pound ref and N/A always show up for a gangbang.
Step 4
Add spice to taste.
Alexander + Claude leapfrogged a lot of work. But there’s still plenty of room for your own judgement and creativity.
For example, when it comes to COT I use the fantastic tools on the CME website which aggregate both futures and options positioning.
I also added leveraged ETF tickers and logic that estimates how much silver there is to buy/sell based on their daily rebalances and even a first pass at computing market impact (see appendix).
Finally, I included a placeholder picture to compute the implied term structure from the SLV options term structure by backing out hard-to-borrow rates.
from moontower data infra
The google sheet is mostly self-explanatory but even if you get stuck just use Gemini in sheets or the Claude extension in a browser to mentor you along.
“Vega wounds, gamma kills” is an esoteric expression that’s still common enough that you can google it and return a bunch of hits. It’s a reasonable acknowledgement of realized vol p/l being quadratic with respect to how large a stock move is.
I’ve recently been cross-posting my writing on how this works on X since they’ve been pushing their Articles functionality.*
* A lot of people (and bots) are boosting these. I am treating these releases as a spaced repetition exercise for long-time readers. Analytics show very high engagement so X must be signal-boosting them. This is a 1-year chart. The recent spike is Articles:
A lot of people cry about the growth of Articles longform on X but twitter is a long way from the community it used to be anyway, so don’t really care as much if I’m burning the house for warmth in the eyes of diehards. Although I don’t think I am since the reason I came to twitter in the first place was to find stuff to read and learn not hot takes. It's different things to different people and when they suppressed Substack it shifted the appeal for me. This is some re-alignment, albeit on their terms. Fine. It's a reasonable negotiation.
The Articles I’ve posted on the theme of non-linearity in options
This last one is about the “gamma” of vega. For OTM options, the vega of the option, its sensitivity to changes in IV, itself changes. We call that second-order sensitivity volga. Volga is to vega as gamma is to delta.
I don’t have a dedicated post on vanna I’ll cover it briefly right now.
Vanna
The definition of vanna you are most familiar with is change in delta due to change vol. You have heard of this because of dealer flow discourse. For example, if dealers are long calls and hedged with short shares, as vol declines on a rally, their long option deltas shrink. If this happens faster than their long gamma increases their net delta, then they will have stock to buy to rebalance to neutral.
But vanna has an alternate definition. One that dominates our understanding of trading skew:
the change in vega for a change in underlying
If you are short puts on a risk reversal as the stock falls, you get shorter vol and vice versa. Your vega changes as the spot moves.
I suspect the “gamma kills” idea is popular because it’s a common experience. Option volume is dominated by near-dated expiries where gamma and theta dominate the p/l. Most people will simply never feel what it’s like to be wrecked or celebrated by volga or by a delta-hedged skew position. They might know what it’s like to get crushed to vega directly, but even that will be less familiar than realized vol-driven performance, given typical trade duration.
But I can tell you that my most memorable p/ls have all had vanna and volga at the scene. 2020 was especially dramatic in this regard as an explosion in vols led to position sizes exploding and finding myself sitting on a growing pile of vega that varied from “increasing in demand” to “panic bid”.
Qualitatively, the repricing of vega is significant because vega is illiquid. You can delta-hedge your way to a replication of a relatively short-dated option. In a sense, the volume in the underlying itself is a form of liquidity for options even if the options themselves are illiquid. But this idea extending to a long-dated option is only theoretical. In practice, if you are short a long-dated straddle that doubles in value, the mark and its accompanying hit to your capital may leave you in a forced position. You don’t have the luxury of manufacturing that vol via delta-hedges for a year.
This will be exacerbated if you were short, say 100k 1-year vega, but because of vol exploding you find that you are now short 200k vega. Maybe you can stomach the p/l hit due to vega, but you might not be able to hold the new position size. If Street Fighter’s Vega had Mortal Kombat finishing moves, they would be called vanna and volga.
The recent silver move has been so crazy that vega p/l has dominated realized p/l (realized p/l is the tug of war between gamma p/l from the equation at the opening of the post and theta). It’s an outstanding case study in how higher-order effects are fundamental to understanding options.
We’ll begin with a classic “trap” trade.
Imagine back on Dec 31st, with SLV at $64.44, you bought put and sold call on the 60/100 risk reversal delta neutral with the plan to hedge the delta at the close each day.
This position starts:
Long vega
Long gamma
Paying theta (you laid out extrinsic option premium)
The 60 put you buy is 59.6% IV, the 100 call you sell is 78.7% IV
The risk reversal would have cost you $2.89 of option premium since the put is much closer to at-the-money.
What happens between when you opened the trade and the snapshot I took this past Tuesday, 1/27/26, when the stock has risen to $97.09 and the options still have over 3 weeks to expiry?
This daily hedged risk reversal has lost $.82 net.
You are short gamma albeit less gamma than you were long when you initiated the trade because the ATM vol is so much higher!
More things to note:
The IV on your long strike: 59.6% → 99.6% or 40 vol points!
The IV on your short strike: 78.7% → 99.4% or 25 vol points.
You won on the vega spread between the options.
So why did you lose money? Was it the realized vol? That seems suspect, after all, you were long gamma at the start of a big move. You’re short gamma now, yes, but it’s not even that much.
The clue is right there in the table:
You went from long 5 cents of vega to short almost 14 cents of vega as your short strike is now at-the-money.
Yes, the vol on your short strike went up much less than the IV of your short strike, BUT it went up when the vega of that strike was much larger than the vega of the strike you were long.
In short, you were getting shorter vol as vol was ripping higher. The vega p/l totally swamps the realized p/l:
from a long option holder point of view of a daily delta-hedged position
Here’s a snapshot from the interim on 1/13/2026, when the stock had rallied almost to the midpoint of the 60 and 100 strikes.
The 60 put you own has gone up over 7 points, and the 100 strike you are short barely budged from the elevated vol from the original skew. You are up $.37 on the hedged position…but your risk is changing quickly. You are now short vega, flat gamma, and collecting theta.
Wait, you are collecting theta without being short gamma.
Technically your gamma is very slightly short, but the point stands — in fact, if the 60 put IV was a bit lower you could even be long gamma and collecting theta.
New option traders will brag about such a favorable greek profile. An experienced trader knows that the ratio is an indication that you are simply short a premium IV and premium IVs happen near the prices where hell breaks loose. As I’ve said many times…the skew just tells you where the bodies are:
In sum,
Despite these options not being “long-dated” their performance has been dominated by IV. In this case, mostly through vanna which is best seen at the interim.
Despite the 60 put vol increasing 7 points, the vega of the option halved as it was now much further from ATM (it went from being a -33 delta put to -9 delta by 1/13/26)
Meanwhile, the 100 call’s vega doubled due to it becoming closer to ATM (it went from a 9 delta call to 21 delta).
Note that volga is not playing much of a role in 100 call vega doubling. The change in option vega can’t be due to IV increasing. Why? Because IV didn’t change on the 100 strike during the rally from $64.44 to $78.60!
From the vol convexity article, we know ATM options have no volga. In fact, ATM vega is insensitive to vol level and holding DTE constant, it only depends on the spot price!
But OTM options have a lot of vega to gain if IV increases since IV ripping higher makes all OTM options look closer to ATM as they are “less far away”. Their delta increases (vanna) and their vega increases (volga). In the above example, the 100 call IV did not rip higher by 1/13/26, so we couldn’t see volga in action. The vol only roofed on the strike once the option was close to ATM.
To give volga its due, we should zoom in on Monday when Feb SLV vol ripped higher on silver popping 10% (before giving back nearly half its gain).
We’ll look at a call nearly 14% OTM with less than a month til expiry.
The $1.33 of hedged option p/l for that call is only partially explained by the initial vega of .033 and a vol change of 26 points. The difference could be explained by the fact that the average vega of the call as vol (and stock) increased was probably closer to .05.
26 vol points x .05 vega = $1.30
Since the stock only rose by 6%, we can safely guess that the 50% increase in the vega of the option is mostly driven by volga.
Gamma is not the only killer. Any position that grows faster than the underlying changes contains risk that is not seen in a snapshot. That delta hedged vertical spread or risk reversal might look gamma, theta, and vega neutral today but that profile gets battered as soon the clock ticks and the waves start coming in. The snapshot neutrality is dangerous because it can easily lull you into thinking your risk is smaller than it really is.
Ask anyone who bought an SLV and nat gas 1×2 call spread because “the skew was fat” or because they are “long gamma, collecting theta” how that’s working out?
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.
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.
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.
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
I don’t know anything material about geopolitics, military strategy, the supply/demand response function for light, sweet crude slated for delivery at Cushing, Oklahoma. I don’t have an opinion now or when I wrote the post.
I only have eyes to see the present. To look at a price and try to reverse engineer how it could make sense. The details are in that post, but the specifics matter less than the approach. In fact, I even mention why my approach is probably exactly wrong, but what a more correct one could look like.
This cuts to the heart of what I think trading is. It’s something pretty light on “opinions”. That’s for VCs and crystal ball investors, me, I’m a donkey.
I try to invert prices to reason about what the point spreads are, then try to find a contradiction. The whole “measurement not prediction” thing*. Measurement is hard enough. Prices tell you things if you can measure. You can separate probabilities from magnitudes. You can know what the consensus is for how correlated assets are to each other. You can divine when the market thinks we will attack Iran. This is all just sitting there.
You can protest that prices are dumb and wrong, but you are only allowed to make such pronouncements from your private jet otherwise, I can’t hear you.
So as oil goes, I have no opinion, but I can pull up a few screens and tell you what one of the smartest, most efficient markets in the world might think. Maybe there are prices in dumber or less liquid or harder-to-access corners of the world that disagree. Trading means different things to different people. I think it’s the art of turning contradictions into cash.
*Related:
I was listening to Citrini chat with Risk of Ruin’s John Reeder when John said:
I have heard Citrini repeat something that George Soros says, which is, I’m not predicting, I’m observing. Paying attention to what’s happening.
You’ll discover Citrini’s key to observing is how he filters, a skill that is increasingly difficult but always growing in value.
As I’ve shared here before, I spun up an investing class for middle and high school kids locally. I am teaching my 12-year-old as it is, so I figured if I formalize it a touch so others could learn as well.
There are a few weeks between each session since there’s a fair amount of prep even with AI helping with:
Claude in PowerPoint was released recently so I gave it a spin. I gave it a stylesheet of colors and fonts as well as an unformatted draft of the lecture, and let it cook. You can see the result below.
The interactive spreadsheet has a bunch of JavaScript behind it
The class we did this week was a lot of fun. There’s even a video to prove it below (I masked any faces. There were 16 kids in attendance). Most importantly, the kids learned a ton. Parents were texting me with their feedback and it felt good to hear their kids’ gears were turning.
For what it’s worth, I think there was a lot of material in here that parents don’t know either but I’ll leave you to guess what some of that might be.
Investment Beginnings — Class 2: The Math of Investing
Class 1 was about building a business.
Class 2 flips the perspective — you’re the investor now.
Someone is asking you for money. What should you pay for shares? What’s the lowest rate you’d lend at? How do you know if it’s a good deal?
This session covers the foundational math that underpins every investment decision you’ll ever make.
What we covered:
✅ The power of compounding (FV = PV × (1 + r)^n)
✅ The lily pad riddle: why most of the action happens at the end
✅ Early Bird vs Late Starter: why starting 10 years earlier beats investing 3x more money
✅ Warren Buffett: 99% of his wealth came after age 50
✅ Total Return vs CAGR: why doubling your money in 10 years is ~7%/yr, not 10%
✅ The Rule of 72: quick trick to estimate how long to double your money
✅ P/E ratio (multiple) and earnings yield (the reciprocal)
✅ The two levers of stock returns: earnings growth vs multiple expansion/contraction
✅ Zoom case study: great earnings, terrible return — how you can pay too much
✅ The asymmetry of losses: why losing 50% requires a 100% gain to recover
Hands-on:
🕹️ Live bidding exercise: students not only bid on shares of Lamorinda Sneaker Co knowing only that it earns $10/share, but quoted the lowest rates they’d lend at.
🕹️ P/E guessing game: guess the real-world multiples for Tesla, Chipotle, Shake Shack, Lululemon, Nike, and more
Homework:
🔨 Inflation Scavenger Hunt — look up prices from the year you were born vs today🔨 Fee Impact Calculator — compare 0.03% vs 1% fees over 40 years
🔨 P/E Return Decomposition — Pick 5 stocks. For each, look up the price and EPS 5 years ago vs today. 1) How much of the total return came from P/E multiple change vs EPS growth? 2) Then compute the current earnings yield (E/P). Compare it to the trailing 5-year CAGR. 3) Using the Rule of 72: if the 5-yr CAGR continued, how long to double your money? If you earned the earnings yield instead, how long to double?
🔨 Compounding Frequency — calculate FV compounded annually vs semi-annually
Resources:
📊 Slides
📈 Spreadsheet (File → Make a copy to get your own editable version; scripts may trigger a security warning — just advance through it)
Full video:
Money Angle For Masochists
Junior Masochists
Let’s review 2 examples from the class that demonstrate how markets are hard because prices are already forward-looking.
The kids learned how to decompose returns into change in earnings vs change in multiple. Or “what happened” vs “the future” or what I sometimes referred to as “sentiment”.
When I asked the class what stock would have been all the rage during Covid (when many of these kids were only 6 years old 🥹), one boy immediately and correctly responded, “ZOOM!”
I pulled up ZM’s price chart:
I asked…”what do you think happened?”
Kids suggested that less people used Zoom as people went back to offices. I explained that ZM’s earnings actually did skyrocket for the past few years so that’s not the culprit behind the horrible return.
It’s not just the revenues that are up (although you can see how revenue growth has slowed). EPS has also skyrocketed.
The multiple just got hammered. Great business, but investors just paid too much for it.
Earnings were up >35x, but the multiple is down 99%.
A handy decomposition:
Price return = (1+ percent change in EPS) * (1 + percent change in multiple) – 1
The point of the formula is that your return depends on changes in fundamentals (actual earnings) AND change in sentiment around future growth prospects.
A quick caveat. This is not complete. Imagine a situation where a company is $5/share and EPS of $1 for a P/E of 5. Over the next year, the company’s earnings don’t grow and the stock price doesn’t change. The price return is zero. But the company did earn $1. It’s assets have grown by 20%. You are economically richer by 20% but if they don’t distribute it by other paying a dividend or buying back shares (which would raise EPS) then the formula above did not account for a more holistic total return.
You could estimate:
Total return = (1+ percent change in EPS) * (1 + percent change in multiple) + earnings yield – 1
That would capture the idea that you are economically better off even if it’s not paid out, although management’s allocation decisions are a matter of concern.
As a class, we stumbled into a situation on the opposite side of the spectrum. A boy mentioned he bought Delta Airlines 5 years ago for ~$35. I pulled up the chart and noticed the stock doubled.
First of all, great teaching moment as we covered rule of 72 minutes earlier so I immediately asked the class, what the annual return must be? Proud dad moment as Zak is the first one to say 14.4% which I know he figured by thinking “72 divided by 10, times 2” which is better than I would have done as I would reach for 70/5.
Mental math aside, I asked our young investor, “Why did Delta do well, did the earnings increase or the multiple?” With zero hesitation, he responds that the earnings haven’t grown. So a perfect anti-Zoom example for the class. Delta Airlines coming out of Covid years had sour vibes but even if the earnings didn’t grow, you could make a nice return on the sentiment and therefore multiple improving.
I did go back after the class to see DAL earnings and stock history and I think it makes more sense that the kid bought the stock just 2 years ago, since that is the point in time where the earnings were about the same to now and the stock was about $35.
A crap business that investors sold too cheap.
For our regular Masochists
Since we are talking fundamentals, a mutual on X pointed out that HRB (H&R Block) has recently gotten trashed and that its shareholder yield is ~15%.
Shareholder yield is dividends + net share repurchase + debt reduction as a percent of market value.
News flash, HRB is not a growth business. It doesn’t re-invest much of its earnings versus just distributing the cash. I do find it amusing that the stock could be trashed along with other AI disruption stories when it has already survived the transition from brick & mortar to the internet, the popularity of TurboTax, and the growth of the standard deduction, relieving a wider proportion of the population from filing. With a P/E of 7 and a management that pays out the earnings you make ~15% if its already crap business stays the same.
Shedding 1/3 of its market cap since the start of the year, the implied vol is unsurprisingly jacked. I’m a little nuts and decided this was enough to launch some puts with the “I’ll take the shares if I’m wrong”. I normally don’t like this mentality, but part of the vol selling attitude is that the stock probably doesn’t have a lot of upside which reduces the regret possibility from “I was right on this stock and all I collected was some put premium”. In other words, if the upside is abridged, that’s a statement about the vol of the stock being lower.
Selling puts for yield is pretty aligned with what I’m trading the stock for in the first place — yield. I’m just taking it in the form of options intead of buying the stock because the option market is giving me that, but if the price falls a lot further well, I’ll have to go for that yield in the form of assigned shares.
Never financial advice, I’m just sharing my thinking aloud. As options go I’m currently short covered calls in silver and short cash-secured puts in HRB and long options on TSLA and IBIT. Overall, vols are on the higher end of their range across the market (outside of bond vols), but there’s always relatively cheap and relatively expensive in any market cross-section.
Inflation Replicator | 
